Daily Lessons
Daily Lessons
for the
Advanced Placement Calculus AB
Classroom
Numerical, Graphical, and Algebraic Analysis of Calculus Concepts
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Numerical, Graphical, and Algebraic Analysis of Functions
Given below are tables of values for different functions. Classify each function by type. Sketch a graph of the function. Then, state as many specific properties, including the equation if possible, of each function as you possibly can.
|x |–5 |–1 |0 |3 |5 |9 |
|F(x) |[pic] |[pic] |–3 |–5 |[pic] |–9 |
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|x |–6 |–4 |–2 |0 |2 |4 |
|G(x) |5 |1 |–3 |1 |5 |9 |
3.
|x |–2 |–1 |0 |1 |2 |3 |
|J(x) |Undefined |–2 |–1 |0 |1 |2 |
5.
|x |–6 |
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|As x → −∞, the graph of f(x) → _________. | |
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|As x → ∞, the graph of f(x) → _________. | |
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|As x → –3 from the left, the graph of f(x) → _________. | |
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|As x → –3 from the right, the graph of f(x) → _________. | |
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|As x → 3 from the left, the graph of f(x) → _________. | |
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|As x → 3 from the right, the graph of f(x) → _________. | |
Based on what you have just seen, how might you informally define what the value of a limit represents in terms of the graph?
[pic]
How is the numerical analysis above related in the graph of the function pictured below?
[pic]
How is the numerical analysis above related in the graph of the function pictured below?
[pic]
How is the numerical analysis above related in the graph of the function pictured below?
Limit Existence Theorem
Limits That Do Not Exist
Example #1
Find each of the following from the graph.
a) [pic]= b. [pic]=
c) f(2) =
d) Does [pic]exist or not? Why or why not?
Example #2
Find each of the following from the graph.
a) [pic]= b. [pic]=
c) f(2) =
d) Does [pic]exist or not? Why or why not?
Example #3
Find each of the following from the graph.
a) [pic]= b. [pic]=
c) f(2) =
d) Does [pic]exist or not? Why or why not?
Based on what you have seen so far, does f(a) have to be defined in order for the[pic]to exist? Draw and explain two different graphs to justify your reasoning. In both graphs, f(a) should be undefined but in one graph, the limit should exist while in the second one, it should not exist.
Understanding the Limit
An Algebraic Approach
Consider the function,[pic], for a moment. The graph of f(x) is pictured below. From the graph, determine the following limits.
|[pic] |Find f(a) using the equation. |Find [pic]from the graph. |
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When a function is defined and continuous at a value, x = a, how can[pic] be found analytically?
Find each of the following limits analytically.
a) [pic] b. [pic]
c. [pic] d. [pic]
e. [pic] f. [pic]
Analytically Finding Limits of Functions at Undefined Values
What happens if we try to evaluate the limits below by the direct substitution method that was used in the previous six examples?
[pic] [pic] [pic]
Just because a function is undefined at a value of x does not mean that a conclusion cannot be reached about the limit. Consider the rational function above. From the graph of the function pictured to the right, what is the value of each limit below?
[pic]
[pic]
[pic]
The task now is to determine how to find these limits analytically. How was it that we found the discontinuities of a rational function in pre-calculus?
We will perform the same algebraic analysis to find the limit of the removable, point discontinuities. Let’s do this Cancellation Process below.
[pic]
Based on our knowledge from pre-calculus, we know that if a rational function has a non-removable infinite discontinuity, graphically a ____________________________ exists. Since the y – values do not approach one specific value from both sides at a __________________________, then the limit does not exist. However, we can determine if the one sided limits approach −∞ or ∞. In order to do this analytically, we will marry the numerical, graphical, and algebraic approaches. For each limit below, determine the sign of the simplified function at the value to the right or the left of x = 1.
[pic] [pic]
|Value of |Simplified function |
|x to the left of x|[pic] |
|= 1 | |
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|0.9 | |
|Value of |Simplified function |
|x to the right of x = |[pic] |
|1 | |
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|1.1 | |
The graph of a function [pic]is pictured to the right. Often, rationalization can be used to evaluate a limit analytically. Find the following limit.
[pic]
Write the equation of the piece-wise defined function pictured to the right.
Constraint of
Equation of Each Piece Each Piece
Consider the function below to find each limit. If a limit does not exist, state why.
[pic]
a) [pic] b) [pic] c) [pic]
Find each of the following limits analytically. Show your algebraic analysis.
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|a. [pic] |
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|b. [pic] |
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|d. [pic] |
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|f. [pic] |
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|g. [pic] |
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|h. [pic] |
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|i. [pic] |
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|j. [pic] |
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|k. [pic] |
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|l. [pic] |m. [pic] |
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If f(x) = 2x2 – 3x + 4, find[pic].
Properties of Limits
Suppose [pic]and[pic]. Find each of the following limits in terms of L and M.
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|1. [pic] |
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|2. [pic] |
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|3. [pic] |
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|4. [pic] |
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|5. [pic] |
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|6. [pic] |
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|7. [pic] |
Graph of f(x) Graph of g(x)
Find each of the following limits applying the properties of limits. If a limit does not exist, state why.
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|[pic] |[pic] |[pic] |
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|[pic] | |[pic] |
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|[pic] | |[pic] |
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Limits of Exponential Functions
Graphical and Analytical Connections
Consider the four exponential functions graphed below. Find the indicated limits for each function based on the graph.
|[pic] |[pic] |
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|[pic] [pic] |[pic] [pic] |
|[pic] |[pic] |
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|[pic] [pic] |[pic] [pic] |
In order to determine a limit as x approaches −∞ or ∞ for an exponential function, you have to determine what the graph will look like. Based on what we have seen above, what are the three possible results of such a limit for an exponential function?
By studying the graphs above, remind yourself of the four rules determining if the function will be a growth or decay function.
1.__________________________________________________________________________________
2.__________________________________________________________________________________
3.__________________________________________________________________________________
4.__________________________________________________________________________________
Determine the limits of each of the following exponential functions.
|1. [pic] |2. [pic] |
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|3. [pic] |4. [pic] |
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|5. [pic] |6. [pic] |
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Limits of Trigonometric and Exponential Functions
An Analytical Approach
We have already looked at how to evaluate limits of trigonometric and exponential functions by direct substitution. Find each of the limits below.
|[pic] |[pic] |[pic] |
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Each of the functions above was defined at the value that θ was approaching. However, we have seen that even in the algebraic world, not all functions are undefined at a value, but their limits do exist. The same is true in the trigonometric world.
Evaluating Trigonometric Limits by Rewriting the Function Using Identities
Let’s consider for a moment the limit below. Try to evaluate this limit by direct substitution.
[pic]
Again, this function is undefined at θ = 0. However, that does not mean that the limit does not exist. In this case, we can often rewrite the function in terms of a single trig ratio using identities in hopes that the new form of the function is not undefined for the approached value of θ. Do this in the space below.
Find each of the following limits by rewriting the function in a form that is defined for the approached value of θ.
|1. [pic] |2. [pic] |3. [pic] |
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It is important to note that as in all cases of evaluating limits, direct substitution should always be tried FIRST. If that does not yield a value, then a simplification of the function can be tried.
Occasionally, it is not even possible to rewrite the function so that it is undefined. There are two special trigonometric limits that can often be employed.
Use your graphing calculator to complete the table of values below for the function[pic] .
|θ |– 0.01 |– 0.001 |– 0.0001 |0.0001 |0.001 |0.01 |
|[pic] | | | | | | |
Based on the values in the table above, what do each of the limits below equal?
[pic] [pic] [pic]
Use your graphing calculator to complete the table of values below for the function[pic] .
|θ |– 0.01 |– 0.001 |– 0.0001 |0.0001 |0.001 |0.01 |
|[pic] | | | | | | |
Based on the values in the table above, what do each of the limits below equal?
[pic] [pic] [pic]
Based on what you have just observed, what inference can you make about the value of the limit [pic], where c is any constant?
Now, in a similar fashion, use your graphing calculator to complete the table of values below for the function[pic] .
|θ |– 0.01 |– 0.001 |– 0.0001 |0.0001 |0.001 |0.01 |
|[pic] | | | | | | |
Based on the values in the table above, what do each of the limits below equal?
[pic] [pic] [pic]
These two special trigonometric functions derived above can often be used to find limits of trigonometric functions that cannot be evaluated by direct substitution nor by rewriting the function using identities.
Find each of the following limits.
|1. [pic] |2. [pic] |
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|3. [pic] |4. [pic] |
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|5. [pic] |
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|5. [pic] |6. [pic] |
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|7. [pic] |8. [pic] |
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Limit-Based Continuity
Graphical and Analytical Approaches
Given the graph of H(x) pictured to the right, list as many limit statements about the graph that you possibly can in the next ten minutes.
In the graph above, there are three values of c for which [pic]does not exist. In order to establish that a limit does not exist at x = c, one of three things must be established. What are these three things?
1.
2.
3.
For the function graphed below, fill in the table with the given information. After filling in the table, write three pieces of information that must be true in order for a function, G(x), to be continuous at x = a.
1.
2.
3.
| |Is the function defined? If|What is the value of[pic]? |What is the value of[pic]? |What is[pic]? |Is G(x) continuous at x = a? |
|x = a |so, what is its value? | | | | |
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|x = –6 | | | | | |
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|x = –3 | | | | | |
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|x = 0 | | | | | |
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|x = 2 | | | | | |
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|x = 6 | | | | | |
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|x = 8 | | | | | |
The graph of the function, G(x), pictured to the right has several x – values at which the function is not continuous. For each of the following x – values, use the three part definition of continuity to determine if the function is continuous or not.
|1. x = –8 |2. x = –6 |3. x = –4 |
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Use the three part definition of continuity to determine if the given functions are continuous at the indicated values of x.
4. [pic] at x = –2 5. [pic] at x = π
6. Consider the function, f(x), to the right to answer the following questions.
a. What two limits must equal in order for f(x) to be
continuous at x = –1?
b. What two limits must equal in order for f(x) to be continuous at x = 3?
c. Determine the values of m and k so that the function is continuous everywhere.
7. Consider the function, g(x), to the right to answer the following questions.
a. What two limits must equal in order for f(x) to be
continuous at x = –2?
b. What two limits must equal in order for f(x) to be continuous at x = 3?
c. Determine the values of m and k so that the function is continuous everywhere.
Intermediate Value Theorem
As we study calculus, we will study several different theorems. The first theorem of investigation is the Intermediate Value Theorem. Together, let’s write the theorem.
Intermediate Value Theorem
Now, investigate the graphs below to determine if the theorem is applicable for these functions on the specified intervals for the values given. Explain why or why not.
|[pic] |[pic] |
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|Does the I.V.T. guarantee a value of c such that f(c) = 2 on the interval|Does the I.V.T. guarantee a value of c such that f(c) = 2 on the |
|[–4, 2]? Why or why not? |interval [–1, 5]? Why or why not? |
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What two conditions must be true to verify the applicability of the Intermediate Value Theorem?
1.__________________________________________________________________________________
2.__________________________________________________________________________________
For each of the following functions, determine if the I.V.T. is applicable or not and state why or why not. Then, if it is applicable, find the value of c guaranteed to exist by the theorem.
|1. [pic] on the interval [–1, 3] for f(c) = [pic] |2. [pic] on the interval [–4, 1] for f(c) = [pic] |
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|3. [pic] on the interval [–1, 1] for f(c) = [pic] |4. [pic]on the interval [3, 5] for |
| |f(c) = –4 |
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Infinite Limits and Limits at Infinity
In our graphical analysis of limits, we have already seen both an infinite limit and a limit at infinity. Let’s consider the equations and the graphs of the two functions below to find the limits that follow.
[pic] [pic]
|Infinite Limits |Limits at Infinity |
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|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic] |
We have already seen how to find infinite limits by marrying a numerical and analytical approach. For the function, f(x) and g(x), whose graphs appear on the previous page, find the infinite limits below.
|[pic] |[pic] |
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|[pic] |[pic] |
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Graphically, an infinite limit will always yield a _____________________________________________. In pre-calculus, we discovered through observation that such a graphical property existed when a factor in the equation would not _________________________________________________. From this point forward, this is NOT a viable justification for the existence of a _________________________________.
Justification of the Existence of a Vertical Asymptote Using Limits
For the function below, find any vertical asymptote(s) that exist. Justify your answer(s) using a limit(s).
[pic]
Now, we must develop an analytical procedure by which we can find limits at infinity. Basically, a limit at infinity describes the end behavior of a function. We have spent a great amount of time talking about end behavior of functions. Find each of the following limits at infinity. Give an explanation of your reasoning for each.
|[pic] |[pic] |[pic] |
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The third example,[pic]will provide us a basis for developing our analytical process by which we can find limits at infinity for all types of rational functions. Before we do that, investigate the two functions below both graphically and numerically.
[pic] [pic]
What does each of these functions have in common algebraically and what do they have in common graphically?
In pre-calculus, we learned three rules for determining the existence of horizontal asymptotes of rational functions. When a rational function had a horizontal asymptote, the end behavior was always such that as x →−∞ or ∞, then the graph of f(x) → the horizontal asymptote. We learned three rules for determining the horizontal asymptote, if one existed, for rational functions. We are about to use the idea of a limit and calculus to find out why those rules are such as they are. For each function below, divide every term in both the numerator and the denominator by the highest power of x that appears in the denominator. Then, evaluate the indicated limit. Does the result of each limit make sense based on the graph that is pictured?
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Let’s see what would happen if our limit at infinity approached −∞.
|[pic] |
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Based on what we have just seen and what we know graphically about the functions above, would approaching −∞ make a difference in the result of our limits?
Graphically, a limit at infinity can yield a _____________________________________________. In pre-calculus, we discovered through observation that such a graphical property existed by comparing the ____________________________________________________________________________________. From this point forward, this is NOT a viable justification for the existence of a _________________________________.
Justification of the Existence of a Horizontal Asymptote Using Limits
For the function below, find the horizontal asymptote if it exists. Justify your answer(s) using a limit(s).
[pic]
The algebraic analysis described above to evaluate a limit at infinity can be used to find limits at infinity for any type of rational function, even [pic], whose graph is pictured to the right.
What is the one thing that you notice is different about the graph of this rational function versus the others that we have investigated in the past?
Use the graph to find each of the following limits.
[pic] [pic]
Perform the same algebraic analysis that we did earlier to find the limits at infinity. The only problem that we will encounter is what to do when x → −∞.
|[pic] |[pic] |
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Look at the graph on the previous page to confirm these results. Then, find the limits below.
|[pic] |[pic] |
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The Difference Quotient
A First Look at the Derivative
Today we are introduced to the concept with which we will spend our greatest amount of time investigating in Calculus AB—the derivative. Let’s draw a picture together.
What does the expression [pic]represent? What does this expression simplify to?
As h, the distance between the x – values, x and (x + h), approaches zero, what happens to the secant line?
What does the limit [pic]represent?
Suppose[pic]. Find[pic].
Your result to the previous limit is defined to be the derivative,[pic], of the function f(x). Now, let’s see what this derivative represents in terms of the graph of f(x).
Your result of[pic]for [pic]is a function in terms of x. The graph of f(x) is pictured below. Complete the chart for the indicated x – values and[pic].
|x – value |Value of[pic] |
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|–4 | |
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|–2 | |
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|–1 | |
Now, use a ruler and draw a tangent line to the graph of f(x) on the grid above at x = –4, x = –2, and x = –1. By investigating the graph, what does it appear that the derivative function [pic] represents in terms of the graph at given values of x?
Definition of the Derivative and What It Represents Graphically
Find the equation of the tangent line to f(x) at each of the points below. Then, draw the graphs of the tangent lines on the grid above where f(x) is graphed.
|Equation of the tangent line at x = –4 |Equation of the tangent line at x = –2 |Equation of the tangent line at x = –1 |
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When you hear “DERIVATIVE,” you think “SLOPE OF THE TANGENT LINE.”
When you hear “SLOPE OF THE TANGENT LINE,” you think “DERIVATIVE.”
Now that we understand what the derivative of a function represents graphically, let’s practice using the limit of the difference quotient, [pic], to find [pic] for each of the functions below.
|[pic] |[pic] |
Notice that [pic] for [pic]was different than [pic]for[pic]. How are they different and why do you suppose this is so?
Find [pic]for the functions given below to find and use[pic].
|[pic] |[pic] |
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|Find the equation of the line tangent to the graph of [pic]at x = 7. |Find the equation of the line tangent to the graph of[pic]at x = 1. |
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Using a graphing calculator, graph each of the functions above and the equation of the tangent line that you found to verify your work.
Over the course of this lesson so far, you have found derivatives of several functions and evaluated that derivative at certain x – values. Look back at your work and complete the table below.
|Equation of Function, f(x) |Equation of Derivative, [pic] |Value of [pic]at the |Find the Value of the Limit |
| | |Indicated value of x |[pic], where a is the value of x. |
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|[pic] | |x = –1 | |
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|[pic] | |x = 7 | |
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What inference can you make that explains what the limit [pic]represents?
Complete the table below, stating what each of the indicated limits finds in terms of the derivative of a function, f(x).
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|Definition of the Derivative|[pic] | |
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|Alternate Form of the |[pic] | |
|Definition of the Derivative| | |
Understanding the Derivative from a Graphical and Numerical Approach
So far, our understanding of the derivative is that it represents the slope of the tangent line drawn to a curve at a point.
Complete the table below, estimating the value of[pic]
at the indicated x – values by drawing a tangent line and
estimating its slope.
| | |Is the function Increasing, | |
| | |Decreasing or at a Relative |Equation of the tangent line at this value of x. |
|x – |Estimation of Derivative |Maximum or Relative Minimum | |
|Value | | | |
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|–7 | | | |
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|–6 | | | |
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|–4 | | | |
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|–1 | | | |
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|7 | | | |
Based on what you observed in the table on the previous page, what inferences can you make about the value of the derivative,[pic], and the behavior of the graph of the function, f(x)?
Numerically, the value of the derivative at a point can be estimated by finding the slope of the secant line passing through two points on the graph on either side of the point for which the derivative is being estimated.
|x |–3 |0 |1 |4 |6 |10 |
|f(x) |2 |1 |–3 |0 |–7 |2 |
| | |Is the function Increasing, | |
| | |Decreasing or at a Relative |Equation of the tangent line at this value of x. |
|x – |Estimation of Derivative |Maximum or Relative Minimum | |
|Value | | | |
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|0 | | | |
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|1 | | | |
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|6 | | | |
The graph of a function, g(x), is pictured to the right. Identify the following characteristics about the graph of the derivative,[pic]. Give a reason for your answers.
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|The interval(s) where [pic] < 0 | |
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|The interval(s) where [pic] > 0 | |
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|The value(s) of x where [pic] = 0 | |
Definition of the Normal Line
Pictured to the right is the graph of[pic].
Draw the tangent line to the graph of f(x) at x = 1. Then, estimate the value of[pic].
Find the equation of the tangent line to the graph of f(x) at x = 1.
The normal line is the line that is perpendicular to the tangent line at the point of tangency. Draw this line and find the equation of the normal line.
The graph of the derivative,[pic], of a function h(x) is pictured below. Identify the following characteristics about the graph of h(x) and give a reason for your responses.
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|The interval(s) where h(x) is increasing | |
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|The interval(s) where h(x) is decreasing | |
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|The value(s) of x where h(x) has a relative maximum. | |
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|The value(s) of x where h(x) has a relative minimum. | |
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|If h(–1) = ½, what is the equation of the tangent line drawn | |
|to the graph of h(x) at x = –1? | |
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|If h(2) = –3, what is the equation of the normal line drawn | |
|to the graph of h(x) at x = 2? | |
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Analytically Finding the Derivative of Polynomial, Polynomial Type, Sine, and Cosine Functions
Consider the function f(x) = 3. What does the graph of this function look like? If a tangent line were drawn to f(x) at any value of x, what would the slope of that tangent line be?
Based on this though process, if f(x) = c, where c is any constant, then[pic].
Shown below are 6 different polynomial, or polynomial–type, functions. Watch as I find the derivative of each function. See if you can figure out the algorithm that I am using for each function.
|Function, f(x) |Derivative, [pic] |
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Based on what you have seen in the table above, you should now be able to infer how to complete the following Power Rule for Differentiation.
[pic]
In order to apply the Power Rule for Differentiation, the equation must be written in “polynomial form.” To what do you suppose “polynomial form” refers?
Find[pic]for each of the following functions. Leave your answers with no negative or rational exponents and as single rational functions, when applicable.
|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |[pic] |
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Remember two trigonometric identities that we will use to find the derivatives of the sine and cosine functions.
cos(a + b) = _________________________________________
sin(a + b) = _________________________________________
Use [pic]to find [pic]for each of the following functions. Your results will show the derivative of the sine and cosine functions.
|f(x) = sin x |f(x) = cos x |
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[pic] [pic]
For each of the following functions, find the equation of the tangent line to the graph of the function at the given point.
|[pic]when x = –1 |[pic] when θ = 0 |
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|[pic] when θ = π |[pic]when x = 2 |
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Given the equation of a function, how might you determine the value(s) at which the function has a horizontal tangent? Explain your reasoning.
At what value(s) of x will the function [pic] have a horizontal tangent?
At what value(s) of θ at which the function[pic]has a horizontal tangent on the interval [0, 2π)?
Connections between F(x) and F’(x) for Polynomial and Trigonometric Functions
|F’(x) |F(x) |
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|Is = 0 | |
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|Is > 0 | |
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|Is < 0 | |
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|Changes from positive to negative | |
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|Changes from negative to positive | |
Pictured below is the graph of a function f(x). Answer the following questions about the derivative.
1. Approximate the value of[pic].
2. At what value(s) of x is[pic] = 0. Justify
your answer.
3. On what open interval(s) is[pic] < 0? Justify your answer.
4. On what open interval(s) is[pic] > 0? Justify your answer.
Pictured below is the graph of[pic] on the interval [–3, 4]. Answer the following questions about f(x).
1. On what open interval(s) is the graph of f(x)
increasing? Justify your reasoning.
2. On what open interval(s) is the graph of f(x) decreasing? Justify your answer.
3. At what value(s) of x does the graph of f(x) have a horizontal tangent? Justify your answer.
4. What is the slope of the tangent line to the graph of f(x) at x = –1? Justify your reasoning.
5. What is the slope of the normal line to the graph of f(x) at x = 4? Justify your reasoning.
For each of the given functions, determine the interval(s) on which f(x) is increasing and/or decreasing. Find all coordinates of the relative extrema. Unless otherwise noted, perform the analysis on all values on [pic]. Provide justification for your answers.
1. [pic]
2. f(x) = 3x5 – 5x3
3. f(θ) = θ + 2sinθ on (0, 2π)
Solidifying the Concept of the Derivative as the Tangent Line
Pictured to the right is the graph of a quadratic function,
[pic].
1. Find [pic]and explain what this value represents in terms of
the graph of the function g(x).
2. Find the equation of the tangent line drawn to the graph of g(x) at x = –4. Sketch a graph of this
tangent line on the grid with the graph of g(x) above.
3. Using the equation of the tangent line, find the value of y when x = –3.9. Then, find the value of
g(–3.9).
4. What do you notice about the values of these two results from question 3? What does this imply about
how the equation of the tangent line might be used?
Pictured to the right is the graph of the function[pic]. Use the graph and the equation to answer questions 5 – 9.
5. Based on the graph, at what value(s) does the graph of g(x) have a
horizontal tangent? Give a reason. Show an algebraic analysis that
supports your answer.
6. On what interval(s) is[pic]< 0? Give a reason for your answer.
7. On what interval(s) is[pic]> 0? Give a reason for your answer.
8. For what value(s) of x is the slope of the tangent line equal to 2? Show your work.
9. Find an equation of the tangent line drawn to the graph of g(x) when x = 4. Then, draw the tangent
line on the grid above.
The table of values below represents values on the graph of the derivative,[pic], of a polynomial function h(x). The zeros indicated in the table are the only zeros of the graph of[pic]. Use the table to answer questions 10 – 15.
|x |−8 |
|[pic] |[pic] |
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|[pic] |[pic] |
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Find the equation of the line tangent to the graph of [pic] when t = [pic].
There is a very valuable lesson that we must learn when we are introduced to the product rule.
|Given the function[pic]. Find [pic]by applying the product rule. |Given the function[pic]. Rewrite the function in polynomial form. Then, |
| |find[pic]. |
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What is the lesson to be learned from the algebraic analysis above?
If [pic], what is the slope of the normal line to the graph of g(x) when x = 2?
Below are graphs of two functions—f(x) and g(x). Let [pic]and let[pic]. Use the graphs to answer the questions that follow.
Graph of f(x) Graph of g(x)
|If[pic], what is the value of [pic]? |If[pic] = 20, what is the value of [pic]? |
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|Find the equation of the line tangent to the graph of P(x) when x = –4. |Find the equation of the line tangent to the graph of R(x) when x = –2. |
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Let f(x) and g(x) be differentiable functions such that the following values are true.
|x |f(x) |g(x) |[pic] |[pic] |
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|4 |1 |7 |2 |–3 |
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|3 |–2 |–3 |–4 |2 |
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|–1 |2 |–2 |1 |–1 |
|Estimate the value of [pic]. |If[pic], what is the value of [pic]? |
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|If[pic], what is the value of[pic]? |Find the equation of the line tangent to the graph of [pic]when x = –1. |
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|If [pic], what is the value of [pic]? |
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Rules for Differentiation
Finding the Derivative of a Quotient of Two Functions
Rewrite the function[pic]as a function in polynomial form. Then, find[pic]
Just as Leibniz was the first to publish a proof of the Product Rule for Differentiation, Isaac Newton was the first to publish a proof of the Quotient Rule of Differentiation using the limit definition of the derivative. Let’s write this rule together in the box below.
Quotient Rule of Differentiation
To show that this rule works, let’s apply this rule to the function [pic]that we rewrote and differentiated as a polynomial-form above.
Find the equation of the tangent line drawn to the graph of[pic]when x = –2.
We will now use the quotient rule to derive the derivative formulas for the remaining trigonometric functions. Rewrite each function in terms of sine and/or cosine and differentiate using the Quotient Rule.
|[pic] |[pic] |
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|[pic] |[pic] |
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Find the equation of the normal line drawn to the graph of [pic] when [pic].
Find the derivative of each of the functions below by applying the quotient rule.
|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |
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Similar to the Product Rule, there is a very valuable lesson that we must learn when we are introduced to the quotient rule.
|Given the function[pic]. Find [pic]by applying the quotient rule. |Given the function[pic]. Simplify f (x). Then, find[pic]by applying the |
| |quotient rule. |
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What is the lesson to be learned from the algebraic analysis above?
Below are graphs of two functions—f(x) and g(x). Let [pic]and let[pic]. Use the graphs to answer the questions that follow.
Graph of f(x) Graph of g(x)
|Find[pic]. |Find the equation of the line tangent to the graph of P(x) when x = 5. |
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|Find [pic] |Find the equation of the line tangent to the graph of R(x) when x = 0. |
Let f(x) and g(x) be differentiable functions such that the following values are true.
|x |f(x) |g(x) |[pic] |[pic] |
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|4 |1 |7 |8 |–2 |
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|3 |–5 |–3 |–4 |6 |
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|2 |2 |–1 |9 |–1 |
|Estimate the value of[pic]. |If[pic], what is the value of[pic]? What does this value say about the graph |
| |of p(x) when x = 4? Give a reason for your answer. |
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|If[pic], what is the value of [pic]? |Find the equation of the line tangent to the graph of [pic]when x = 3. |
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Rules for Differentiation
Finding the Derivative of a Composite Function
Rewrite the function[pic]as a function in polynomial form. Then, find[pic]
Leibniz was the first of the two great calculus developers to use the Chain Rule to differentiate composite functions. Let’s write this rule together in the box below.
Chain Rule of Differentiation of Composite Functions
To show that this rule works, let’s apply this rule to the function [pic]that we rewrote and differentiated as a polynomial-form above.
Find the slope of the normal line to the graph of[pic]when [pic].
Find the derivative of each of the following functions by applying the chain rule.
|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |[pic] |
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Now that you now “The Big Three” rules of differentiation—product, quotient, and chain—let’s see how the three can be incorporated with each other. Find the derivative of each of the following functions.
|[pic] |
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|[pic] |
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Given the graph of H(x) pictured to the right, find the equation of the tangent line to the graph of [pic]when x = –4.
Let f(x) and g(x) be differentiable functions such that the following values are true.
|x |f(x) |g(x) |[pic] |[pic] |
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|4 |1 |7 |8 |–2 |
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|3 |–5 |4 |–4 |6 |
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|2 |2 |–1 |0 |–1 |
|Is the graph of [pic]increasing, decreasing or at a relative maximum or |If[pic], what is the value of[pic]? |
|minimum when x = 3? Give a reason for your answer. | |
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|If[pic], what is the value of [pic]? What does this value say about the graph of q(x) when x = 4? Give a reason for your answer. |
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Rules for Differentiation
Finding the Derivative of the Natural Exponential and Logarithmic Functions
Differentiation Rule for Natural Exponential Functions
Find the derivative of each of the following functions.
|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |
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Differentiation Rule for Natural Logarithmic Functions
Find the derivative of each of the following functions.
|[pic] |[pic] |
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|[pic] |[pic] |
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Finding Values of Derivatives Using the Graphing Calculator
For each of the functions below, find the value of [pic]at the indicated value of x using the graphing calculator. Then, determine if the function is increasing, decreasing, has a horizontal tangent or has a vertical tangent. Give a reason for your answer.
|Function |Value of[pic] |Is f(x) increasing or decreasing, or does f(x) have a horizontal or a|
| | |vertical tangent? |
|1. |a = –2 | |
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|[pic] | | |
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|2. |a = 1 | |
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|[pic] | | |
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|3. |a = [pic] | |
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|[pic] | | |
|4. |a = π | |
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|5. |a = 0 | |
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|[pic][pic] | | |
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|6. |a = 1 | |
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|[pic] | | |
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When the value of the derivative of a function is positive, we say that the function is increasing. When the value of the derivative of a function is negative, we say that the function is decreasing. When speaking of quantities increasing or decreasing, they do so at a certain rate. We already understand the derivative to be the SLOPE OF THE TANGENT LINE. Slope is a rate. Therefore, the derivative of a function actually represents the RATE AT WHICH A FUNCTION IS CHANGING.
| |The number of people entering a concert can be modeled by the function[pic], where t represents the number of hours after the gates are open. |
|7. | |
| |Find the values of [pic]and[pic]. Using correct units, explain what each value represents in the context of this problem. |
|a. | |
| |How many people have entered the concert 2 hours after the gates are opened? Is the number of people entering increasing or decreasing at this |
|b. |time? Justify your answer. |
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| |After being poured into a cup, coffee cools so that its temperature, T(t), is represented by the function [pic], where t is measured in minutes |
|8. |and T(t) is measured in degrees Fahrenheit. |
| |What is the temperature of the coffee 5 minutes after it has been poured into the cup? |
|a. | |
| |Is the temperature decreasing faster 1 minute after it is poured or 3 minutes after it is poured? Give a reason for your answer. |
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The Relationship between Continuity and Differentiability
In this lesson, our goal is to establish a relationship between a function being continuous at a value of x and a function being differentiable at the same value. In other words, if a function is continuous at a particular value of x, does that imply that it is also differentiable. Or, if a function is differentiable, does that mean that it must also be continuous. Let’s investigate three functions.
Consider the function [pic]at x = 2. Answer the questions that follow.
On the grid to the right, sketch a graph of f(x) from your graphing calculator.
Based on the graph, if f(x) continuous at x = 2? Explain your reasoning.
Find the value of[pic]to determine if f(x) is differentiable at x = 2.
Consider the function [pic]at x = 0. Answer the questions that follow.
On the grid to the right, sketch a graph of f(x) from your graphing calculator.
Based on the graph, if f(x) continuous at x = 0? Explain your reasoning.
Find the value of[pic]to determine if f(x) is differentiable at x = 0.
Consider the function [pic]at x = 0. Answer the questions that follow.
On the grid to the right, sketch a graph of f(x) from your graphing calculator.
Based on the graph, if f(x) continuous at x = 0? Explain your reasoning.
Find the value of[pic]to determine if f(x) is differentiable at x = 0.
Based on what we have seen, does continuity imply differentiability or does differentiability imply continuity?
In order for a function to be differentiable at a value of x, then two things must be true:
1.___________________________________________________________________________________
2.___________________________________________________________________________________
Consider the function [pic]to answer the following questions.
Is g(x) continuous at x = 3? Show the complete analysis.
Is g(x) differentiable at x = 3? Show the complete analysis.
For what values of k and m will the function below be both continuous and differentiable at x = 3?
[pic]
For what values of a and b will the function below be differentiable at x = 1?
[pic]
Implicit Differentiation
Much of our algebraic study of mathematics has dealt with functions. In pre-calculus, we talked about two different types of equations that relate x and y—explicit and implicit. Explicit equations in x and y are such that y is isolated on one side of the equation, and it equals an expression that is totally in terms of x. When I look at an explicitly defined equation, I can EXPLICITLY tell if it represents a function or not. In other words, when I see[pic], I know that for every x, there is only one y, making this equation a function. When I see[pic], I know that for every x there are two y values, making this equation not a function.
Implicit equations are very different. Typically, they do not have y isolated on one side of the equation. Often, there is a power on the y term(s) in the equation and both y’s and x’s may appear throughout the equation. For example, the equation, x2 + y2 – 2x + 4y +16 = 0, is that of a circle, if you will remember. Calculus can even be applied to implicitly defined equations. In this lesson, we will see how to differentiate those equations that are implicitly defined.
We will utilize an alternate notation for the derivative. Instead of[pic], we will use [pic]. As calculus was developed by two different men, a blend of their notations has been accepted. Let’s think about how we differentiate[pic]. Then, let’s differentiate the implicit form of this equation, x2 + y2 = 25.
Consider the graph of the circle to the right. Find the equation of the circle in implicit form below.
Now, implicitly differentiate the equation of the circle in the space below
Complete the table below finding the value of [pic]at each of the indicated points. Then, draw the graphical representation, the tangent line, on the graph at each indicated point.
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|(0, 2) | |
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|(3, 3) | |
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|(8, –2) | |
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|(3, –7) | |
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|(6, 2) | |
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|(–2 , –2) | |
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|(6, –6) | |
Find [pic]for each of the following implicitly defined equations.
|[pic] |[pic] |
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|[pic] |[pic] |
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For what values of x will the curve [pic]have a horizontal tangent? Show your work and explain your thinking.
In terms of y, describe the values of x for which the curve [pic]will have a vertical tangent? Show your work and explain your thinking.
|Given the curve[pic], find [pic]. |Given the curve[pic], find [pic]. |
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Related Rates
Implicitly differentiate the following formulas with respect to time. State what each rate in the differential equation represents
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|[pic] | |
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|Surface Area of a Sphere | |
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|[pic] | |
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|Volume of a Sphere | |
|3. | |
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|[pic], where c is | |
|is constant | |
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|4. | |
|[pic], where r is | |
|constant | |
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|Volume of a Cylinder | |
|5. | |
|[pic] | |
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|6. | |
|[pic] | |
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|Volume of a Cone | |
Air is leaking out of an inflated balloon in the shape of a sphere at a rate of 230π cubic centimeters per
minute. At the instant when the radius is 4 centimeters, what is the rate of change of the radius of the
balloon?
|1. Identify all of the variables involved in the problem. | |
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|2. Identify which, if any, of the variables in the problem that remain | |
|constant. | |
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|3. Identify the rate(s) that are given and the rate that you wish to find. | |
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|4. Write an equation, often a geometric formula or trigonometric equation, | |
|that relates all of the variables in the problem for which a rate is given or| |
|for which a rate is to be determined. Substitute any value that represents a| |
|variable that is constant throughout the problem. It is important to keep in| |
|mind that you can have only one more variable than you have rates. You may | |
|have to make a substitution that relates one variable in terms of another. | |
|5. Implicitly differentiate both sides of the equation with respect to time.| |
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|6. Substitute all instantaneous rates and values of the variable and solve | |
|for the remaining rate or variable. | |
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A stone is dropped into a calm body of water, causing ripples in the form of concentric circles. The radius
of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the
total area of the disturbed water changing?
|1. Identify all of the variables involved in the problem. | |
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|2. Identify which, if any, of the variables in the problem that remain | |
|constant. | |
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|3. Identify the rate(s) that are given and the rate that you wish to find. | |
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|4. Write an equation, often a geometric formula or trigonometric equation, | |
|that relates all of the variables in the problem for which a rate is given or| |
|for which a rate is to be determined. . Substitute any value that | |
|represents a variable that is constant throughout the problem. It is | |
|important to keep in mind that you can have only one more variable than you | |
|have rates. You may have to make a substitution that relates one variable in | |
|terms of another. | |
|5. Implicitly differentiate both sides of the equation with respect to time.| |
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|6. Substitute all instantaneous rates and values of the variable and solve | |
|for the remaining rate or variable. | |
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Water is leaking out of a cylindrical tank at a rate of 3 cubic feet per second. If the radius of the tank is 4 feet, at what rate is the depth of the water changing at any instant during the leak?
|1. Identify all of the variables involved in the problem. | |
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|2. Identify which, if any, of the variables in the problem that remain | |
|constant. | |
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|3. Identify the rate(s) that are given and the rate that you wish to find. | |
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|4. Write an equation, often a geometric formula or trigonometric equation, | |
|that relates all of the variables in the problem for which a rate is given or| |
|for which a rate is to be determined. Substitute any value that represents a| |
|variable that is constant throughout the problem. It is important to keep in| |
|mind that you can have only one more variable than you have rates. You may | |
|have to make a substitution that relates one variable in terms of another. | |
|5. Implicitly differentiate both sides of the equation with respect to time.| |
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|6. Substitute all instantaneous rates and values of the variable and solve | |
|for the remaining rate or variable. | |
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A cone has a diameter of 10 inches and a height of 15 inches. Water is being poured into the cone so that
the height of the water level is changing at a rate of 1.2 inches per second. At the instant when the radius
of the expose surface area of the water is 2 inches, at what rate is the volume of the water changing?
|1. Identify all of the variables involved in the problem. | |
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|2. Identify which, if any, of the variables in the problem that remain | |
|constant. | |
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|3. Identify the rate(s) that are given and the rate that you wish to find. | |
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|4. Write an equation, often a geometric formula or trigonometric equation, | |
|that relates all of the variables in the problem for which a rate is given or| |
|for which a rate is to be determined. Substitute any value that represents a| |
|variable that is constant throughout the problem. It is important to keep in| |
|mind that you can have only one more variable than you have rates. You may | |
|have to make a substitution that relates one variable in terms of another. | |
|5. Implicitly differentiate both sides of the equation with respect to time.| |
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|6. Substitute all instantaneous rates and values of the variable and solve | |
|for the remaining rate or variable. | |
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A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second.
a. How fast is the top of the ladder moving down the wall when the base of the ladder is 7 feet from
the wall?
b. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at
which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.
c. Find the rate at which the angle formed by the ladder and the wall of the house is changing when the
base of the ladder is 9 feet from the wall.
An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles past the antenna, the rate at which the distance between the antenna and the plane is changing is 240 miles per hour. What is the speed of the plane?
The radius of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the surface
area of the sphere when the radius is 6 inches.
A spherical balloon is expanding at a rate of 60π cubic inches per second. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 inches.
Analytical Connections between[pic],[pic], and[pic]
Let’s begin by filling in the following charts about the relationships that exist between the graphs of a function and its first derivative.
|F’(x) |F(x) |
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|Is = 0 or is undefined | |
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|Is > 0 | |
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|Is < 0 | |
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|Changes from positive to negative | |
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|Changes from negative to positive | |
Analytically find all intervals where f(x) = [pic]+ 1 is increasing/decreasing or has a relative maximum or minimum. Sketch a graph using a graphing calculator on the grid to the right to verify your analytical results.
For each of the functions below, determine the interval(s) where the graph is increasing, decreasing, has a relative maximum, and/ or has a relative minimum. Show your analysis and justify your answers.
|1. [pic] |2. [pic] |
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The FIRST DERIVATIVE of a function identifies intervals where a function is increasing, decreasing, has a relative maximum or has a relative minimum.
In a similar fashion, the SECOND DERIVATIVE identifies intervals where a function is concave up, concave down, or has a point of inflection.
Since the SECOND DERIVATIVE is the FIRST DERIVATIVE of[pic], then the same relationships that exist between F(x) and[pic] must exist between[pic] and[pic].
With these relationships in mind, complete the following table.
|F’’(x) |F(x) |F’(x) |
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|Is = 0 or is undefined | | |
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|Is > 0 | | |
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|Is < 0 | | |
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|Changes from positive to negative | | |
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|Changes from negative to positive | | |
On the previous page, you found that the first derivative of the function f(x) = [pic]+ 1 was the function[pic]. Find[pic]and perform a sign analysis to determine intervals of concavity and point(s) of inflection. Again, verify your analytical results by looking at the graph on the previous page.
For each of the functions below, determine the interval(s) where the graph is concave up, is concave down, and determine the coordinates of the point(s) of inflection. Show your analysis and give justification for your answers.
|1. [pic] |2. [pic] |
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THE SECOND DERIVATIVE TEST
An Alternate Way to Identify Relative Maximums and Minimums using the 2nd Derivative
Use the second derivative test to find all coordinates of relative extrema for each function.
|1. f(x) = x4 – 4x3 + 2 |2. [pic], on the interval (0, 2π) |
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Connecting the Graphs of[pic],[pic], and[pic]
Given below is the graph of a function, F(x). State all of the conclusions that you can state about the graphs of[pic] and[pic]. Justify each of your conclusions.
Graph of F(x)
|Conclusions about[pic] |Conclusions about[pic] |
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Given below is the graph of a function,[pic]. State all of the conclusions that you can state about the graphs of F(x) and[pic]. Justify each of your conclusions.
Graph of[pic]
|Conclusions about F(x) |Conclusions about[pic] |
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Given below is the graph of a function,[pic]. State all of the conclusions that you can state about the graphs of F(x) and[pic]. Justify each of your conclusions.
Graph of[pic]
|Conclusions about F(x) |Conclusions about[pic] |
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Calculator Active Questions
The function [pic]is the first derivative of a twice differentiable function, f(x).
a. On the interval 0 < x < 10, find the x – value(s) where f(x) has a relative maximum.
Justify your answer.
b. On the interval 0 < x < 10, find the x – value(s) where f(x) has a relative minimum.
Justify your answer.
c. On the interval 0 < x < 10, find the x – value(s) where f(x) has a point of inflection.
Justify your answer.
On the interval 0 < x –2 D. x > 2 only E. x < –2 or x > 2
The second derivative of the function f is given by[pic]. The graph of [pic]is shown to the right. For what values of x does the graph of [pic] have a relative maximum?
A. j and k only
B. a and b only
C. a only
D. 0 only
E. a and 0 only
If h(x) is a twice differentiable function such that [pic]for all values of x, then at what value(s) does the graph of g(x) have a relative maximum if[pic]?
A. x = 3 and x = –3 B. x = 3 only
C. x = 9 only D. x = –3 only
E. g(x) does not have a relative maximum
A table of function values for a twice differentiable function, f(x), is pictured to the right. Which of the following statements is/are true if f(x) has only one zero on the –3 < x < 3?
I. [pic]on the interval –3 < x < 3.
II. f(x) has a zero between x = 1 and x = 3.
III. [pic] on the interval –3 < x < 3.
A. I only B. I and II only C. III only
D. II and III only E. I, II and III
f, [pic], and [pic]Relationships and The Extreme Value Theorem
The graph given to the right is the graph of[pic], the first derivative of a differentiable function, f. Use the graph to answer the questions below.
1. On the interval [0, 8], are there any values where f(x) is not
differentiable? Give a reason for your answer.
2. On what interval(s) is[pic]> 0? < 0? Give reasons for your
answers.
3. On what intervals is f increasing? Decreasing? Give reasons for your answers.
4. What is the value of[pic]? 5. What is the value of[pic]?
Explain your reasoning. Explain your reasoning.
6. If [pic]and f(2) = –3, what is the equation of the normal line to the graph of g at x = 2?
7. Consider the function[pic]. Given the table of information below, answer the
questions that follow.
|x |< –1 |
|3. [pic] on [-3, 6] |4. [pic] on [–1, 3] |
The Derivate as a Rate of Change
Mean Value Theorem and Rolle’s Theorem
Consider the values of a differentiable function, f(x), in the table below to answer the questions that follow. Plot the points and connect them on the grid below.
|x |0 |
|Interval | |
|[2, 8] | |
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For each of the functions below, determine whether Rolle’s Theorem is applicable or not. Then, apply the theorem to find the values of c guaranteed to exist.
|1. [pic]on the interval [–3, 0] |2. [pic]on the interval [–4, –1] |
Rolle’s Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate of change is equal to zero.
The Mean Value Theorem is similar. In fact, Rolle’s Theorem is a specific case of what is known in calculus as the Mean Value Theorem.
Mean Value Theorem
Consider the function [pic] . The graph of h(x) is pictured below. Does the M.V.T. apply on the interval [–1, 5]? Explain why or why not.
Does the M.V.T. apply on the interval [1, 5]? Why or why not?
Graphically, what does the M.V.T. guarantee for the function
on the interval [1, 5]? Draw this on the graph to the left.
Apply the M.V.T. to find the value(s) of c guaranteed for h(x) on the interval [1, 5]
Explain why you cannot apply the Mean Value Theorem for [pic] on the interval [−1, 1].
Find the equation of the tangent line to the graph of [pic]on the interval (0, π) at the point which is guaranteed by the mean value theorem.
The Mean Value Theorem guarantees that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there is guaranteed to exist a value of c where the instantaneous rate of change at x = c is equal to the average rate of change of f on the interval [a, b].
Applying Theorems in Calculus
Intermediate Value Theorem, Extreme Value Theorem, Rolle’s Theorem, and Mean Value Theorem
Before we begin, let’s remember what each of these theorems says about a function.
Intermediate Value Theorem
Extreme Value Theorem
Rolle’s Theorem
Mean Value Theorem
The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table below shows the rate as measured every 3 hours for a 24-hour period.
|t |0 |3 |6 |9 |
|(hours) | | | | |
|1 |6 |4 |2 |5 |
|2 |9 |2 |3 |1 |
|3 |10 |−4 |4 |2 |
|4 |–1 |3 |6 |7 |
The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table above gives values of the functions and their first derivatives at selected values of x. The function h is given by the equation [pic].
a. Find the equation of the tangent line drawn to the graph of h when x = 3.
b. Find the rate of change of h for the interval 1 < x < 3.
c. Explain why there must be a value of r for 1 < r < 3 such that h(r) = –2.
d. Explain why there is a value of c for 1 < c < 3 such that [pic].
Particle Motion Problems
Particle motion problems deal with particles that are moving along the x – or y – axis. Thus, we are speaking of horizontal or vertical movement. The position, velocity, or acceleration of a particle’s motion are DEFINED by functions, but the particle DOES NOT move along the graph of the function. It moves along an axis. Most of the time, we speak of movement along the x – axis. In units 6 and 7, particle motion is revisited. At that time, we will deal more with vertical motion. For the time, we will focus on horizontal motion of particles.
In this lesson, we develop the ideas of velocity and acceleration in terms of position. We will speak of two types of velocities and accelerations. Let’s define average and instantaneous velocity in the box below.
Average and Instantaneous Velocity
A particle’s position is given by the function[pic], where p(t) is measured in centimeters and t is measured in seconds. Answer the following questions.
What is the average velocity on the interval t = 1 to t = 3 seconds? Indicate appropriate units of measure.
What is the instantaneous velocity of the particle at time t = 1.5. Indicate appropriate units of measure.
Before we proceed, a connection needs to be made. When given a function, f(x), how did we find the slope of the secant line on the interval from x = a to x = b? In terms of position of a particle, to what does the slope of the secant line correspond? To what does the instantaneous velocity correspond?
Average and Instantaneous Acceleration
A particle’s position is given by the function[pic], where p(t) is measured in centimeters and t is measured in seconds. Answer the following questions.
What is the average acceleration on the interval t = 1 to t = 3 seconds? Indicate appropriate units of measure.
What is the instantaneous acceleration of the particle at time t = 1.5.
In summary, let’s correlate the concepts of position, velocity, and acceleration to what we already know about a function and its first and second derivative.
corresponds with
corresponds with
corresponds with
Let’s summarize our relationships between position, velocity and acceleration below.
|Velocity |Position (Motion of the |
| |Particle) |
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|Is = 0 or is undefined | |
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|Is > 0 | |
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|Is < 0 | |
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|Changes from positive to negative | |
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|Changes from negative to positive | |
|Acceleration |Velocity |
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|Is = 0 or is undefined | |
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|Is > 0 | |
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|Is < 0 | |
|Changes from positive to negative | |
|Changes from negative to positive | |
The graph below represents the position, s(t), of a particle which is moving along the x axis.
■ At which point(s) is the velocity equal to zero? Justify your answer.
■ At which point(s) does the acceleration equal zero? Justify your answer.
■ On what interval(s) is the particle’s velocity positive? Justify your answer.
■ On what interval(s) is the particle’s velocity negative? Justify your answer.
■ On what interval(s) is the particle’s acceleration positive? Justify your answer.
■ On what interval(s) is the particle’s acceleration negative? Justify your answer.
Five Commandments of Particle Motion
Suppose the velocity of a particle is given by the function[pic] for t > 0, where t is measured in minutes and v(t) is measured in inches per minute. Answer the questions that follow.
a. Find the values of [pic]and[pic]. Based on these values, describe the speed of the particle at t = 3.
b. On what interval(s) is the particle moving to the left? Right? Show your analysis and justify your
answer.
1998 AP Calculus AB #3 (Modified)
The graph of the velocity v(t), in feet per second, of a car traveling on a straight road, for 0 < t < 50 is shown below. A table of values for v(t), at 5 second intervals of time, is also shown to the right of the graph.
a. During what interval(s) of time is the acceleration of the car positive? Give a reason for your
answer.
b. Find the average acceleration of the car over the interval 0 < t < 50. Indicate units of measure.
c. Find one approximation for the acceleration of the car at t = 40. Show the computations you used
to arrive at your answer. Indicate units of measure.
d. Is the speed of the car increasing or decreasing at t = 40? Give a reason for your answer.
2000 AP Calculus AB #2 (Partial)
Two runners, A and B, run on a straight racetrack for 0 < t < 10 seconds. The graph below, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of Runner B is given by the function v defined by [pic].
a. Find the velocity of Runner A and the velocity of Runner B at t = 2 seconds. Indicate units of
measure.
b. Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate
units of measure.
2002 AP Calculus AB #3 (Partial)
An object moves along the x – axis with initial position x(0) = 2. The velocity of the object at time t > 0 is given by the function [pic].
a. What is the acceleration of the object at time t = 4?
b. Consider the following two statements.
Statement I: For 3 < t < 4.5, the velocity of the object is decreasing.
Statement II: For 3 < t < 4.5, the speed of the object is decreasing.
Are either or both of these statements correct? For each statement, provide a reason why it is correct
or not correct.
2003 AP Calculus AB #2 (Partial)
A particle moves along the x – axis so that its velocity at time t is given by
[pic].
a. Find the acceleration of the particle at t = 2. Is the speed of the particle increasing at t = 2? Explain
why or why not.
b. Find all times in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.
More on Particle Motion
Finding Net and Total Distance
The graph below represents the velocity, v(t) which is measured in meters per second, of a particle moving along the x – axis.
At what value(s) of t does the particle have no acceleration on the interval (0, 10)? Justify your answer.
Express the acceleration, a(t), as a piecewise-defined function on the interval (0, 10).
For what value(s) of t is the particle moving to the right? To the left? Justify your answer.
Find the average acceleration of the particle on the interval [1, 8]. Show your work.
Definition of Net Distance:
Definition of Total Distance:
If a particle is moving in the same direction the entire amount of time, what can be said about the net distance and the total distance?
To Find the Net Distance a Particle Travels on an Interval
To Find the Total Distance a Particle Travels on an Interval
The position of a particle is given by the function [pic]where p(t) is measures in centimeters. Find the net and total distance the particle travels from t = 1.5 seconds to t = 4 seconds.
The position of a particle is given by the function [pic]where p(t) is measures in feet. Find the net and total distance the particle travels from t = 0.5 minutes to t = 1.5 minutes.
The position of a particle is given by the function [pic]where p(t) is measures in feet. Find the net and total distance the particle travels from t = [pic] minutes to t = [pic] minutes.
NO CALCULATOR PERMITTED
A particle moves along the x – axis so that its position at any time t > 0 is given by the function [pic], where p is measured in feet and t is measured in seconds.
a. Find the average velocity on the interval t = 1 and t = 2 seconds. Give your answer using correct
units.
b. On what interval(s) of time is the particle moving to the left? Justify your answer.
c. Using appropriate units, find the value of [pic] and[pic]. Based on these values, describe the
motion of the particle at t = 3 seconds. Give a reason for your answer.
d. What is the maximum velocity on the interval from t = 0 to t = 3 seconds. Show the analysis
that leads to your conclusion.
e. Find the total distance that the particle moves on the interval [1, 5]. Show and explain your analysis.
CALCULATOR PERMITTED
A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v is a differentiable function of t. Selected values of v(t) for 0 < t < 40 are shown in the table below
|t | |
|(min) |0 |
|[pic] |[pic] |
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We learned that [pic]and [pic]. Similarly, write what the anti-derivatives of sine and cosine are.
[pic]_________________ [pic]_________________
Find each of the following anti-derivatives.
|[pic] |[pic] |
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|[pic] |[pic] |
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Use the given information about [pic]and [pic]to find f(x).
|1. [pic] [pic] f (2) = 10 |2. [pic] [pic] f (0) = 0 |
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An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by the differential equation
[pic],
where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted, at t = 0.
a. Find the value of the differential equation above when t = 3. Using correct units of measure,
explain what this value represents in the context of this problem.
b. Find an equation for h(t), the height of the shrubs at any year t. Then, determine how tall the shrubs
are when they are sold.
A particle moves along the x – axis at a velocity of [pic], for t > 0. At time t = 1, its position is 4.
a. What is the acceleration of the particle b. What is the position of the particle
when t = 9? when t = 9?
Riemann Sums
A Graphical Approach to Approximating the Definite Integral
Calculating Riemann sums is a way to estimate the area under a curve for a graphed function on a particular interval. In this activity, you will learn to calculate four types of Riemann sums: Left Hand, Right Hand, Midpoint, and Trapezoidal Sums.
Approximation #1 – Left Hand Riemann Sum with intervals of length 2 units
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #2 – Left Hand Riemann Sum with intervals of length 1 unit
Again, consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length1 unit. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 0, x = 7, and the x-axis.
Approximation #3 – Left Hand Riemann Sum with intervals of length ½ unit
Again, consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit. Place the upper left hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 0, x = 7, and the x-axis.
Approximation #4 – Right Hand Riemann Sum with intervals of length 2 units
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #5 – Right Hand Riemann Sum with intervals of length 1 unit
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #6 – Right Hand Riemann Sum with intervals of length ½ unit
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length ½ unit. Place the upper right hand vertex of the rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #7 – Midpoint Riemann Sum with intervals of length 2 units
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 2 units. Place the midpoint of each rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #8 – Midpoint Riemann Sum with intervals of length 1 unit
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into rectangles of length 1 unit. Place the midpoint of each rectangle on the curve each time. Then, calculate the area of each rectangle and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #9 – Trapezoidal Riemann Sum with intervals of length 2 units
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into right trapezoids of length 2 units. Place the upper left and right hand vertices of the trapezoids on the curve each time. Then, calculate the area of each trapezoid and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
Approximation #10 – Trapezoidal Riemann Sum with intervals of length 1 unit
Let’s consider for a moment the function f(x) = [pic]. Graph this function on the axes provided below.
On the interval [1, 7] subdivide the area bounded by the graph of the function and the x-axis into right trapezoids of length 1 unit. Place the upper left and right hand vertices of the trapezoids on the curve each time. Then, calculate the area of each trapezoid and sum the areas to approximate the area of the region under the curve bounded by f(x) = [pic], x = 1, x = 7, and the x-axis.
| | |Do you think this is an over or under |
|Type of Riemann Sum |Numerical Approximation |approximation of the area? |
|Left Hand Riemann Sum | | |
|Intervals of 2 units | | |
|Left Hand Riemann Sum | | |
|Intervals of 1 unit | | |
|Left Hand Riemann Sum | | |
|Intervals of ½ unit | | |
|Right Hand Riemann Sum | | |
|Intervals of 2 units | | |
|Right Hand Riemann Sum | | |
|Intervals of 1 unit | | |
|Right Hand Riemann Sum | | |
|Intervals of ½ unit | | |
|Midpoint Riemann Sum | | |
|Intervals of 2 units | | |
|Midpoint Riemann Sum | | |
|Intervals of 1 unit | | |
|Trapezoidal Riemann Sum | | |
|Intervals of 2 units | | |
|Trapezoidal Riemann Sum | | |
|Intervals of 1 unit | | |
Given the table of values below, approximate each definite integral by finding the indicated Riemann Sum.
|x |0 |
|[pic] |[pic] |
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Properties of Definite Integrals
Given the integral statements, write what you think each is equivalent to. Be prepared to explain your reasoning with the rest of the class.
|1. [pic] |
|2. Given that a < c < b, [pic]= |
|3. If [pic] then [pic] = |
|4. Given that b < a, then [pic]= |
|5. If k is a constant, then [pic]= |
|6. [pic]= |
|7. Given that f(x) is an even function, [pic]= |
|[pic] |
|8. Given that f(x) is an odd function, [pic]= |
|[pic] |
If [pic] and [pic], determine the value of each of the following integrals using the properties of definite integrals. Explain how you arrived at your answer for each.
|[pic] |[pic] |
|[pic] |[pic] |
|[pic] |[pic], if f(x) is an even function |
|[pic], if f(x) is an odd function | |
Pictured to the right is the graph of a function f(x).
|What is the value of [pic]? |
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|What is the value of [pic]? |What is the value of [pic]? |
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|If F(0) = 5, what is the value of F(3), where F is the anti-derivative of f(x)? |If F(–2) = –2, what is the value of F(2), where F is the anti-derivative of |
| |f(x)? |
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The Fundamental Theorem of Calculus, Particle Motion, and Average Value
Three Things to Always Keep In Mind: (1) [pic], where v(t) represents the velocity
and p(t) represents the position.
(2) [pic]The Net Distance the particle travels on the
interval from t = a to t = b. If v(t) > 0 on the interval
(a, b), then it also represents the Total Distance.
(3) [pic]The Total Distance the particle travels on
the interval (a, b), whether or not v(t) > 0. To be safe,
always do this integral when asked to find total distance
when given velocity.
1. The velocity of a particle that is moving along the x – axis is given by the function v(t) = 3t2 + 6.
a. If the position of the particle at t = 4 is 72, what is the position when t = 2?
b. What is the total distance the particle travels on the interval t = 0 to t = 7?
2. The velocity of a particle that is moving along the x – axis is given by the function[pic].
a. If the position of the particle at t = 1.5 is 2.551, what is the position when t = 3.5?
b. What is the total distance that the object travels on the interval t = 1 to t = 5?
The graph of the velocity, measured in feet per second, of a particle moving along the x – axis is pictured below. The position, p(t), of the particle at t = 8 is 12. Use the graph of v(t) to answer the questions that follow.
a. What is the position of the particle at t = 3?
b. What is the acceleration when t = 5? c. What is the net distance the particle travels from
t = 0 to t = 10?
d. What is the total distance the particle travels from t = 0 to t = 10?
The table above shows values of the velocity, V(t) in meters per second, of a particle moving along the x – axis at selected values of time, t seconds.
a. What does the value of [pic]represent?
b. Using a left Riemann sum of 6 subintervals of equal length, estimate the value of [pic]. Indicate
units of measure.
c. Using a right Riemann sum of 6 subintervals of equal length, estimate the value of [pic]. Indicate
units of measure.
d. Using a midpoint Riemann sum of 3 subintervals of equal length, estimate the value of [pic].
Indicate units of measure.
e. Using a trapezoidal sum of 6 subintervals of equal length, estimate the value of [pic].
Indicate units of measure.
f. Find the average acceleration of the particle from t = 3 to t = 9. For what value of t, in the table, is this
average acceleration approximately equal to v’(t)? Explain your reasoning.
Interpretations and Applications of the Derivative and the Definite Integral
[pic]The rate at which that amount is changing
For example, if water is being drained from a swimming pool and R(t) represents the amount of water, measured in cubic feet, that is in a swimming pool at any given time, measured in hours, then [pic]would represent the rate at which the amount of water is changing.
[pic]
What would the units of [pic]be?__________________________
[pic]
In the context of the example situation above, explain what this value represents: [pic].
The table given below represents the velocity of a particle at given values of t, where t is measure in minutes.
|t minutes |0 |5 |
|[pic] | |To what is this equivalent? |
The average value of a function, f(x), on
an interval [a, b] is defined to be:
Find the average value of the function [pic]on the interval 1 < x < 3.
Find the average value of the function f(x) = 2 – 4x on the interval 2 < x < 6.
A ski resort uses a snow machine to control the snow level on a ski slope. Over a 24-hour period the volume of snow added to the slope per hour is modeled by the equation[pic]. The rate at which the snow melts is modeled by the equation[pic]. Both S(t) and M(t) have units of cubic yards per hour and t is measured in hours for 0 < t < 24. At time t = 0, the slope holds 50 cubic yards of snow.
a. Compute the total volume of snow added to the mountain over the first 6-hour period.
b. Find the value of [pic] and [pic]. Using correct units of measure, explain what each
represents in the context of this problem.
c. Is the volume of snow increasing or decreasing at time t = 4? Justify your answer.
d. How much snow is on the slope after 5 hours? Show your work.
e. Suppose the snow machine is turned off at time t = 10. Write, but do not solve, an equation
that could be solved to find the time t = K when the snow would all be melted.
2003 AB #6 Part b
[pic]
[pic]
The graph pictured to the right represents the graph of the derivative,[pic], of a function on the interval [–4, 5]. The graph consists of straight line segments and a semi-circle.
a. Find the value of [pic].
b. Find the value of [pic].
c. If f(0) = –3, what is the value of f(3)?
d. If f(1) = 4, what is the value of f(5)?
The Second Fundamental Theorem of Calculus
Functions Defined by Integrals
Statement and proof of the Second Fundamental Theorem of Calculus.
Given the functions, f(t), below, use [pic] to find F(x) and F’(x) in terms of x.
|1. f(t) = t3 |2. f(t) = 4t – t2 |3. f(t) = cos t |
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Given the functions, f(t), below, use [pic] to find F(x) and F’(x) in terms of x.
|4. f(t) = t3 |5. f(t) = cos t |6. f(t) = [pic] |
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Given to the right is the graph of f(t) which consists of three line segments and one semicircle. Additionally, let the function g(x) be defined to be[pic].
1. Find g(–2).
2. Find g(2).
3. Find g(6). 4. Find[pic].
5. Find[pic]. 6. Find[pic]. Give a reason for your
answer.
Pictured to the right is the graph of g(t) and the function f(x) is defined to be[pic].
1. Find the value of f(0).
2. Find the value of f(4). 3. Find the value of f(3).
4. Find[pic]. Give a reason for your 5. Find[pic]. Give a reason for your
answer. answer.
6. Approximate the value of[pic]. Explain why this is a good approximation.
Let F(x) = [pic], where the graph of f(t) is shown to the right. Answer the following questions.
1. Complete the following table for values of F(x).
|x |2 |3 |5 |6 |9 |
|F(x) | | | | | |
2. On what interval(s) is f(t) positive? 3. On what interval(s) is f(t) negative?
4. On what interval(s) is F(x) increasing? 5. On what interval(s) is F(x) decreasing?
Justify your answer. Justify your answer.
6. On what interval(s) is F(x) concave up? Justify your answer.
7. On what interval(s) is F(x) concave down? Justify your answer.
Solving Differential Equations
Examples of Variable Separable Differential Equations
Given below are differential equations with given initial condition values. Find the particular solution for each differential equation.
|1. [pic] and f(–1) = 2 |2. [pic] and f(0) = 2 |
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|3. [pic]and f(0) = 2 |4. [pic]and f(1) = –3 |
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|5. [pic]and f(0) = 0 |6. [pic]and f(2) = 0 |
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The velocity of a particle is given by the function v(t) = 3t + 6 and the position at t = 2 is 3.
a. Is the particle moving to the left or right at t = 2? Justify your answer.
b. What is the position of the particle, s(t), for any time t > 0?
c. Does the particle ever change directions? Justify your answer.
d. Find the total distance traveled by the particle for t = 1 to t = 4.
A particle moves along the x-axis so that, at any time t > 0, its acceleration is given by a(t) = 6t + 6. At time t = 0, the velocity of the particle is –9 and its position is –27.
a. Find v(t), the velocity of the particle at any time t.
b. For what value(s) of t > 0 is the particle moving to the right? Justify your answer.
c. Find the net distance traveled by the particle over the interval [0, 2].
d. Find the total distance traveled by the particle over the interval [0, 2].
Slope Fields
Graphical Representations of Solutions to Differential Equations
A slope field is a pictorial representation of all of the possible solutions to a given differential equation.
Remember that a differential equation is the first derivative of a function, [pic]or [pic]. Thus, the solution to a differential equation is the function, f(x) or y.
There is an infinite number of solutions to a differential equation. Why?
For the AP Exam, you are expected to be able to do the following four things with slope fields:
1.________________________________________________________________________________
________________________________________________________________________________
2.________________________________________________________________________________
________________________________________________________________________________
3.________________________________________________________________________________
________________________________________________________________________________
4.________________________________________________________________________________
________________________________________________________________________________
#1 Sketch a slope field for a given differential equation.
Given the differential equation below, compute the slope for each point
Indicated on the grid to the right.
Then, make a small mark that
approximates the slope
through the point.
Given the differential equation below, compute the slope for each point
Indicated on the grid to the right.
Then, make a small mark that
approximates the slope
through the point.
#2 Given a slope field, sketch a solution curve through a given point.
To the right is pictured the slope field that you developed for the differential equation
on the previous page.
Sketch the solution curve through the point (1, -1).
To do this, you find the point and then follow the slopes as you connect the lines.
#3 Match a slope field to a differential equation.
Since the slope field represents all of the particular solutions to a differential
equation, and the solution represents the ANTIDERIVATIVE of a differential
equation, then the slope field should take the shape of the antiderivative of dy/dx.
Match the slope fields to the differential equations on the next page.
A. B. C.
D. E. F.
G. H. I.
J.
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| |1. [pic] | |2. [pic] |
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| |3. [pic] | |4. [pic] |
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| |5. [pic] | |6. [pic] |
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| |7. [pic] | |8. [pic] |
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| |9. [pic] | |10. [pic] |
#4 Match a slope field to a solution to a differential equation.
When given a slope field and a solution to a differential equation, then the slope
field should look like the solution, or y.
Match the slope fields below to the solutions on the next page.
A. B. C.
D. E. F.
G. H. I.
J.
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| |1. [pic] | |2. [pic] |
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| |3. [pic] | |4. [pic] |
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| |5. [pic] | |6. [pic] |
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| |7. [pic] | |8. [pic] |
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| |9. [pic] | |10. [pic] |
Shown below is a slope field for which of the following differential equations?
Finding the Area between Two Curves
An Application of Integration
Graph the function[pic]and find the value of [pic]. Using one color, shade the region for which this value represents the area.
Graph the function [pic] on the same grid above and then find the value of [pic]. Using a different color, shade the region for which this value represents the area.
What do you suppose you would do to find the area of the region that is located in between the graphs of f(x) and g(x)? Find this value.
Now, find the value of the definite integral below if [pic]and [pic]. Show your work.
[pic]
What do you notice about this value?
This brings about the general way that we will find the area between two curves.
Find the area of the shaded region, R, that is bounded by y = sin((x) and y = x3 – 4x.
Find the area of Region R.
Find the equation of line l if it is tangent to the graph of f(x) at (0, 3).
At what ordered pair, other than (0, 3), does the graph of line l intersect the graph of f(x)?
Find the area of Region S.
Identify the points of intersection of f(x) and g(x).
Find the area of Region R.
Find the area of Region S.
Volumes of Solids of Revolution
Region #1 y = x2 y = x + 2
1. Find the volume when the region is rotated about the x – axis.
2. Find the volume when the region is rotated about the line y = 4.
3. Find the volume when the region is rotated about the line y = –2.
Region #2 y = [pic] y = x – 2
x = y2 x = y + 2
1. Find the volume when the region is rotated about the y – axis.
2. Find the volume when the region is rotated about the line x = 4.
3. Find the volume when the region is rotated about the line x = 7.
Region #3 y = x2 y = 4x – x2
x = [pic] x = [pic]
1. Find the volume when the region is rotated about the x – axis.
2. Find the volume when the region is rotated about the y – axis.
3. Find the volume when the region is rotated about the line x = –2.
4. Find the volume when the region is rotated about the line y = 6.
Volumes of Solids with Known Cross Sections
|Cross Sections that are Squares |Cross Sections that are Isosceles Right Triangles |
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|Cross Sections that are Equilateral Triangles |Cross Sections that are Semicircles |
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Region R is bounded by y = sin((x) and y = x3 – 4x.
Find the volume of the solids formed whose cross sections are the shapes indicated below. The cross sections are perpendicular to the x – axis.
|a. Cross sections are equilateral triangles |b. Cross sections are semi-circles |
|c. Cross sections are isosceles right triangles |d. Cross sections are squares. |
|e. Cross sections are rectangles whose height is |f. Cross sections are rectangles whose height |
|twice the length of the base. |is one-third the length of the base. |
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More on Area, Volumes of Solids of Revolution, Volumes of Solids with Known Cross Sections, and Functions Defined By Integrals
1. Let R be the region bounded by the graphs
of y = ln x and the line y = x – 2 as shown to
the right.
a) Find the area of R.
b) Find the volume of the solid generated when R is rotated about the horizontal line y = –3.
c) Write, but do not evaluate, an integral expression that can be used to find the
volume of the solid generated when R is rotated about the y-axis.
d) Find the volume of the solid whose base is region R that is formed by cross sections that are semi-circles that are perpendicular to the x – axis.
2. Let f and g be the functions given by
[pic] and [pic]. Let
R be the region in the first quadrant enclosed by
the y-axis and the graphs of f and g, and let S be
the region in the first quadrant enclosed by the
graphs of f and g shown to the right.
a) Find the area of R.
b) Find the area of S.
c) Find the volume of the solid generated when S is revolved about the horizontal line y = –1.
d) Find the volume of the solid whose base is the cross section area of region S and is formed by squares that are perpendicular to the x-axis.
e) Find the volume of the solid whose base is the cross section area of region S and is formed by equilateral triangles that are perpendicular to the x – axis.
3. Let f and g be the functions given by [pic] and
[pic] for 0 < x < 1. The graphs of f and g are
shown in the figure to the right.
a) Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by squares that are perpendicular to the x-axis.
b) Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by semi-circles that are perpendicular to the x-axis.
c) Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by equilateral triangles that are perpendicular to the x-axis.
d) Find the volume of the solid whose base is the cross section of the region bounded by the graphs of f and g and is formed by right isosceles triangles that are perpendicular to the x-axis.
Functions Defined by Integrals
1. Find [pic]if [pic].
(a) e (b) 2e (c) e – 1 (d) 3e (e) 4e
2. Let [pic]. Find h’(1).
(a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic]
Pictured to the right is the graph of f, which consists of two semi-circles and one line segment on the interval [0, 17]. Let [pic].
(a) Find the values of g(8), g’(8) and g’’(8).
(b) On what interval(s) is the graph of g(x) concave down? Justify your answer.
(c) On what interval(s) is the graph of g(x) increasing? Justify your answer.
(d) Find all values on the open interval (0, 17) at which g has a relative minimum. Justify your answer.
(e) What are the x – coordinates of each point of inflection of g(x)? Justify your answer.
-----------------------
[pic]
[pic]
[pic]
Possible Graph of [pic]
Graph of f(x)
Graph of f(x)
Graph of [pic]
[pic]
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
At this point, place a star next to the approximation that you feel is the most accurate to the actual area.
In our next lesson, we will learn how to find the EXACT area of a region between the graph of a function and the x – axis.
In the space below, we will come back to this to find the exact area once we complete the next lesson in order to see which approximation to the left is the most accurate.
The acceleration of a particle moving along the x – axis at time t is given by a(t) = 6t – 2. If the velocity is 25 when t = 3 and the position is 10 when t = 1, then the position x(t) =
A. 9t2 + 1
B. 3t2 – 2t + 4
C. t3 – t2 + 4t + 6
D. t3 – t2 + 9t – 20
E. 36t3 – 4t2 – 77t + 55
A particle moves along the x – axis so that its velocity is given by the function[pic]. On the interval 0 < t < 10, how many times does the particle change directions?
A. One
B. Three
C. Four
D. Five
E. Seven
[pic]
[pic]
[pic]
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