Chapter 3
AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 1
DEFINITION DEVELOPMENT
Average rate of change of f on [a,b]
Slope of the secant line though curve f on [a, b]
Standard:
Alternate:
Instantaneous rate of change of f at [pic]
Slope of the line tangent to the curve of f at [pic]
Derivative of f at [pic]
Standard:
Alternate:
1. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [-1, 2]
b. Using the standard definition of a derivative at [pic] …
Find the instantaneous rate of change in f at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic]
2. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [0, 8]
b. Using the alternate definition of a derivative at [pic] …
Find the instantaneous rate of change in f at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic].
3. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [0, 2]
b. Using the standard definition of a derivative at [pic] …
Find the instantaneous rate of change in f at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic].
4. Consider the function [pic]where[pic]
a. Find the average rate of change in g on the interval [-2, 5]
b. Using the standard definition of a derivative at [pic] …
Find the instantaneous rate of change in g at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic].
Details and Summary
What is the derivative of f at x = a?
DEF (Sandard): The derivative of the function [pic]at the point [pic]is [pic], provided it exists.
DEF (Alternate): The derivative of the function [pic]at the point [pic]is [pic], provided it exists.
Is the function differentiable at x = a?
DEF (Std): A function f is differentiable at x = a if [pic]exists.
If the limit does not exists then we say that the function is not differentiable at x = a.
DEF (alt): A function f is differentiable at x = a if [pic]exists.
If the limit does not exists then we say that the function is not differentiable at x = a.
AB.Q103.LESSON 1 – HW:
1. If [pic]and [pic], find an equation of (a) the tangent line, and (b) the normal line to the graph of [pic]at the point where [pic].
2. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [-2, 4]
b. Find the instantaneous rate of change in [pic] at [pic].
Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].
c. Write an equation of the tangent line to the graph of [pic]at [pic]
3. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [-2, 4]
b. Find the instantaneous rate of change in [pic] at [pic].
Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].
c. Write an equation of the tangent line to the graph of [pic]at [pic]
4. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [-4, 0]
b. Find the instantaneous rate of change in [pic] at [pic].
Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].
c. Write an equation of the tangent line to the graph of [pic]at [pic]
5. Consider the function [pic].
a. Find [pic]using the standard definition of the derivative at x = a.
b. Find [pic]using the alternate definition of the derivative at x = a.
6. Let [pic]. Prove that p is or is not continuous at [pic].
AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 2
1. The function [pic]is shown below. This function consists of line segments (in gray) and other non-linear curves (in black). Integer coordinates have been highlighted.
[pic]
A) Find or estimate [pic]
B) Find or estimate [pic]
C) Find or estimate [pic]
D) Find or estimate [pic]
E) Find or estimate [pic]
F) Find or estimate [pic]
G) Find or estimate [pic]
H) Find or estimate [pic]
I) Find or estimate [pic]
J) Find or estimate all values of x where [pic]has a horizontal tangent.
2. Estimating the derivative with “an average on a small neighborhood.”
The coordinates of f for various values of x are given.
x |0 |0.5 |1.0 |1.5 |2.0 |2.5 |3.0 |3.5 |4.0 | |f |-12 |-15 |-16 |-15 |-12 |-7 |0 |9 |20 | |
Assuming a smooth curve representation of [pic].
[pic]
A) Eyeball estimate [pic]
B) Eyeball estimate [pic]
C) Use a standard estimation of [pic]. Show your work.
D) Use a standard estimation of [pic]. Show your work.
3. Consider the function [pic]where [pic]
a. Find the average rate of change in f on the interval [-1, 1]
b. Using the alternate definition of a derivative at [pic] …
Find the instantaneous rate of change in f at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic].
4. Consider the function [pic]
A. Use the standard definition of a derivative at [pic]to prove that p is or is not differentiable at [pic]. In other words, prove that the function p does or does not have a derivative at [pic].
B. Classify the behavior of [pic]at [pic].
C. Provide a formal proof for whether or not [pic]is continuous at [pic].
LOGIC:
THM:
PROOF:
AB.Q103.LESSON 2 – HW:
1. Consider the function [pic]where[pic]
a. Find the average rate of change in f on the interval [-1, 2]
b. Using the standard definition of a derivative at [pic] …
Find the instantaneous rate of change in f at [pic] (Precede your answer with Lagrange notation).
c. Write an equation of the tangent line to the graph of [pic]at [pic]
2. Consider the function [pic]
a. Prove that b is or is not continuous at [pic]
b. Using the standard definition of a derivative at [pic], prove that f is not differentiable [pic]
c. Explain how you could have used part (a) to prove that f is not differentiable [pic]
3. Use the table below to estimate a) [pic] and b) [pic]
t |0.00 |0.56 |0.92 |1.19 |1.30 |1.39 |1.57 |1.74 |1.98 |2.18 |2.41 |2.64 |3.24 | |f(t) |1577 |1512 |1448 |1384 |1319 |1255 |1191 |1126 |1062 |998 |933 |869 |805 | |
4. Using the definition for a derivative at [pic],
prove that [pic]is or is not differentiable at [pic].
Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].
5. Prove that [pic] is or is not continuous at [pic].
AB.Q103.NOTES: Chapter 2.4, 3.1, 3.2 LESSON 3
Definition: The derivative of the function [pic]with respect to the variable [pic]is the function [pic]whose value at [pic] is [pic], provided it exists.
NO ALTERNATE DEFINITION FOR [pic]
1. Consider the function[pic].
A. Find [pic].
B. Write an equation of the tangent line to the graph of f(x) at [pic] and [pic].
C. Write an equation of the normal to the graph of f(x) at [pic] and [pic].
D. Find the points on the graph of f where the slope of the tangent line is parallel to [pic].
2. Graph [pic]from [pic]
The graph of [pic]shown here is made of line segments joined end to end. Graph the function’s derivative.
[pic] [pic]
3. Graph [pic] from [pic]
Sketch a possible graph of a continuous function f that has domain [-3, 3], where [pic]and the equation of [pic]is shown below.
[pic]
[pic]
AB.Q103.LESSON 3 – HW:
1. The graph of the function [pic]shown here is made of line segments joined end to end.
Graph [pic]and state its domain.
[pic] [pic]
2. Sketch the graph of a continuous function with domain [-2,2],[pic], and [pic].
[pic]
3. Using the information from problem 2, write an equation of the line tangent to f at [pic]
4. Find [pic] for [pic] using the appropriate definition.
5. Find the value of x for which the tangent to [pic] is horizontal.
6. Find [pic] for [pic] using the appropriate definition.
7. Find [pic] for [pic] using the appropriate definition.
8. Consider the function [pic]where[pic].
Find [pic].
9. Using the definition for a derivative at [pic],
prove that [pic]is or is not differentiable at [pic].
Classify [pic]as either smooth, a corner, a cusp, a vertical line tangent, or discontinuous at [pic].
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- chapter 3 developmental psychology quizlet
- mcgraw hill algebra1 chapter 3 lesson 8
- chapter 3 psychology quizlet test
- psychology chapter 3 quiz answers
- developmental psychology chapter 3 quizlet
- strategic management chapter 3 quizlet
- psychology chapter 3 exam
- psychology chapter 3 test questions
- quizlet psychology chapter 3 quiz
- chapter 3 psychology quiz
- developmental psychology chapter 3 test
- quizlet psychology chapter 3 answers