MAT 271 - Arizona State University



MAT 270

Chapter 1-4

Section 1.1 Tangent lines and the approximate length of a curve

Equation of the tangent line through the point [pic] having slope m is

[pic].

The slope of a tangent line, which passes through [pic] and [pic]is [pic]

The middle point of a line segment joining the points [pic] and [pic]is [pic]

Examples

1. Find the equation of a line through (1, 2) and (5, 7).

Solution: [pic], the equation of the line is [pic]or [pic]

2. Estimate the slope of the tangent line to [pic] at [pic]

Solution: We have the given point [pic], we need to choose several second points close to the given point and look at the convergence of the slope. As an example we choose another point (1.1, 2.21) for the slope [pic]. Observe the following table

|Second point |(1.1, 2.21) |(1.01, 2.0201) |(0, 1) |(0.9, 1.81) |(0.99,1.9801) |

|Slope m |2.1 |2.01 |1 |1.9 |1.99 |

We may conclude that the estimated slope is 2. The point (0, 1) is not a good choice as it is little far from the given point ((1, 2)

3. Estimate the arc length of the curve [pic] for [pic]

Solution: We have on the curve two end points (-1, 2) and (1, 2) and the middle point (0, 1). Our first estimation is [pic]

Next on the curve we take middle point (-.5, 1.25) of (-1, 2) and (0, 1) and middle point (.5, 1.25) for (0, 1) and (1, 2). Our second estimate is [pic]

Exercises

Estimate the slope of [pic] at x = a

1. [pic] at x = 2

2. [pic] at x = 1

3. [pic] at x = 0

Estimate the arc length (First and second estimate)

1. [pic] between [pic]

2. [pic] between [pic]

3. [pic] between [pic]

Section 1.2 The concept of limit

The graph of [pic] is not defined at x = 2 but it has limiting value 4 at that point. If you graph and trace at x = 2 you will find a hole. On the other hand the graph of [pic] has a vertical asymptote at x = 2. This graph does not have a hole at x = 2 and limit at x = 2 also does not exist. We examine the situation below:

Test the value of [pic] close to 2 from left side

|x |2 |1.9 |1.99 |1.999 |1.9999 |

|[pic] |- |3.9 |3.99 |3.999 |3.9999 |

And the value of [pic] close to 2 from right side

|x |2 |2.1 |2.01 |2.001 |2.0001 |

|[pic] |- |4.1 |4.01 |4.001 |4.0001 |

In both the cases the value approaches (converse to) 4. The first case is called the left hand limit and the second case is the right hand limit.

Now let us look at the second example for [pic]

Test the value of [pic] close to 2 from left side

|x |2 |1.9 |1.99 |1.999 |1.9999 |

|[pic] |- |-76.1 |-796.01 |-7996.001 |-79996.0001 |

And the value of [pic] close to 2 from right side

|x |2 |2.1 |2.01 |2.001 |2.0001 |

|[pic] |- |84.1 |804.01 |8004.001 |80004.0001 |

In the first case the value approaches to [pic], on the other hand in second case the value approaches to [pic]. Thus the limit does not exist.

Exercises

1. Test the limit of [pic] for x = 0

2. Test the limit of [pic] for x = 0

3.

Section 1.3 Computation of limits

Remember we have the following formulas

[pic]

Examples

1. [pic] 2. [pic]

3. [pic] is not a number, limit does not exist.

4. [pic] 5. [pic]

6. [pic] 7. [pic]

8. [pic] 9. [pic]

10. [pic] 11. [pic]

12. [pic] 13. [pic] 14. [pic]

15. [pic] 16. [pic] 17. [pic]

18. [pic]

Squeeze Theorem Suppose that [pic] for all [pic], except possibly at [pic] and that

[pic], for some real number l, then it follows that [pic]

19. [pic] by squeeze theorem

20. [pic] is a piece-wise defined function. You need to find the following results.

a) [pic] b) [pic] c) [pic]

21. [pic] is a piece-wise defined function. You need to find the following results.

a) [pic] b) [pic] c) [pic]

22. [pic] 23. [pic] 24. [pic]

Section 1.4 Continuity

If [pic], we say that [pic] is continuous at [pic]. Otherwise the function is discontinuous or not continuous. The limit of the function exists if [pic], a finite value.

Examples

1. [pic] is a piece-wise defined function. You need to test the continuity.

Solution: [pic], the function is continuous at [pic]. And [pic], limit does not exist means the function is discontinuous at x = 1.

2. Remove discontinuity if any of [pic]

Solution: [pic], limit exists. For continuity we must have [pic].

4. Determine the value(s) of a, and b that makes the given function continuous.

[pic]

Solution: [pic]

For continuity [pic]

Exercises

5. Determine the value(s) of a, and b that makes the given function continuous.

[pic]

6. Determine the value(s) of a, and b that makes the given function continuous.

[pic]

Intermediate Value Theorem (IVT)

Suppose that f is continuous on [a, b] and W is any number between [pic] and [pic], then there is a number [pic] for which [pic]

Corollary: Suppose that f is continuous on [a, b] and [pic] and [pic] have opposite signs, then there is at least a number [pic] for which [pic], the number c is called the zero of the function[pic].

Examples

Use intermediate value theorem to verify that [pic]has a zero in the interval. Then use bisection method to find an interval of length 1/32 that contains the zero.

1. [pic]

Solution: [pic], there is a zero in [2, 3]

[pic], there is a zero in [2.5, 3]

[pic], there is a zero in [2.5, 2.75]

[pic], there is a zero in [2.625, 2.75]

[pic], there is a zero in [2.625, 2.6875]

[pic], there is a zero in [2.625, 2.65625] having length 2.65625-2.625 = 0.03125 = 1/32

Exercise

2. [pic]

Section 1.5 Limits involving infinity; Asymptotes

1. [pic] (DNE) 2. [pic] 3. [pic]

4. [pic] 5. [pic], has a slant asymptote

6. [pic], K is positive constant.

7. [pic] 8. [pic] 9. [pic]

Section 1.6 Formal Definition of Limit

For a function f defined in some open interval containing a (but not necessarily at a), we say that [pic], if given any number [pic], there is another number [pic], such that [pic] guarantees that [pic]

Exercises

Determine relation between [pic] and [pic]to prove the following limits:

1. [pic] 2. [pic] 3. [pic]

4. [pic]

Chapter 2

Section 2.1 Tangent Lines and Velocity

You need to know the following results:

Equation of a straight line through (0, b) having slope m is [pic]

Equation of a straight line through two given points [pic]is [pic] where [pic] = slope

Now if [pic] be two points on the graph of the function [pic], then line joining the given two points is called a secant line and has the equation [pic] where [pic]. On the other hand if the point Q approaches to P then the limiting position of the secant line called the tangent line to the function at the point P. In this case the slope of the tangent line is given by [pic]. The slope of the secant line is denoted by [pic].

As a general case we use the following notations and definitions

Consider [pic] and two points on the function [pic] then [pic] and [pic]

The equation of a tangent line to [pic] at x = a is [pic]

Examples

1. Find the slope of a secant line passing through the points (1, 2) and (0, 1) on [pic].

2. Find the slope of a tangent line to [pic] at [pic]

3. Find the equation of the tangent line to [pic] at [pic]

4. Find the equation of the tangent line to [pic] at [pic]

Velocity

Suppose [pic] gives the position of a particle at time t moving along a straight line. For [pic] and [pic], the average velocity [pic] and the velocity of the particle at [pic] is [pic] which is also known as instantaneous velocity or rate of change of displacement at [pic].

5. [pic], find average velocity

6. Find instantaneous velocity of [pic]

7. Show that [pic]does not have a tangent line at [pic]

8. Compute the slope of the secant line for

[pic] at i) [pic] ii) [pic]

9. Compute the slope of the secant line for

[pic] at i) [pic] ii) [pic]

10. Compute the slope of the secant line for

[pic] at i) [pic] ii) [pic]

11. Compute the slope of the secant line for

[pic] at i) [pic] ii) [pic]

12. Compute the slope of the secant line for

[pic] at i) [pic] ii) [pic]

13. Find the equation of the tangent line to [pic] at i) [pic]

ii) [pic]

14. Find the equation of the tangent line to [pic] at i) [pic]

ii) [pic]

15. Find the equation of the tangent line to [pic] at i) [pic] ii) [pic]

16. Find the equation of the tangent line to [pic] at i) [pic] ii) [pic]

17. Find the equation of the tangent line to [pic] at i) [pic] ii) [pic]

18. For the given position functions calculate average velocity

a) [pic]

b) [pic]

c) [pic]

19. For the given position functions calculate instantaneous velocity

a) [pic]

b) [pic]

c) [pic]

Section 2.2 The Derivative

Definition: The derivative of the function [pic] at [pic] is defined as [pic] provided the limit exists. If this limit exists we say that [pic]is differentiable at [pic]. The general form derivative at every point of [pic], we write the limit definition of derivative [pic]

Examples:

1. [pic]

2. [pic]

3. [pic]

4. The graph of [pic] is given below, plot the graph of [pic]

1 3 5

Theorem: If [pic]is differentiable at [pic], then [pic]is continuous at [pic]. The converse is false in general.

The point(s) of nondifferentiability

1. A hole or a jump

2. Vertical tangency

3. A cusp or a corner

5. Find the point of nondifferentiability from example 4 above.

Section 2.3 The Power Rule for Derivative

You may use the limit definition of derivative to verify the following results. We use [pic]

Formulas

1. [pic], the derivative of y = c, a constant is zero

2. [pic], the derivative of y = x is 1.

3. [pic], for all real numbers x.

If [pic] are displacement and velocity functions respectively then the acceleration function [pic].

Examples:

Find the derivative of the following functions with respect to x (1-9)

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic], find [pic]

11. Find the equation of the tangent line to the curve [pic] at x = 1

12. Find the equation of the normal line to the curve [pic] at x = 1

Section 2.4 The Product and Quotient Rule for Derivative

For two differentiable functions [pic] and [pic] the product rule is [pic] or in another form [pic] and the quotient rule is [pic] or in another form [pic]

Examples

Find the derivative of the following functions w. r. to x (use product and quotient rules)

1. [pic] 2. [pic] 3. [pic]

Section 2.5 The Chain Rule for Derivative

For derivative of functions like [pic] is not so difficult to perform as we can expand the function to [pic] and then take the derivative. But functions like [pic] is very time consuming to expand and then take the derivative. So need a convenient method to perform derivative of such functions.

We introduce the chain rule here [pic], if g is differentiable at x and f is differentiable at g(x).

Examples

1. [pic] and [pic]. But using chain rule

[pic]

2. [pic], then [pic]

3. [pic], then

[pic]

Theorem: If f is differentiable at all x and has an inverse function [pic]

Then [pic]

Example Given that [pic], find [pic] and show that [pic] does not exist.

Solution: Set [pic] and solve for [pic], now [pic], where [pic]

But for [pic]: set [pic] and solve for [pic], then [pic] DNE.

4. Find derivative of [pic] and compute [pic], where

[pic]

5. Find derivative of [pic]

6. Find derivative of [pic]

Section 2.6 Derivatives of Trigonometric Functions

Formulas:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

Examples

Find derivative of the given functions w. r. to given variables

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] show that the function is continuous and differentiable for all x

14. Higher order derivatives

a) [pic] determine [pic]

b) [pic] determine [pic]

Application:

15. Suppose that. [pic] measures the displacement (in inches) of a weight suspended from a spring t seconds after it is released and that [pic], find the velocity at any time t and determine the maximum velocity.

16. Draw a simple circuit with capacitance 1 farad, the inductance 1 Henry and impressed voltage [pic] volts at time t. The model for a total charge [pic]in the circuit at time t is [pic] coulombs. The current is defined to be the rate of change of the charge with respect to time t and so is given by [pic] amperes. Compute the current at t = 0 and t = 1.

Section 2.7 Derivatives of Exponential and Logarithmic Functions

Formulas:

[pic] [pic] [pic]

Note: [pic]is a constant

Examples: Find derivative of the following functions

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. Given [pic] and [pic] show that [pic] and [pic]

Application:

14. If the value of a 100-dollar investment doubles every year, its value after t years is given by [pic]. Find the rate of investment value, and the relative rate of change.

Solution: The rate of investment is [pic] and relative rate of change is [pic]

15. The velocity of a hanging weight is [pic] where vertical displacement is given by [pic], [pic] are constants. Find [pic]

Solution: [pic]

Section 2.8 Implicit Derivatives and Inverse Trigonometric Functions

Formulas:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

Examples

1. Given [pic], determine [pic] when [pic]

2. Find the equation of the tangent line to curve of [pic] at (2, 1)

3. Find the equation of the tangent line to curve of [pic] at (2, -2)

4. Given [pic], find [pic] when [pic]

Find the derivative of the following functions:

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic], simplify the answer.

Section 2.9 The Mean Value Theorem

First of all we discuss the special case of Mean Value Theorem.

The Rolle’s Theorem: Suppose that f is continuous on the interval [pic], differentiable on [pic]and [pic]then there is a number [pic]such that [pic]

The Mean Value Theorem: Suppose that f is continuous on the interval [pic], differentiable on [pic]then there is a number [pic]such that [pic]

Constant Theorem: Suppose that [pic]for all x on an open interval I, then f must be a constant, that is [pic], a constant

Corollary: Suppose that [pic]for all x in some open interval I, then for some constant c, [pic]

Chapter 3 Application of Differentiation

Section 3.1 Linear Approximations and Newton’s Method

We have the linear approximation [pic] and the Newton’s method [pic]. , to apply Newton’s method effective one needs an intelligent guess for [pic].

Examples

1. Find the linear approximation of [pic] at [pic] and thus approximate the values of [pic] and [pic]

2. Find a zero of [pic] using Newton’s Method with an initial guess [pic]

3.

Section 3.2 Indeterminate Forms and L’Hopital Rule

We have noticed the following indeterminate forms [pic]. The L’Hopital rule is applicable for [pic] if this limit has the form [pic]. The L’Hopital rule is as follows [pic]. One may apply the rule repeatedly as long as the indeterminate form appears.

Examples with the above indeterminate forms:

1. [pic]

2. [pic], does not exist (DNE)

3. [pic]

4. [pic], DNE as the result is not a finite number

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic] DNE

Section 3.3 Maximum and Minimum Values

For a function [pic]if [pic] or [pic] is undefined for all c in the domain of [pic], then c is called the critical number. If a function [pic] has either maximum or minimum value(s) at x = c, then c must be a critical number. The converse is not true in general.

Theorem: A continuous function [pic] defined on a closed, bounded interval [pic] attains both an absolute maximum and absolute minimum.

i) [pic]is the absolute maximum of f in S if [pic]

ii) [pic]is the absolute minimum of f in S if [pic]

Examples:

1. Find all critical numbers of [pic]

2. Find the absolute extrema of [pic] on [-2, 4]

3. Find the absolute extrema of [pic] on [0, 4]

4. Find the local extrema of [pic]

5. Find the local extrema of [pic]

Section 3.4 Increasing and Decreasing Functions

Under the following conditions the function will be called increasing [pic]or decreasing [pic]on some open interval.

i) If [pic] for all [pic], then [pic] on I.

ii) If [pic] for all [pic], then [pic] on I.

Examples:

1. Graph the function [pic]showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing.

2. Graph the function [pic]showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing

3. Graph the function [pic]showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing

4. Graph the function [pic]showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing

Section 3.5 Concavity and Second Derivative Test

Under the following conditions the function will be called concave up [pic]or concave down [pic]on some open interval.

i) If [pic] for all [pic], then [pic] then f is concave up [pic] on the open interval I.

ii) If [pic] for all [pic], then [pic] then f is concave up [pic] on the open interval I.

Examples:

1. Graph the function [pic], showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing and open interval(s) where the function is concave up or concave down.

2. Graph the function [pic], showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing and open interval(s) where the function is concave up or concave down.

3. Graph the function [pic], showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing and open interval(s) where the function is concave up or concave down.

4. Graph the function [pic], showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing and open interval(s) where the function is concave up or concave down.

5. Graph the function [pic], showing all maximum, minimum values, open interval(s) where the function is increasing or decreasing and open interval(s) where the function is concave up or concave down.

6. Sketch the graph of the function [pic]with following properties: [pic]for all [pic] and [pic], [pic] for [pic], [pic] for all [pic], [pic] for all [pic]

Section 3.7 Optimization

Examples:

1. Find a positive number such that the sum of the number and its reciprocal is as small as possible.

2. A farmer with 750 feet fencing wants to enclose a rectangular area and divide it into four pens with fencing parallel to one side of the rectangle. What is the total largest area for the four pens?

3. A box with an open top is to be constructed from a square cardboard 3 feet wide by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

4. If 1200 square cm of material is available to make a box with square base and an open top, find the maximum volume of the box.

5. Find the point(s) (x, y) on the curve [pic]that is (are) farthest from (1, 0).

Solution: Let (x, y) be any point on the ellipse. Then distance of this point from (1, 0) is [pic]. Let us write [pic] for simplicity. Now maximize D, use [pic]. [pic]. Using derivative you will find critical number [pic], then [pic] , and [pic], so [pic], which is maximum since [pic].

6. A right circular cylinder is inscribed in a sphere of radius r. To find the largest possible surface area of such a cylinder.

Solution: Let us consider the sphere with radius r. If x is the radius

of the base of the cylinder and y is the vertical distance of center

from diameter of the base of the cylinder then

[pic]

The surface area of the cylinder [pic]

= [pic]

Now find derivative of S, which is [pic].

The critical numbers are [pic]

Observe that at [pic], S is maximum. Then y = 0.52573r and

max S=3.23607 r.

Section 3.8 Related Rates

Examples:

1. An oil tanker has an accident and oil pours at the rate of 150 gallons per minute. Suppose that the oil spreads onto the water in a circle at a thickness of [pic]. Given that [pic]gallons. Determine the rate at which the radius of the spill is increasing when the radius reaches 500 ft.

2. A car is traveling at 50 mph due south at a point 1/2 mile north of an intersection. A police car is traveling at 40 mph due west at a point 1/4 mile east of the same intersection. At that instant, the radar in the police car measurers the rate at which the distance between the cars is changing. What does the radar gun register?

3. A 10-foot ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at the rate of 2 ft/s, how fast is the bottom of the ladder sliding away from the wall when the top of the ladder is 8 ft off the ground.

Section 3.9 Rates of Change in Economics and Sciences

Examples:

1. Cost function [pic], where x is the number of units produced. The marginal cost function is defined as [pic] and average cost is defined as [pic]. Find the extreme marginal cost and extreme average cost.

2. For the demand function [pic], where p is the price, the elasticity is defined as [pic]. For the demand function [pic], where p is the price per unit. Find elasticity when p = 10, and find p for which E < -1.

Chapter 4 Integration

Section 4.1 Antiderivatives

A function F is called an antiderivative of f on an interval I if [pic] for all x in I.

An engineer who can measure the variable rate at which water is leaking from a tank wants to know the amount leaked over a certain time period. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. A physicist who knows the velocity of a particle might wish to know its position at a given time. In all these cases, the problem is to find a function F(x) whose derivative is a known function f(x). When such a function exists, we say that F(x) is the antiderivative of f(x).

If F(x) is the antiderivative of f(x) on an interval I, then the most general antiderivative of f(x) is the function F(x) + c, where c is an arbitrary constant.

In this section we use the following formulas and symbols:

The symbol [pic]is used for finding antiderivative. The antiderivative of the function [pic] with respect to x is symbolized as [pic]

Formulas:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. [pic] 9. [pic] 10. [pic]

Examples:

Find the most general antiderivative of the following functions.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

Exercises

1. [pic] 2. [pic]

3. [pic] 4. [pic]

Find the function f:

5. Given [pic]

6. Given [pic]

7. Given [pic]

8. Given [pic]

9. Given [pic]

10. Given [pic]

Section 4.2 Areas and Distances

The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:

[pic]

Examples

1. Let A be the area of the region that lies under the graph of [pic] between x = 0 to x = 2.

a) Use right endpoints; find an expression for A as a limit. Do not evaluate the limit

b) Use left endpoints; find an expression for A as a limit. Do not evaluate the limit

c) Use middle points, find an expression for A as a limit. Do not evaluate the limit

Solution: a) [pic], [pic], [pic]

2. Find an expression for the area under the graph of f as a limit. Do not evaluate the limit. Given [pic], [pic]. Use all three rules as in example1.

Section 4.3 Definite Integrals

We have the following results to use in this section:

[pic]

[pic]

Examples and Exercises

1. Evaluate the Riemann sum for [pic] taking the sample points to be right endpoints and [pic]

Solution: [pic], [pic]

2. Evaluate [pic]

Solution: [pic]

3. Use the property: If [pic] for [pic], then [pic] to estimate [pic]

Solution: [pic]

Where the absolute max [pic] and the absolute min [pic] as the function is decreasing on [0, 1].

4. Prove that [pic]

5. Express [pic] as limit of Riemann sum. Do not evaluate the limit.

Section 4.4 The Fundamental Theorem of Calculus

We have the following results to use in this section:

The fundamental theorem of calculus, Part 1: If f is continuous function on [a, b], then the function g defined by [pic] is continuous on [a, b] and differentiable on (a, b) and [pic]

Corollary: For [pic] then [pic]

The fundamental theorem of calculus, Part 2: If f is continuous function on [a, b], then [pic]. Where F is any antiderivative of f, that is a function such that [pic]

Examples and Exercises

Use part 1 of fundamental theorem of calculus to find the derivative of the function.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

Use part 2 of fundamental theorem of calculus to find the integral.

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

Find the derivative:

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. If [pic] where [pic] find [pic].

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