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1 Unit 5- Applications of the Derivative (Part II)

The Extreme Value Theorem In the previous unit, we investigated heavily how to locate relative extrema of the graph of a function by using the derivative. In pre-calculus, we talked about the difference between relative and absolute extrema. In the space below, distinguish between the two.

Definitions of Relative and Absolute Extrema of a Function

Pictured below are the graphs of f and g. Answer the questions about these two functions.

Graph of f(x)

Graph of g(x) Identify the coordinates of the relative extrema of f.

Identify the coordinates of the relative extrema of g.

On the domain of f, what are the coordinates of the absolute extrema of f?

On the domain of g, what are the coordinates of the absolute extrema of g?

On the domain of the given function, did the absolute extrema occur at the function’s relative extrema?


Graph of f(x)

Graph of g(x) On the interval −2 <x< 3, what are the absolute extrema of f?

On the interval −4 <x< 5, what are the absolute extrema of g?

On the interval −4 <x< 1, what are the absolute extrema of f?

On the interval −2 <x< 6, what are the absolute extrema of g?

When the domain is restricted to a particular closed interval, at what three places that the absolute extrema could exist?

The Extreme Value Theorem (E. V. T.):


Consider the cubic function 296)( 23 xxxxf to answer the following questions. a. Determine the intervals where f is increasing and decreasing. Justify your answers. b. Determine the coordinates of the relative extrema of f. Justify your answers. Pictured below is a graph of the function f on the closed interval −4 <x< 1.

Identify the absolute maximum of f on the closed interval −4 <x< −1.

Identify the absolute minimum of f on the closed interval −4 <x< −1.

Identify the absolute maximum of f on the closed interval −4 <x< 1.

Identify the absolute minimum of f on the closed interval −4 <x< 1.


Use the extreme value theorem to locate the absolute extrema of the function 296)( 23 xxxxf on the given closed intervals. Your algebraic results should concur with your graphical conclusions from the previous page.

Interval: −4 <x< −1

Interval: −4 <x< 1

For each of the following functions, state specifically why the E. V. T. is or is not applicable on the giveninterval.

Interval: −5 <x< 0

323)(

xxxH

32)( xxxG

)7ln()( xxf


Given the functions below, determine the absolute extreme values of the function on the given interval, provided the extreme value theorem is applicable. If it is not, state specifically why it is

not.1. 232)( 23 xxxxf on [–1, 3]

2. xxxg cossin)( 2 on 22

x

3. 32

2)( xxf on [-3, 6] 4. 4ln)( 2 xxh on [–1, 3]


Day #39 Homework

1. For which of the following functions is the Extreme Value Theorem NOT APPLICABLE on the interval [a, b]? Give a reason for your answer. Graph I Graph II Graph III

For exercises 2 – 4, determine the critical numbers for each of the functions below.

5. Given the function below, use a calculator to help determine the absolute extrema on the given interval. )1ln(sin)( xxxf on the interval [1, 6]

2. 3. )4ln()( 2 xxg 4. 3 3)( xxh


For exercises 6 – 9, determine the absolute extreme values on the given interval. You should do each of these independent from a calculator.

6. 23 3)( xxxf on the interval [–1, 3] 7. 3 2)( xxg on the interval [–3, 6]

8. 2

)(

x

xxh on the interval [–4, 0] 9. xxxf 23)( 32

on the interval [–1, 1]


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. In calculus, the derivative has many interpretations. One of the most important interpretations is that the derivative represents the Rate of Change of a Function. When speaking of rate of change, there are two rates of change that can be found that are associated with a function—average rate of change and instantaneous rate of change. Average Rate of Change of f(x) on an Interval Instantaneous Rate of Change of f(x) at a Point Find the average rate of change of f(x) on the Is the instantaneous rate of change of f at x = 4 interval [2, 12]. greater than the rate of change at x = 6? Justify.

x 0 2 4 6 8 10 12 14 16

f(x) 1 5 8 10 11 10 8 5 1


Rolle’s Theorem

Consider the function, f(x), presented on the previous page. Does Rolle’s Theorem apply on the following intervals? Explain why or why not? 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. 429)( xxxg on the interval [–3, 0] 2. 2

2sin)(

x

xxg on the interval [–4, –1]

Interval [2, 14]

Interval [2, 8]


Rolle’s Theorem guarantees that if a function is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there is guaranteed to exist a value of con (a, b) 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 functionx

xh5

3)( . 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 2)( 32

xxf on the interval [−1, 1]. Find the equation of the tangent line to the graph of 1sin2)( xxxf 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].


Day #40 Homework

For the exercises 1 – 5, determine whether Rolle’s Theorem can be applied to the function on the indicated interval. If Rolle’s Theorem can be applied, find all values of c that satisfy the theorem.

1. xxxf 4)( 2 on the interval 0 <x< 4 2. )3()4()( 2 xxxf on the interval –4 <x< 3

3. 24)( xxf on the interval –3 <x< 7

4. xxf sin)( on the interval 0 <x< 2π

5. xxf 2cos)( on the interval3

23

x


For exercises 6 – 9, determine whether the Mean Value Theorem can be applied to the function on the indicated interval. If the Mean Value Theorem can be applied, find all values of c that satisfy the theorem.

6. xxxxf 2)( 23 on –1 <x< 1

7. 3)( xxf on 3 <x< 7

8. x

xxf

2)(

on 2

21 x

9. xxxh 2coscos2)( on 0 <x< π


Using the graph of the function, f(x), pictured below, and given the intervals in the table below, determine if Rolle’s or Mean Value Theorem, whichever is indicated, can be applied or not. Give reasons for your answers.

10.

[–5, –1] Rolle’s

Theorem

11.

[–2, 8] Rolle’s

Theorem

12.

[–1, 8] Mean Value

Theorem


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 (hours)

0 3 6 9 12 15 18 21 24

R(t) (gallons per

hour)

9.6

10.4

10.8

11.2

11.4

11.3

10.7

10.2

9.6

a. Estimate the value of )5('R , indicating correct units of measure. Explain what this valuemeans about R(t). b. Using correct units of measure, find the average rate of change of R(t) from t = 3 to t = 18. c. Is there some time t, 0 <t< 24, such that )(' tR = 0? Justify your answer.


The total order and transportation cost C(x), measured in dollars, of bottles of Pepsi Cola is approximated by the function

3

1000,10)(

x

x

xxC ,

where x is the order size in number of bottles of Pepsi Cola in hundreds. Answer the following questions. a. Is there guaranteed a value of r on the interval 0 <r< 3 such that the average rate of change of cost is equal to )(' rC ? Give a reason for your answer. b. Is there a value of r on the interval 3 <r< 6 such that 0)(' rC . Give a reason for your answer and if such a value of r exists, then find that value of r. c. For 3 <x< 9, what is the greatest cost for order and transportation?


A car company introduces a new car for which the number of cars sold, S, is modeled by the function

2

95300)(

ttS ,

where t is the time in months. a. Find the value of )5.2('S . Using correct units, explain what this value represents in the context of this problem. b. Find the average rate of change of cars sold over the first 12 months. Indicate correct units of measure and explain what this value represents in the context of this problem. c. Is it possible that a value of c for 0 <c< 12 exists such that )(' cS is equal to the average rate of change? Give a reason for your answer and if such a value of c exists, find the value.


x f(x) )(' xf g(x) )(' xg

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

6))(()( xgfxh . 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 5)(' ch .


Day #41 Homework For the functions in exercises 1 and 2, determine if the Mean Value Theorem holds true for 0 <c< 5? Give a reason for your answer. If it does hold true, find the guaranteed value(s) of c. [CALC] 3. Administrators at a hospital believe that the number of beds in use is given by the function

50sin20)(10

ttB ,

where t is measured in days. [CALC] a. Find the value of )7('B . Using correct units of measure, explain what this value means in the context of the problem. b. For 12 <t< 20, what is the maximum number of beds in use?

1. 32)(

21 xxf

2. xxxg 2sin2)(


4. For t> 0, the temperature of a cup of coffee in degrees Fahrenheit t minutes after it is poured is

modeled by the function ttF )91.0(9368)( . Find the value of )4('F . Using correct units of measure, explain what this value means in the context of the problem. [CALC] For questions 5 – 8, use the table given below which represents values of a differentiable function g on the interval 0 <x< 6. Be sure to completely justify your reasoning when asked, citing appropriate theorems, when necessary. 5. Estimate the value of )5.2('g . 6. If one exists, on what interval is there guaranteed to be a value of c such that g(c) = –1? Justify your reasoning. 7. If one exists, on what interval is there guaranteed to be a value of c such that 0)(' cg ? Justify your reasoning. 8. If one exists, on what interval is there guaranteed to be a value of c such that 4)(' cg ? Justify your reasoning.

x 0 2 3 4 6

g(x) –3 1 5 2 1


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 tetp t sin)( , 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 tetp t sin)( , 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)

Is = 0 or is undefined

Is > 0

Is < 0

Changes from positive to negative

Changes from negative to positive

Acceleration Velocity

Is = 0 or is undefined

Is > 0

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 2)4)(2()( tttv 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 v(3) and )3('v . 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.

2.

3.

4.

5.

1.


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 32

24)(

t

ttv .

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.



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 ttv

3sin)( .

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.


Day #43 Homework

A particle moves along the x axis such that its position, for t> 0, is given bythe function p(t) = e2t – 5t. Use this information to complete exercises 1 – 4. 1. What are the values of )2('p and )2(''p ? Explain what each value represents. 2. Based on the values found in part (a), what can be concluded about the speed of the particle at t = 2? Give a reason for your answer. 3. On what interval(s) of t is the particle moving to the left? To the right? Justify your answers. 4. Does the particle ever change directions? Justify your answer. 5. The graph of v(t), the velocity of a moving particle, is given below. What conclusions can be made about the movement of the particle along the x – axis and the acceleration, a(t), of the particle for t> 0? Give reasons for your answers. 6. If the position of a particle is defined by the function x(t) =

ttt 249 23 for t> 0, is the speed of the particle increasing or decreasing when t = 2.5? Justify your answer.


The position of a particle is given by the function tettp 2)32()( for t> 0. Answer questions 7 – 9. 7. What is the average velocity from t = 1 to t = 3? 8. Find an equation for v(t), the velocity of the particle. 9. For what value(s) of t will the v(t) = 0?



A particle moves along the x – axis so that its velocity at time t is given by

2

2sin)1()( tttv .

10. Find the acceleration of the particle at t = 2. Is the speed of the particle increasing at t = 2? Explain why or why not. 11. 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 ttttp 862)( 23 where p(t) is measured 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 tetp t 8)( 2 where p(t) is measured 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 tttp sin2)( where p(t) is measured in feet. Find the net

and total distance the particle travels from t = 6 minutes to t =

45 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

134)( 23 ttttp , 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 )3('p and )3(''p . 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 = 1 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

a. Find the average acceleration on the interval 5 <t< 20. Express your answer using correct units of measure. b. Based on the values in the table, on what interval(s) is the accelerationof the plane guaranteed to equal zero on the open interval 0 <t< 40? Justify your answer. c. Does the data represent velocity values of the plane moving away from its point of origin or returning to its point of origin? Give a reason for your answer. d. The function f, defined by

407

10sin3cos6)( tttf , is used to model the velocity of the plane, in

miles per minute, for 0 <t< 40. According to this model, what is the acceleration of the plane at t = 23? What does this value indicate about the velocity at t = 23? Justify your answer, indicating units of measure.

t (min)

0

5

10

15

20

25

30

35

40

v(t) (miles

per min)

7.0

9.2

9.5

7.0

4.5

2.4

2.4

4.3

7.3


Day #44 Homework The function whose graph is pictured below, represents the velocity, v(t), of a particle for t = 0 to t = 9 seconds moving along the x – axis. Use the graph to complete exercises 1 – 4. 1. On what interval(s) is the particle moving to the right? Left? Justify your answer. 2. On what interval(s) is the particle slowing down? Speeding up? Justify your answer. 3. At what value(s) of t is the particle momentarily stopped and changing directions? Justify your answer. 4. On what interval of the time is the acceleration 0? Justify your answer.


The graph below represents the position, p(t), of a particle that is moving along the x – axis on the interval 0 <t< 6. Use the graph to complete exercises 5 – 9. p(t) is measured in centimeters and t is measured in seconds. 5. For what interval(s) of time is the particle moving to the right? Justify your answer. 6. For what interval(s) of time is the particle moving to the left? Justify your answer. 7. Express the velocity, v(t), as a piecewise-defined function on the interval 0 <t< 6. 8. At what value(s) of t is the velocity undefined on the interval 1 <t< 6? Graphically justify your reasoning. 9. Find the average velocity of the particle on the interval 1 <t< 6.


A particle moves along the x – axis so that at any time 0 <t< 5, the velocity, in meters per second, is given by

the function tttv 2cos)2()( 2 . Use a graphing calculator to complete exercises 10 – 12. 10. On the interval 0 <t< 5, at how many times does the particle change directions? Give a reason for your answer. 11. Using appropriate units, what is the value of )2('v . Describe the motion of the particle at this time. Justify your answer. 12. Using appropriate units, what is the average acceleration between t = 1 and t = 3.5 seconds?


Jeff leaves his house riding his bicycle toward school. His velocity v(t), measured in feet per minute, on the interval 0 <t< 15, for t minutes, is shown in the graph to the right. Use the graph to complete exercises 13 – 16. 13. Find the value of )4('v . Explain, using appropriate units, what this value represents. 14. On the interval 0 <t< 5, is there any interval of time at which a(t) = 0? Explain how you know. 15. On the interval 0 <t< 5, does Rolle’s Theorem guaranteed that there will be a value of t such that a(t) = 0? Justify your answer. 16. At some point, Jeff realizes that he forgot something at home and has to turn around. After how many minutes does he turn around? Give a reason for your answer.



AP Free Response and Multiple Choice Practice

NO CALCULATOR

A particle moves along the x – axis with velocity at time t> 0 given by tetv 11)( . a. Find the acceleration of the particle at t = 3. b. Is the speed of the particle increasing at t = 3? Give a reason for your answer. c. Find all values of t at which the particle changes direction. Justify your answer.

d. The function tetp t 1)( models the position of the particle for t> 0. Find the total distance that particle traveled on the time interval 0 <t< 3.


NO CALCULATOR A car is traveling on a straight road. For 0 <t< 24 seconds, the car’s velocity, v(t), in meters per second, is modeled by the piecewise-linear function defined by the graph below. a. For what interval(s) of time does the car have zero acceleration?Show the work and explain the analysis that leads to your answer. b. For each value of )4('v and )20('v , find the value or explainwhy it does not exist. Indicate units of measure. c. Let a(t) be the car’s acceleration at time t in meters per secondper second. For 0 <t< 24, write a piecewise-defined function for a(t). d. Find the average rate of change of v over the interval 8 <t< 20. Does the Mean Value Theorem guarantee a value of c, for 8 <c< 20, such that )(' cv is equal to this average rate of change? Why or why not?


You must show your work to earn credit for the following. You will need to use a calculator for these. If

2sin)( xxf , then there exists a number c on the interval

23

2 x that satisfies the conclusion of the

Mean Value Theorem. Which of the following values could be c? (A)

32 (B)

43 (C)

65 (D) π (E)

23

A particle moves along a line so that at time t, where 0 <t<π, its position is given by 10cos4)(2

2 ttts .

What is the velocity of the particle when its acceleration is zero? (A) –5.19 (B) 0.74 (C) 1.32 (D) 2.55 (E) 8.13


The graph of the function xxxxy cos276 23 changes concavity at x = (A) –1.58 (B) –1.63 (C) –1.67 (D) –1.89 (E) –2.33 If y = 2x – 8, what is the minimum value of the product of xy? (A) –16 (B) –8 (C) –4 (D) 0 (E) 2


Solving Optimization Problems

General Approach to Solving Optimization Problems

Example 1 A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? What is the maximum volume?

1.

2.

3.

4.

5.


Example 2 A box is to be built from a rectangular piece of cardboard that is 25 cm wide and 40 cm long by cutting out a square from each corner and then bending up the sides. Find the size of the corner square with will produce a container that will hold the most amount of soup. Example 3 A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1 ½ inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? Example 4 A rectangle is bounded by the x and y axes and the graph of y = 3 – ½x. What length and width should the rectangle have so that its area is a maximum?


Example 5 A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. Example 6 The profit P (in thousands of dollars) for a company spending an amount of s (in thousands of dollars) on

advertising is 4006 23101 ssP . Find the amount of money the company should spend on advertising in

order to yield a maximum profit.


Example 7 Determine the point on the line y = 2x + 3 so that the distance between the line and the point (1, 2) is a minimum.

Example 8 A rectangle ABCD with sides parallel to the coordinate axes is inscribed in the region enclosed by the graph of y = –4x2 + 4 as shown in the figure below. Find the x and y coordinates of the point C so that the area of the rectangle is a maximum.


Day #45 Homework

1. Find the point on the graph of 8)( xxf so that the point (2, 0) is closest to the graph. 2. A rancher has 200 total feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should each corral be so that the enclosed area will be a maximum? 3. The area of a rectangle is 64 square feet. What dimensions of the rectangle would give the smallest perimeter?


4. A rectangle is bound by the x – axis and the graph of a semicircle defined by 225 xy . What length and width should the rectangle have so that its area is a maximum?

5. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure below). Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.


6. Find the maximum volume of a box that can be made by cutting squares from the corners of an 8 inch by 15 inch rectangular sheet of cardboard and folding up the sides. 7. The volume of a cylindrical tin can with a top and bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what much the height, in inches, of the can be?

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