Derivative Apps Homework Packet

CALCULUS I Assignment Problems

Applications of Derivatives

Paul Dawkins

Calculus I

© 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

Table of Contents Preface ............................................................................................................................................. 2 Applications of Derivatives ........................................................................................................... 2

Introduction .............................................................................................................................................. 2 Rates of Change ......................................................................................................................................... 3 Critical Points ............................................................................................................................................ 3 Minimum and Maximum Values .............................................................................................................. 7 Finding Absolute Extrema...................................................................................................................... 10 The Shape of a Graph, Part I ................................................................................................................... 12 The Shape of a Graph, Part II ................................................................................................................. 19 The Mean Value Theorem ...................................................................................................................... 24 Optimization ........................................................................................................................................... 25 More Optimization Problems................................................................................................................. 27 Indeterminate Forms and L’Hospital’s Rule ......................................................................................... 29 Linear Approximations .......................................................................................................................... 31 Differentials............................................................................................................................................. 32 Newton’s Method .................................................................................................................................... 33 Business Applications ............................................................................................................................ 34

http://tutorial.math.lamar.edu/terms.aspx

Calculus I

© 2007 Paul Dawkins 2 http://tutorial.math.lamar.edu/terms.aspx

Preface Here are a set of problems for my Calculus I notes. These problems do not have any solutions available on this site. These are intended mostly for instructors who might want a set of problems to assign for turning in. I try to put up both practice problems (with solutions available) and these problems at the same time so that both will be available to anyone who wishes to use them.

Applications of Derivatives

Introduction Here are a set of problems for which no solutions are available. The main intent of these problems is to have a set of problems available for any instructors who are looking for some extra problems. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of topics in this chapter that have problems written for them. Rates of Change Critical Points Minimum and Maximum Values Finding Absolute Extrema The Shape of a Graph, Part I The Shape of a Graph, Part II The Mean Value Theorem Optimization Problems More Optimization Problems L’Hospital’s Rule and Indeterminate Forms Linear Approximations Differentials Newton’s Method Business Applications


Calculus I


Rates of Change As noted in the text for this section the purpose of this section is only to remind you of certain types of applications that were discussed in the previous chapter. As such there aren’t any problems written for this section. Instead here is a list of links (note that these will only be active links in the web version and not the pdf version) to problems from the relevant sections from the previous chapter. Each of the following sections has a selection of increasing/decreasing problems towards the bottom of the problem set.

Differentiation Formulas Product & Quotient Rules Derivatives of Trig Functions Derivatives of Exponential and Logarithm Functions Chain Rule

Related Rates problems are in the Related Rates section.

Critical Points Determine the critical points of each of the following functions. Note that a couple of the problems involve equations that may not be easily solved by hand and as such may require some computational aids. These are marked are noted below. 1. ( ) 3 28 18 240 2R x x x x= − − +

2. ( ) 4 3 22 16 20 7f z z z z= − + −

3. ( ) 5 6 790

78 12 25g z z z z= − − +

4. ( ) 4 3 23 20 132 672 4g t t t t t= − − + − Note : Depending upon your factoring skills this may require some computational aids. 5. ( ) 2 3 4 515

210 15h x x x x x= − + − Note : Depending upon your factoring skills this may require some computational aids. 6. ( ) 3 24 7 1P w w w w= − − −

7. ( ) 3 27 3 15A t t t t= − + −


Calculus I


8. ( ) 2 3 44 2 6 3a t t t t= − − −

9. ( ) 4 3 23 20 6 120 5f x x x x x= − + + + Note : This problem will require some computational aids. 10. ( ) 5 4 310 15h v v v v= + + −

11. ( ) ( ) ( )5 43 2 1g z z z= − +

12. ( ) ( ) ( )24 22 8R q q q= + −

13. ( ) ( ) ( )23 22 1f t t t= − +

14. ( )2 2 13 5

w wf ww+ +

=−

15. ( ) 2

3 41th t

t−

=+

16. ( )2

2 3 8y yR y

y y−

=+ +

17. ( ) 3 7Y x x= −

18. ( ) ( )2

3 325f t t t= −

19. ( ) ( )25 2 8h x x x= +

20. ( ) ( ) 32 26 4Q w w w= − −

21. ( ) ( )47sin 2tQ t = −

22. ( ) ( )3cos 2 5g x x x= −


Calculus I


23. ( ) ( )7cos 2f x x x= +

24. ( ) ( )6sin 2 12h t t t= +

25. ( ) ( )3

5cos zw z =

26. ( ) ( )tan 4U z z z= −

27. ( ) ( ) ( )cos sinh x x x x= −

28. ( ) ( ) ( )2cos cos 2h x x x= −

29. ( ) ( ) ( )2 4cos cosf w w w= −

30. ( ) 14 3wF w += e

31. ( ) 2 1 zg z z −= e

32. ( ) ( ) 2

3 2 xA x x= − e

33. ( ) ( ) 286 1 t tP t t −= + e

34. ( ) 2 23 2 4x xf x + −= −e e

35. ( ) 2 24 8 2z z z zf z − −= +e e

36. ( ) 3 26 8y yh y −= e

37. ( ) 3 22 4t t tg t + −= e

38. ( ) ( )2ln 3Z t t t= + +

39. ( ) ( )2ln 1G r r r= − +


Calculus I


40. ( ) ( )2 6 ln 8 1A z z z= − + +

41. ( ) ( )24ln 2f x x x x= − + +

42. ( ) ( ) ( )ln 4 2 ln 4g x x x= + − +

43. ( ) ( ) ( )2ln 1 ln 4h t t t t= − + + −

44. The graph of some function, ( )f x , is shown. Based on the graph, estimate the location of

all the critical points of the function.






Calculus I


Minimum and Maximum Values 1. Below is the graph of some function, ( )f x . Identify all of the relative extrema and absolute

extrema of the function.

2. Below is the graph of some function, ( )f x . Identify all of the relative extrema and absolute



Calculus I







Calculus I


4. Sketch the graph of ( ) 123f x x= − and identify all the relative extrema and absolute extrema

of the function on each of the following intervals. (a) ( ),−∞ ∞

(b) [ ]3,2−

(c) [ )4,1−

(d) ( )0,5

5. Sketch the graph of ( ) ( )22 1g x x= − + and identify all the relative extrema and absolute

extrema of the function on each of the following intervals. (a) ( ),−∞ ∞

(b) [ ]0,3

(c) [ ]1,5−

(d) [ ]1,1−

(e) [ )1,3

(f) ( )2, 4

6. Sketch the graph of ( ) 3 xh x −= e and identify all the relative extrema and absolute extrema of

the function on each of the following intervals. (a) ( ),−∞ ∞

(b) [ ]1,3−

(c) [ ]6, 1− −

(d) ( ]1,4

7. Sketch the graph of ( ) ( )cos 2h x x= + and identify all the relative extrema and absolute

extrema of the function on each of the following intervals. Do, all work for this problem in radians.

(a) ( ),−∞ ∞

(b) ,3 4π π −

(c) , 22π

π −

(d) 1 ,12


Calculus I


8. Sketch the graph of a function on the interval [ ]3,9 that has an absolute maximum at 5x =

and an absolute minimum at 4x = . 9. Sketch the graph of a function on the interval [ ]0,10 that has an absolute minimum at 5x =

and an absolute maximums at 0x = and 10x = . 10. Sketch the graph of a function on the interval ( ),−∞ ∞ that has a relative minimum at

7x = − , a relative maximum at 2x = and no absolute extrema. 11. Sketch the graph of a function that meets the following conditions :

(a) Has at least one absolute maximum. (b) Has one relative minimum. (c) Has no absolute minimum.

12. Sketch the graph of a function that meets the following conditions :

(a) Graphed on the interval [ ]2,9 .

(b) Has a discontinuity at some point interior to the interval. (c) Has an absolute maximum at the discontinuity in part (b). (d) Has an absolute minimum at the discontinuity in part (b).


(a) Graphed on the interval [ ]4,10− .

(b) Has no relative extrema. (c) Has an absolute maximum at one end point. (d) Has an absolute minimum at the other end point.


(a) Has a discontinuity at some point. (b) Has an absolute maximum and an absolute minimum. (c) Neither absolute extrema occurs at the discontinuity.

Finding Absolute Extrema For each of the following problems determine the absolute extrema of the given function on the specified interval. 1. ( ) 4 3 22 16 20 7f z z z z= − + − on [ ]2,6−


Calculus I


2. ( ) 4 3 22 16 20 7f z z z z= − + − on [ ]2,4−

3. ( ) 4 3 22 16 20 7f z z z z= − + − on [ ]0,2

4. ( ) 3 4 520 280 75 12Q w w w w= + + − on [ ]3,2−

5. ( ) 3 4 520 280 75 12Q w w w w= + + − on [ ]1,8−

6. ( ) 5 6 790

78 12 25g z z z z= − − + on [ ]1,1−

7. ( ) 4 3 23 20 132 672 4g t t t t t= − − + − on [ ]5,8− Note : Depending upon your factoring skills this may require some computational aids. 8. ( ) 4 3 23 20 132 672 4g t t t t t= − − + − on [ ]2,8− Note : Depending upon your factoring skills this may require some computational aids. 9. ( ) 3 214 11 4 3V x x x x= + − + on [ ]1,1−

10. ( ) 2 3 44 2 6 3a t t t t= − − − on [ ]2,1−

11. ( ) 2 38 3 7h x x x x= + + − on [ ]1,5−

12. ( ) 4 3 23 20 6 120 5f x x x x x= − + + + on [ ]1,5− Note : This problem will require some computational aids. 13. ( ) 5 4 310 15h v v v v= + + − on [ ]3,2−

14. ( ) ( ) ( )5 43 2 1g z z z= − + on [ ]1,3−

15. ( ) ( ) ( )24 22 8R q q q= + − on [ ]4,1−

16. ( ) 2

3 41th t

t−

=+

on [ ]2,4−

17. ( )2

2

6 91

x xg xx x

+ +=

+ + on [ ]6,0−


Calculus I


18. ( ) ( )2

3 325f t t t= − on [ ]2,6

19. ( ) ( )2

252 1F x t t t= + + + on [ ]2,1−

20. ( ) ( ) 32 26 4Q w w w= − − on [ ]125,−

21. ( ) ( )3cos 2 5g x x x= − on [ ]0,6

22. ( ) ( )33 10sin ws w w= − on [ ]10,38

23. ( ) ( )7cos 2f x x x= + on [ ]5,4−

24. ( ) ( ) ( )cos sinh x x x x= − on [ ]15, 5− −

25. ( ) 2 1 zg z z −= e on 51

2 2,−

26. ( ) ( ) 286 1 t tP t t −= + e on [ ]1,3−

27. ( ) 5 9 1 3 6x xf x + −= + +e e on [ ]1,0−

28. ( ) 3 26 8y yh y −= e on [ ]12 ,1−

29. ( ) ( )2ln 3Z t t t= + + on [ ]2,2−

30. ( ) ( )24ln 2f x x x x= − + + on [ ]1,9−

31. ( ) ( ) ( )2ln 1 ln 4h t t t t= − + + − on [ ]1,3

The Shape of a Graph, Part I For problems 1 – 4 the graph of a function is given. Determine the open intervals on which the function increases and decreases.


Calculus I


1.

2.

3.

4.


Calculus I


For problems 5 – 7 the graph of the derivative of a function is given. From this graph determine the open intervals in which the function increases and decreases. 5.

6.


Calculus I


7.

For problems 8 – 10 The known information about the derivative of a function is given. From this information answer each of the following questions.

(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither.

8.

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1 0 3 0 8 0

0 on ,1 , 3,8 0 on 1,3 , 8,

f f f

f x f x

′ ′ ′= = =

′ ′< −∞ > ∞

9.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )2 0 0 0 3 0 6 0

0 on 0,3 , 6, 0 on , 2 , 2,0 , 3,6

g g g g

g x g x

′ ′ ′ ′− = = = =

′ ′< ∞ > −∞ − −

10.

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )1 0 2 0 5 0

0 on , 1 , 1,2 0 on 2,5 , 5,

h h h

h x h x

′ ′ ′− = = =

′ ′< −∞ − − > ∞

For problems 11 – 28 answer each of the following.

(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither.

11. ( ) 3 215 63 3f t t t t= − + +

12. ( ) 2 3 420 8 4g x x x x= + + −


Calculus I


13. ( ) 3 28 18 24 10Q w w w w= − − −

14. ( ) 5 4 35

4 20 7f x x x x= + − −

15. ( ) 2 35 4 9 3P x x x x= − − −

16. ( ) 5 4 36 5R z z z z= + − +

17. ( ) 2 3 41 12 9 2h z z z z= − − −

18. ( ) ( )7 sin 4Q t t t= − + on 3 3

2 2,−

19. ( ) ( )26 20cos zf z z= − on [ ]0,22

20. ( ) ( )324cos 8 2xg x x= + + on [ ]30,25−

21. ( ) ( )9 5sin 2h w w w= − on [ ]5,0−

22. ( ) ( )5 7h x x x= +

23. ( ) ( )( )2

2 310 2W z w w= − +

24. ( ) ( ) 32 28 4f t t t= − −

25. ( )3 21

3 3x x xf x − −= e

26. ( ) ( )2 38 zh z z −= − e

27. ( ) ( )2ln 5 8A t t t= + +

28. ( ) ( )23 ln 1g x x x x= − + + +

29. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly one relative extrema?


Calculus I


(b) What is the minimum degree of a polynomial that has exactly two relative extrema? (c) What is the minimum degree of a polynomial that has exactly three relative extrema? (d) What is the minimum degree of a polynomial that has exactly n relative extrema?

30. For some function, ( )f x , it is known that there is a relative minimum at 4x = − . Answer

each of the following questions about this function. (a) What is the simplest form that the derivative of this function? Note : There really are many

possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.

(b) Using your answer from (a) determine the most general form that the function itself can take.

(c) Given that ( )4 6f − = find a function that will have a relative minimum at 4x = − . Note

: There are many possible answers here so just give one of them. 31. For some function, ( )f x , it is known that there is a relative maximum at 1x = − . Answer

each of the following questions about this function. (a) What is the simplest form that the derivative of this function? Note : There really are many



(c) Given that ( )1 3f − = find a function that will have a relative maximum at 1x = − . Note

: There are many possible answers here so just give one of them. 32. For some function, ( )f x , it is known that there is a critical point at 3x = that is neither a

relative minimum or a relative maximum. Answer each of the following questions about this function.

(a) What is the simplest form that the derivative of this function? Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative.


(c) Given that ( )3 2f = find a function that will have a critical point at 3x = that is neither

a relative minimum or a relative maximum. Note : There are many possible answers here so just give one of them.

33. For some function, ( )f x , it is known that there is a relative maximum at 1x = and a

relative minimum at 4x = . Answer each of the following questions about this function. (a) What is the simplest form that the derivative of this function? Note : There really are many



Calculus I



(c) Given that ( )1 6f = and ( )4 2f = − find a function that will have a relative maximum

at 1x = and a relative minimum at 4x = . Note : There are many possible answers here so just give one of them.

34. Given that ( )f x and ( )g x are increasing functions will ( ) ( ) ( )h x f x g x= − always be

an increasing function? If so, prove that ( )h x will be an increasing function. If not, find

increasing functions, ( )f x and ( )g x , so that ( )h x will be a decreasing function and find a

different set of increasing functions so that ( )h x will be an increasing function.

35. Given that ( )f x is an increasing function. There are several possible conditions that we can

impose on ( )g x so that ( ) ( ) ( )h x f x g x= − will be an increasing function. Determine as

many of these possible conditions as you can. 36. For a function ( )f x determine a set of conditions on ( )f x , different from those given in

#15 in the practice problems, for which ( ) ( ) 2h x f x= will be an increasing function.

37. For a function ( )f x determine a single condition on ( )f x for which ( ) ( ) 3h x f x= will

be an increasing function. 38. Given that ( )f x and ( )g x are positive functions. Determine a set of conditions on them

for which ( ) ( ) ( )h x f x g x= will be an increasing function. Note that there are several

possible sets of conditions here, but try to determine the “simplest” set of conditions.

39. Repeat #38 for ( ) ( )( )

f xh x

g x= .

40. Given that ( )f x and ( )g x are increasing functions prove that ( ) ( )( )h x f g x= will also

be an increasing function.


Calculus I


The Shape of a Graph, Part II For problems 1 & 2 the graph of a function is given. Determine the open intervals on which the function is concave up and concave down. 1.

2.

For problems 3 – 5 the graph of the 2nd derivative of a function is given. From this graph determine the open intervals in which the function is concave up and concave down. 3.


Calculus I


4.

5.


Calculus I



(a) Determine the open intervals on which the function is concave up and concave down. (b) Determine the inflection points of the function.

6. ( ) 3 29 24 6f x x x x= + + −

7. ( ) 4 3 22 120 84 35Q t t t t t= − − − +

8. ( ) 5 4 33 20 40h z z z z= − +

9. ( ) 4 3 25 2 18 108 12g w w w w w= − − + −

10. ( ) 4 5 610 360 20 3g x x x x x= + + + −

11. ( ) ( )2

49 3 160sin xA x x x= − − on [ ]20,10−

12. ( ) ( ) 23cos 2 14f x x x= − − on [ ]0,6

13. ( ) ( )21 2 sin 2h t t t= + − on [ ]2,4−

14. ( ) ( )138R v v v= −

15. ( ) ( )( )251 3g x x x= − +

16. ( ) 4x xf x −= −e e

17. ( ) 2 wh w w −= e

18. ( ) ( )2 2ln 1A w w w= − +


(a) Identify the critical points of the function. (b) Determine the open intervals on which the function increases and decreases. (c) Classify the critical points as relative maximums, relative minimums or neither. (d) Determine the open intervals on which the function is concave up and concave down. (e) Determine the inflection points of the function.


Calculus I


(f) Use the information from steps (a) – (e) to sketch the graph of the function.

19. ( ) 2 310 30 2f x x x= − +

20. ( ) 3 314 4G t t t= + −

21. ( ) 4 3 24 18 9h w w w w= + − −

22. ( ) 3 4 510 10 3g z z z z= + +

23. ( ) 6 5 49 20 10f z z z z= − + +

24. ( ) ( )3 5sin 2Q t t t= − on [ ]1, 4−

25. ( ) ( )1 1

2 3cosg x x x= + on [ ]25,0−

26. ( ) ( )134h x x x= −

27. ( ) 2 1f t t t= +

28. ( ) ( )45 27A z z z= −

29. ( ) 4 6w wg w = −e e

30. ( )21

413 tP t t −= e

31. ( ) ( )31 xg x x −= + e

32. ( ) ( )2ln 1h z z z= + +

33. ( ) ( )22 8ln 4f w w w= − +

34. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly two inflection points. (b) What is the minimum degree of a polynomial that has exactly three inflection points.


Calculus I


(c) What is the minimum degree of a polynomial that has exactly n inflection points. 35. For some function, ( )f x , it is known that there is an inflection point at 3x = . Answer each

of the following questions about this function. (a) What is the simplest form that the 2nd derivative of this function? . (b) Using your answer from (a) determine the most general form that the function itself can

take. (c) Given that ( )0 6f = − and ( )3 1f = find a function that will have an inflection point at

3x = . For problems 36 – 39 ( )f x is a polynomial. Given the 2nd derivative of the function classify, if

possible, each of the given critical points as relative minimums or relative maximum. If it is not possible to classify the critical point(s) clearly explain why they cannot be classified. 36. ( ) 23 4 15f x x x′′ = − − . The critical points are : 3x = − , 0x = and 5x = .

37. ( ) 3 24 21 24 68f x x x x′′ = − − + . The critical points are : 2x = − , 4x = and 7x = .

38. ( ) 2 323 18 9 4f x x x x′′ = + − − . The critical points are : 4x = − , 1x = − and 3x = .

39. ( ) 2 3 4216 410 249 60 5f x x x x x′′ = − + − + . The critical points are : 1x = , 4x = and

5x = .

40. Use ( ) ( ) ( )3 41 1f x x x= + − for this problem.

(a) Determine the critical points for the function. (b) Use the 2nd derivative test to classify the critical points as relative minimums or relative

maximums. If it is not possible to classify the critical point(s) clearly explain why they cannot be classified.

(c) Use the 1st derivative test to classify the critical points as relative minimums, relative maximums or neither.

41. Given that ( )f x and ( )g x are concave down functions. If we define

( ) ( ) ( )h x f x g x= + show that ( )h x is a concave down function.

42. Given that ( )f x is a concave up function. Determine a condition on ( )g x for which

( ) ( ) ( )h x f x g x= + will be a concave up function.


Calculus I


43. For a function ( )f x determine conditions on ( )f x for which ( ) ( ) 2h x f x= will be a

concave up function. Note that there are several sets of conditions that can be used here. How many of them can you find?

The Mean Value Theorem For problems 1 – 4 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval. 1. ( ) 3 24 3f x x x= − + on [ ]0,4

2. ( ) 215 2Q z z z= + − on [ ]2, 4−

3. ( ) 2 91 th t −= − e on [ ]3,3−

4. ( ) [ ]1 cosg w wπ= + on [ ]5,9

For problems 5 – 8 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval. 5. ( ) 3 2 8f x x x x= − + + on [ ]3,4−

6. ( ) 3 22 7 1g t t t t= + + − on [ ]1,6

7. ( ) 2 6 3tP t t= − −e

8. ( ) ( )29 8sin xh x x= − on [ ]3, 1− −

9. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]2,5− , that

( )5 14f = and that ( ) 10f x′ ≤ . What is the smallest possible value for ( )2f − ?

10. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]6, 1− − , that

( )6 23f − = − and that ( ) 4f x′ ≥ − . What is the smallest possible value for ( )1f − ?


Calculus I


11. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]3,4− , that

( )3 7f − = and that ( ) 17f x′ ≤ − . What is the largest possible value for ( )4f ?

12. Suppose we know that ( )f x is continuous and differentiable on the interval [ ]1,9 , that

( )9 0f = and that ( ) 8f x′ ≥ . What is the largest possible value for ( )1f ?

13. Show that ( ) 7 5 32 3 14 1f x x x x x= + + + + has exactly one real root.

14. Show that ( ) 3 26 2 4 3f x x x x= − + − has exactly one real root.

15. Show that ( ) 420 xf x x −= − e has exactly one real root.

Optimization 1. Find two positive numbers whose sum of six times one of them and the second is 250 and whose product is a maximum. 2. Find two positive numbers whose sum of twice the first and seven times the second is 600 and whose product is a maximum. 3. Let x and y be two positive numbers such the sum is 175 and ( )( )3 4x y+ + is a maximum.

4. Find two positive numbers such that the sum of one and the square of the other is 200 and whose product is a maximum. 5. Find two positive numbers whose product is 400 and such that the sum of twice the first and three times the second is a minimum. 6. Find two positive numbers whose product is 250 and such that the sum of the first and four times the second is a minimum. 7. Let x and y be two positive numbers such that ( )2 100y x + = and whose sum is a minimum.

8. Find a positive number such that the sum of the number and its reciprocal is a minimum. 9. We are going to fence in a rectangular field and have 200 feet of material to construct the fence. Determine the dimensions of the field that will enclose the maximum area.


Calculus I


10. We are going to fence in a rectangular field. The cost of material of each side is $6/ft, $9/ft, $12/ft and $14/ft respectively. If we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area. 11. We are going to fence in a rectangular field that encloses 75 ft2. Determine the dimensions of the field that will require the least amount of fencing material to be used. 12. We are going to fence in a rectangular field that encloses 200 m2. If the cost of the material for of one pair of parallel sides is $3/ft and cost of the material for the other pair of parallel sides is $8/ft determine the dimensions of the field that will minimize the cost to build the fence around the field. 13. Show that a rectangle with a fixed area and minimum perimeter is a square. 14. Show that a rectangle with a fixed perimeter and a maximum area is a square. 15. We have 350 m2 of material to build a box whose base width is four times the base length. Determine the dimensions of the box that will maximize the enclosed volume. 16. We have $1000 to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides cost $10/cm2 and the material for the bottom cost $15/cm2 determine the dimensions of the box that will maximize the enclosed volume. 17. We want to build a box whose base length is twice the base width and the box will enclose 80 ft3. The cost of the material of the sides is $0.5/ft2 and the cost of the top/bottom is $3/ft2. Determine the dimensions of the box that will minimize the cost. 18. We want to build a box whose base is a square, has no top and will enclose 100 m3. Determine the dimensions of the box that will minimize the amount of material needed to construct the box. 19. We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 in3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. 20. We want to construct a cylindrical can whose volume is 105 mm3. The material for the wall of the can costs $3/mm2, the material for the bottom of the can costs $7/mm2 and the material for the top of the can costs $2/mm2. Determine the dimensions of the can that will minimize the cost of the materials needed to construct the can. 21. We have a piece of cardboard that is 30 cm by 16 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.


Calculus I


22. We have a piece of cardboard that is 5 in by 20 in and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume. 23. A printer needs to make a poster that will have a total of 500 cm2 that will have 3 cm margins on the sides and 2 cm margins on the top and bottom. What dimensions of the poser will give the largest printed area? 24. A printer needs to make a poster that will have a total of 125 in2 that will have ½ inch margin on the bottom, 1 inch margin on the right, 2 inch margin on the left and 4 inch margin on the top. What dimensions of the poser will give the largest printed area?

More Optimization Problems 1. We want to construct a window whose bottom is a rectangle and the top of the window is an equilateral triangle. If we have 75 inches of framing material what are the dimensions of the window that will let in the most light? 2. We want to construct a window whose middle is a rectangle and the top and bottom of the window are equilateral triangles. If we have 4 feet of framing material what are the dimensions of the window that will let in the most light? 3. We want to construct a window whose middle is a rectangle, the top of the window is a semicircle and the bottom of the window is an equilateral triangle. If we have 1500 cm of framing material what are the dimensions of the window that will let in the most light? 4. Determine the area of the largest rectangle that can be inscribed in a circle of radius 5. 5. Determine the area of the largest rectangle whose base is on the x-axis and the top two corners lie on semicircle of radius 16. 6. Determine the area of the largest rectangle whose base is on the x-axis and the top two corners lie 24y x= − .

7. Find the point(s) on 2 2

14 36x y

+ = that are closest to ( )0,1 .

8. Find the point(s) on 2 8x y= − that are closest to ( )5,0 .

9. Find the point(s) on 22y x= − that are closest to ( )0, 3− .


Calculus I


10. A 6 ft piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into a rectangle with one side twice the length of the other side. Determine where, if anywhere, the wire should be cut to minimize the area enclosed by the two figures. 11. A 250 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into circle. Determine where, if anywhere, the wire should be cut to maximize the area enclosed by the two figures. 12. A 250 cm piece of wire is cut into two pieces. One piece is bent into an equilateral triangle and the other will be bent into circle. Determine where, if anywhere, the wire should be cut to minimize the area enclosed by the two figures. 13. A 4 m piece of wire is cut into two pieces. One piece is bent into a circle and the other will be bent into a rectangle with one side three times the length of the other side. Determine where, if anywhere, the wire should be cut to maximize the area enclosed by the two figures. 14. A line through the point ( )4,1− forms a right triangle with the x-axis and y-axis in the 2nd

quadrant. Determine the equation of the line that will minimize the area of this triangle. 15. A line through the point ( )3,3 forms a right triangle with the x-axis and y-axis in the 1st

quadrant. Determine the equation of the line that will minimize the area of this triangle. 16. A piece of pipe is being carried down a hallway that is 14 feet wide. At the end of the hallway there is a right-angled turn and the hallway narrows down to 6 feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 17. A piece of pipe is being carried down a hallway that is 9 feet wide. At the end of the hallway there is a right-angled turn and the hallway widens up to 21 feet wide. What is the longest pipe (always keeping it horizontal) that can be carried around the turn in the hallway? 18. Two poles, one 15 meters tall and one 10 meters tall, are 40 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used? 19. Two poles, one 2 feet tall and one 5 feet tall, are 3 feet apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used.? 20. Two poles, one 15 meters tall and one 10 meters tall, are 40 meters apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the angle formed by the two pieces of wire at the stake is a maximum?


Calculus I


21. Two poles, one 34 inches tall and one 17 inches tall, are 3 feet apart. A length of wire is attached to the top of each pole and it is staked to the ground somewhere between the two poles. Where should the wire be staked so that the angle formed by the two pieces of wire at the stake is a maximum? 22. A trough for holding water is to be formed as shown in the figure below. Determine the angle θ that will maximize the amount of water that the trough can hold.

23. A trough for holding water is to be formed as shown in the figure below. Determine the angle θ that will maximize the amount of water that the trough can hold.

Indeterminate Forms and L’Hospital’s Rule Use L’Hospital’s Rule to evaluate each of the following limits.

1. 3 2

3 24

6 32lim5 4x

x xx x x→−

+ −+ +

2. 6

3lim4w

w

w

−

−→−∞ +e

e

3. ( )( )0

sin 6lim

sin 11t

tt→

4. 2

3 21

8 9lim2 5 6x

x xx x x→

+ −− − +

5. 3 2

4 3 22

7 16 12lim4 4t

t t tt t t→

− + −− +


Calculus I


6. 2

2

4 1lim3 7 4w

w ww w→−∞

− ++ −

7. 2 6

2 7lim4y

y

yyy→∞

−+ee

8. ( )

( )

2

20

2cos 4 4 2lim

sin 2 2x

x xx x x→

− −− −

9. 2 6 2

3 23

3 12lim6 9x

x xx x x

+

→−

+ −+ +

e

10. ( )

( )6

sinlim

ln 5z

zzπ

→ −

11.

2

0

0limx

x t dt

x→

∫ e

12. 2lim ln 1

3ww

w→∞

−

13. ( ) ( )0

lim ln sint

t t+→

14. 2lim

z

zz→−∞

e

15. 7lim sin

xx

x→∞

16. ( )22

0lim lnz

z z+→

17. 1

0limx

xx+→

18. 1

0limt

t tt+→

+ e


Calculus I


19. 1

2lim 3x

x xx−

→∞ − e

20. Suppose that we know that ( )f x′ is a continuous function. Use L’Hospital’s Rule to show

that,

( ) ( ) ( )0

lim2h

f x h f x hf x

h→

+ − −′=

21. Suppose that we know that ( )f x′′ is a continuous function. Use L’Hospital’s Rule to show

that,

( ) ( ) ( ) ( )20

2limh

f x h f x f x hf x

h→

+ − + −′′=

Linear Approximations For problems 1 – 4 find a linear approximation to the function at the given point. 1. ( ) ( )cos 2f x x= at x π=

2. ( ) ( )2ln 5h z z= + at 2z =

3. ( ) 2 32 9 3g x x x x= − − − at 1x = −

4. ( ) ( )sin tg t = e at 4t = −

5. Find the linear approximation to ( ) ( )sin 1h y y= + at 0y = . Use the linear approximation to

approximate the value of ( )sin 2 and ( )sin 15 . Compare the approximated values to the exact

values.

6. Find the linear approximation to ( ) 5R t t= at 32t = . Use the linear approximation to

approximate the value of 5 31 and 5 3 . Compare the approximated values to the exact values. 7. Find the linear approximation to ( ) 1 xh x −= e at 1x = . Use the linear approximation to

approximate the value of e and 4−e . Compare the approximated values to the exact values.


Calculus I


For problems 8 – 10 estimate the given value using a linear approximation and without using any kind of computational aid. 8. ( )ln 1.1

9. 8.9 10. ( )sec 0.1

Differentials For problems 1 – 5 compute the differential of the given function. 1. ( ) 6 3 23 8 9 4f x x x x x= − + − −

2. ( )2 cos 2u t t=

3. ( )cos zy = e 4. ( ) ( ) ( )sin 3 cos 1g z z z= − −

5. ( ) 4 6 xR x x −= + e

5. Compute dy and y∆ for ( )siny x= as x changes from 6 radians to 6.05 radians.

6. Compute dy and y∆ for ( )2ln 1y x= + as x changes from -2 to -2.1.

7. Compute dy and y∆ for 1

2y

x=

− as x changes from 3 to 3.02.

8. Compute dy and y∆ for 14 xy x= e as x changes from -10 to -9.99.

9. The sides of a cube are found to be 6 feet in length with a possible error of no more than 1.5 inches. What is the maximum possible error in the surface area of the cube if we use this value of the length of the side to compute the surface area?


Calculus I


10. The radius of a circle is found to be 7 cm in length with a possible error of no more than 0.04 cm. What is the maximum possible error in the area of the circle if we use this value of the radius to compute the area? 11. The radius of a sphere is found to be 22 cm in length with a possible error of no more than 0.07 cm. What is the maximum possible error in the volume of the sphere if we use this value of the radius to compute the volume? 12. The radius of a sphere is found to be ½ foot in length with a possible error of no more than 0.03 inches. What is the maximum possible error in the surface area of the sphere if we use this value of the radius to compute the surface area?

Newton’s Method

For problems 1 – 3 use Newton’s Method to determine 2x for the given function and given value

of 0x .

1. ( ) 37 8 4f x x x= − + , 0 1x = −

2. ( ) ( ) ( )cos 3 sinf x x x= − , 0 0x =

3. ( ) 2 37 xf x −= − e , 0 5x =

For problems 4 – 8 use Newton’s Method to find the root of the given equation, accurate to six decimal places, that lies in the given interval. 4. 5 6x = in [ ]1,2

5. 3 22 9 17 20 0x x x− + + = in [ ]1,1−

6. 3 43 12 4 3 0x x x− − − = in [ ]3, 1− −

7. ( )4cosx x=e in [ ]1,1−

8. 22 2 xx −= e in [ ]0,2


Calculus I


For problems 9 – 12 use Newton’s Method to find all the roots of the given equation accurate to six decimal places. 9. 3 22 5 10 4 0x x x+ − − = 10. 4 3 24 54 92 105 0x x x x+ − − + =

11. ( )232 cosx x−− =e

12. ( ) ( )ln 2cosx x=

13. Suppose that we want to find the root to 3 27 8 3 0x x x− + − = . Is it possible to use 0 4x =

as the initial point? What can you conclude about using Newton’s Method to approximate roots from this example? 14. Use the function ( ) ( ) ( )2cos sinf x x x= − for this problem.

(a) Plot the function on the interval [ ]0,9 .

(b) Use 0 4x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (c) Use 0 5x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (d) Use 0 6x = to find one of the roots of this function to six decimal places. Did you get the

root you expected to? (e) What can you conclude about choosing values of 0x to find roots of equations using

Newton’s Method. 15. Use 0 0x = to find one of the roots of 5 32 7 3 1 0x x x− + − = accurate to six decimal places.

Did we chose a good value of 0x for this problem?

Business Applications 1. A company can produce a maximum of 2500 widgets in a year. If they sell x widgets during the year then their profit, in dollars, is given by, ( ) 2 31

3500,000,000 1,540,000 1450P x x x x= − + −


Calculus I


How many widgets should they try to sell in order to maximize their profit? 2. A company can produce a maximum of 25 widgets in a day. If they sell x widgets during the day then their profit, in dollars, is given by, ( ) 2 31

33000 40 11P x x x x= − + − How many widgets should they try to sell in order to maximize their profit? 3. A management company is going to build a new apartment complex. They know that if the complex contains x apartments the maintenance costs for the building, landscaping etc. will be, ( ) 2 32736 211 1

5 50 15070,000C x x x x= + − + The land they have purchased can hold a complex of at most 400 apartments. How many apartments should the complex have in order to minimize the maintenance costs? 4. The production costs of producing x widgets is given by,

( ) 90,0002000 4C x xx

= + +

If the company can produce at most 200 widgets how many should they produce to minimize the production costs? 5. The production costs, in dollars, per day of producing x widgets is given by, ( ) 2 3400 3 2 0.002C x x x x= − + + What is the marginal cost when 20x = and 75x = ? What do your answers tell you about the production costs? 6. The production costs, in dollars, per month of producing x widgets is given by,

( ) 2

8,000,00010,000 14C x xx

= + −

What is the marginal cost when 80x = and 150x = ? What do your answers tell you about the production costs? 7. The production costs, in dollars, per week of producing x widgets is given by,


Calculus I


( ) 2 365,000 4 0.2 0.00002C x x x x= + + − and the demand function for the widgets is given by, ( ) 5000 0.5p x x= − What is the marginal cost, marginal revenue and marginal profit when 2000x = and 4800x = ? What do these numbers tell you about the cost, revenue and profit? 8. The production costs, in dollars, per week of producing x widgets is given by,

( ) 2 56,000800 0.008C x xx

= + +

and the demand function for the widgets is given by, ( ) 2350 0.05 0.001p x x x= − − What is the marginal cost, marginal revenue and marginal profit when 175x = and 325x = ? What do these numbers tell you about the cost, revenue and profit?


Documents

Derivative Apps Homework Packet