1. Find the local and absolute extreme values of the function on the given interval.
,
2. Find all the critical numbers of the function.
3. Find the maximum and minimum points of the function.
4. Find the maximum and
...
1. Find the local and absolute extreme values of the function on the given interval.
,
2. Find all the critical numbers of the function.
3. Find the maximum and minimum points of the function.
4. Find the maximum and minimum points of the function.
5. The graph of the first derivative of a function f is shown below. At what values of x does f
have a local maximum or minimum?
6. Estimate the extreme values of the function. Round the answers to the nearest hundredth.Stewart - Calculus 8e ET Chapter 4 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
7. A steel pipe is being carried down a hallway 14 ft wide. At the end of the hall there is a
right-angled turn into a narrower hallway 7 ft wide. What is the length of the longest pipe that can
be carried horizontally around the corner?
8. A fence ft tall runs parallel to a tall building at a distance of ft from the building. What is the
length of the shortest ladder that will reach from the ground over the fence to the wall of the
building? Round the result to the nearest hundredth.
9. A car braked with a constant deceleration of 40 , producing skid marks measuring 60 ft
before coming to a stop. How fast was the car traveling when the brakes were first applied?
10. A company estimates that the marginal cost (in dollars per item) of producing items is
. If the cost of producing one item is $560 find the cost of producing 400 items.
11. Given .
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.
12. Determine where the graph of the function is concave upward and where it is
concave downward on the interval . Also, find all inflection points of the function.
13. A city’s population (in thousands) t years from now is estimated by
What will the population of the city be in the long run? Hint: FindStewart - Calculus 8e ET Chapter 4 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
14. Sketch the graph of the function on using the curve-sketching guidelines.
15. Find two numbers whose difference is 170 and whose product is a minimum.
16. The owner of a ranch has 1,800 yd of fencing with which to enclose a rectangular piece of grazing
land situated along a straight portion of a river. If fencing is not required along the river, what are
the dimensions of the largest area he can enclose? What is the area?
17. Use Newton’s method to find the zero of to within 0.00001 by solving the
equation using
–2 –1 1 2 x
4 3 2 1
–1
–2
–3
–4
y
18. Use Newton’s method to approximate the zero of between and
using . Continue until two successive approximations differ by less than 0.00001.
19. Approximate the zero of in to within 0.00001.
20. Find the position function of a particle moving along a coordinate line that satisfies the given
conditions.
, s (0) = 5, v (0) = 0Stewart - Calculus 8e ET Chapter 4 Form A
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. Max , Min
2. integers
3. 0
4.
5. 3, 5, 8, 10
6.
7. 29.13 ft
8.
9. 69.28 mi/h
10.
11. (a) Increasing on decreasing on and
(b) Rel. max. , rel. min.
12. CU on , CD on , IP
13. 26,000
14.
1 x
0.5
y
15. –85, 85
16. Dimensions: 900 yd 450 yd, Maximum area: 405,000
17. –0.32219
18. –0.32219
19. 1.6717
20.Stewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
1. Find the critical numbers of the function.
2. Find all the critical numbers of the function.
3. Estimate the absolute maximum value of the function to two decimal places on the
interval .
4. Find the maximum and minimum points of the function.Stewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
5. The graph of the second derivative of a function f is shown. State the x-coordinates of
the inflection points of f.
6. Estimate the extreme values of the function. Round the answers to the nearest hundredth.
7. For what values of a and b is (2, 2.5) is an inflection point of the curve ? What
additional inflection points does the curve have?
8. Sketch the curve. Find the equation of the slant asymptote.
9. What is the minimum vertical distance between the parabolas and ?
10. The average cost of producing x units of a commodity is given by the equation
.
Find the marginal cost at a production level of 1,255 units.Stewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
11. Find the maximum area of a rectangle that can be circumscribed about a given rectangle with
length L = 8 and width W = 3.
12. An aircraft manufacturer wants to determine the best selling price for a new airplane. The
company estimates that the initial cost of designing the airplane and setting up the factories in
which to build it will be million dollars. The additional cost of manufacturing each plane can
be modeled by the function
where x is the number of aircraft produced and m is the manufacturing cost, in millions of dollars.
The company estimates that if it charges a price p (in millions of dollars) for each plane, it will be
able to sell
.
Find the cost function.
13. A baseball team plays in a stadium that holds 56,000 spectators. With ticket prices at $9, the
average attendance had been 32,000. When ticket prices were lowered to $8, the average
attendance rose to 36,000. How should ticket prices be set to maximize revenue? Assume the
demand function is linear.
14. The manager of a -unit apartment complex knows from experience that all units will be
occupied if the rent is per month. A market survey suggests that, on the average, one
additional unit will remain vacant for each increase in rent. What rent should the manager
charge to maximize revenue?Stewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. Use Newton's method to approximate the indicated root of in the interval ,
correct to six decimal places.
Use as the initial approximation.
16. The quantity demanded per month of an item is related to the unit price by the demand equation
,
where p is measured in dollars and x is measured in units of a thousand. How many items must be
sold by the manufacturer to maximize its revenue?
Hint: Recall that the revenue is given by
17. Consider the function . (a) Find the intervals on which f is increasing or
decreasing. (b) Find the relative maxima and relative minima of f.
18. The altitude (in feet) attained by a model rocket t sec into flight is given by the function
.
When is the rocket ascending, and when is it descending? What is the maximum altitude attained
by the rocket?
19. Sketch the graph of the function using the curve-sketching guidelines.
20. A woman is on a lake in a rowboat located one mile form the closest point P of a straight shoreline
(see the figure). She wishes to get to point Q, 8 miles along the shore from P, by rowing to a point
R between P and Q and then walking the rest of the distance. If she can row at a speed of 3 mph
and walk at a speed of 4 mph, how should she pick the point R to get to Q as quickly as possible?
How much time does she require?
8 miStewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2. integers
3. 2.92
4. 0
5. 2, 4, 9
6.
7.
8.
9.
10. 18.64
11. 60.5
12.
13.
14.
15. 1.283782
16. 5,000
17. (a) decreasing on and , increasing on ; (b) relative minimum:
relative maximum:Stewart - Calculus 8e ET Chapter 4 Form B
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
18. The rocket is ascending on
The rocket is descending on .
Maximum altitude attained is approximately 65 ft.
19.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x
0.1
0.2
–0.1
–0.2
y
20.Stewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Select the correct answer for each question.
____ 1. Find all the critical numbers of the function.
a.
b.
c.
d.
e. None of these
____ 2. Find the absolute maximum value of on the interval .
a. 7
b. 5
c. 8
d. 6
e. 4
____ 3. Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval.
,
a.
b.
c.
d. 25
4
e. None of theseStewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 4. The function satisfies the hypotheses of Rolle’s Theorem on the interval
. Find all values of c that satisfy the conclusion of the theorem.
a. 2, 4
b. 2
c. 3, 4
d. 3
____ 5. The function satisfies the hypotheses of Rolle’s Theorem on the interval
. Find all values of c that satisfy the conclusion of the theorem.
a.
b.
c.
d.
____ 6. The function satisfies the hypotheses of Rolle’s Theorem on the interval
. Find all values of c that satisfy the conclusion of the theorem.
a.
b. 4
c.
d.
____ 7. The function satisfies the hypotheses of Rolle’s Theorem on the interval .
Find all values of c that satisfy the conclusion of the theorem.
a.
b.
c.
d.Stewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 8. The function satisfies the hypotheses of the Mean Value Theorem on the
interval . Find all values of c that satisfy the conclusion of the theorem.
a. 3
b. 4
c. 2, 3
d. 2, 4
____ 9. The function satisfies the hypotheses of the Mean Value Theorem on the interval
Find all values of c that satisfy the conclusion of the theorem.
a.
b.
c.
d.
____ 10. How many real roots does the equation have in the interval ?
a. at most five real roots
b. no real roots
c. at most one real root
d. at most two real roots
e. at most three real rootsStewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 11. Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval.
Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
,
a.
,
b.
,
c.
,
d.
,
e. None of these
____ 12. Given .
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.
a. (a) Increasing on , decreasing on and
(b) Rel. max. ,
rel. min.
b. (a) Increasing on and decreasing on and
(b) Rel. max. ,
rel. min.
c. (a) Increasing on and , decreasing on and
(b) Rel. max. ,
rel. min.
d. (a) Increasing on and , decreasing on
(b) Rel. max. ,
rel. min.Stewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 13. A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and
before deploying her parachute is given by
where is measured in meters per second. What is her terminal velocity? Hint: Evaluate
a. 54 m/sec
b. 7 m/sec
c. 47 m/sec
d. 87 m/sec
____ 14. Evaluate the limit using l’Hôpital’s Rule.
a. 0
b. 36
c. 18
d. 108
____ 15. Find the value of the limit.
a.
b.
c.
d.
e.Stewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 16. Sketch the graph of the function using the curve-sketching guidelines.
a.
–5 –4 –3 –2 –1 1 2 3 4 5 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
c.
–5 –4 –3 –2 –1 1 2 3 4 5 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
b.
–5 –4 –3 –2 –1 1 2 3 4 5 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
d.
–5 –4 –3 –2 –1 1 2 3 4 5 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
yStewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 17. A production editor decided that a promotional flyer should have a 1-in. margin at the top and the
bottom, and a -in. margin on each side. The editor further stipulated that the flyer should have
an area of 72 . Determine the dimensions of the flyer that will result in the maximum printed
area on the flyer.
a. 6 in. 12 in.
b. 3 in. 24 in.
c. in. in.
d. in. in.
____ 18. A piece of wire m long is cut into two pieces. One piece is bent into a square and the other is
bent into an equilateral triangle. How should the wire be cut for the square so that the total area
enclosed is a minimum?
Round your answer to the nearest hundredth.
a.
b.
c.
d.
e.Stewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 19. The size of the monthly repayment k that amortizes a loan of A dollars in N years at an interest rate
of r per year, compounded monthly, on the unpaid balance is given by
The value of r can be found by performing the iteration
A family secured a loan of $360,000 from a bank to finance the purchase of a house. They have
agreed to repay the loan in equal monthly installments of $2476 over 25 years. Find the interest
rate on this loan. Round the rate to one decimal place.
a. 8.7%
b. 7.7%
c. 6.7%
d. 5.7%
____ 20. Find f.
,
a.
b.
c.
d.
e. None of theseStewart - Calculus 8e ET Chapter 4 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. B
2. E
3. D
4. D
5. B
6. C
7. B
8. A
9. A
10. C
11. A
12. C
13. C
14. D
15. C
16. D
17. A
18. B
19. C
20. BStewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Find the critical number(s), if any, of the function .
a. 3
b. 3 and 3
c. 9
2 and 9 2
d. 9
2
____ 2. The graph of the derivative of a continuous function f is shown. On what intervals is f
decreasing?
.
a.
b.
c.
d.
e. None of theseStewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 3. Find the intervals where is increasing and where it is decreasing.
a. increasing: , decreasing:
b.
increasing: , decreasing:
c. increasing: , decreasing:
d.
increasing: , decreasing:
____ 4. Let P (x) and Q (x) be polynomials. Find
if the degree of P (x) is 2 and the degree of Q (x) is 6.
a. 9
b. 5
c. – 4
d. 0
e. 4
____ 5. Find the limit.
a.
b.
c.
d.Stewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 6. A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and
before deploying her parachute is given by
where is measured in meters per second. What is her terminal velocity? Hint: Evaluate
a. 54 m/sec
b. 8.1 m/sec
c. 87 m/sec
d. 62.1 m/sec
____ 7. Evaluate the limit.
a. 9
b. 0
c. –9
d. 2
9
____ 8. Evaluate the limit using l’Hôpital’s Rule.
a. 6
b. –6
c. 12
d. 27Stewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 9. Evaluate the limit.
a. 2
b.
c. 0
d.
e.
____ 10. Evaluate the limit.
a. –5
b.
c.
d. 1
e. 0
____ 11. Find the limit.
a.
b.
c.
d. 5
e. 0Stewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 12. Find the slant asymptote of the graph of using the curve-sketching guidelines.
a.
–12 –10 –8 –6 –4 –2 2 4 6 x
8 6 4 2
10
12
14
–2
–4
y
c.
–6 –4 –2 2 4 6 8 10 12 x
8 6 4 2
10
12
14
–2
–4
y
b.
–12 –10 –8 –6 –4 –2 2 4 6 x
4 2
–2
–4
–6
–8
–10
–12
–14
y
d.
–6 –4 –2 2 4 6 8 10 12 x
4 2
–2
–4
–6
–8
–10
–12
–14
y
____ 13. What can you say about point of inflation for ?
a. point of inflation goes away from the origin
b. point of inflation goes away from
c. point of inflation gets closer to the origin
d. point of inflation gets closer toStewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. Identify any transitional values of at which the basic shape of the curve changes.
a.
b.
c.
d.
____ 15. Find two positive numbers whose product is and whose sum is a minimum.
a. 6, 24
b. 2, 72
c.
d. 3, 48
e.
____ 16. A rectangular beam will be cut from a cylindrical log of radius inches. Suppose that the
strength of a rectangular beam is proportional to the product of its width and the square of its
depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.
a.
in, in
b.
in, in
c. in, in
d.
in, in
e.
in, inStewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 17. Find the dimensions of the rectangle enclosed in the semicircle with the largest
possible area.
–12 –8 –4 4 8 12 x
8 4
12
–4
–8
–12
y
a. in. in.
b. 5 in. 7 in.
c. in. in.
d. in. in.
____ 18. A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey
indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60
per week. Find the demand function.
a.
b.
c.
d.
e.Stewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 19. The size of the monthly repayment k that amortizes a loan of A dollars in N years at an interest rate
of r per year, compounded monthly, on the unpaid balance is given by
The value of r can be found by performing the iteration
A family secured a loan of $360,000 from a bank to finance the purchase of a house. They have
agreed to repay the loan in equal monthly installments of $2342 over 25 years. Find the interest
rate on this loan. Round the rate to one decimal place.
a. 4.1%
b. 5.1%
c. 6.1%
d. 7.1%
____ 20. Find f.
,
a.
b.
c.
d.
e. None of theseStewart - Calculus 8e ET Chapter 4 Form D
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. D
2. A
3. D
4. D
5. C
6. A
7. B
8. C
9. D
10. B
11. B
12. C
13. C
14. D
15. C
16. A
17. D
18. B
19. C
20. CStewart - Calculus 8e ET Chapter 4 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Find the critical number(s), if any of the function Select the correct answer.
a.
b.
c.
0,
d.
0,
____ 2. Find the critical number(s), if any, of the function . Select the correct answer.
a. 10
b. 5
c. 10 and 10
d. 5 and 5
3. Find the absolute maximum and absolute minimum values, if any, of the function
on [0, 9]
4. Find the critical number(s) of the function.
____ 5. Use the Second Derivative Test to find the relative extrema, if any, of the function
. Select the correct answer.
a. Rel. max. ,
rel. min.
b. Rel. max. ,
rel. min.
c. Rel. max. ,
rel. min.
d. Rel. max. ,
rel. min.
6. How many points of inflection are on the graph of the function?Stewart - Calculus 8e ET Chapter 4 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
7. Given .
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.
8. Find the limit.
____ 9. A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and
before deploying her parachute is given by
where is measured in meters per second. What is her terminal velocity? Hint: Evaluate
Select the correct answer.
a. 54 m/sec
b. 7 m/sec
c. 47 m/sec
d. 87 m/sec
10. Evaluate the limit using l’Hôpital’s Rule.
____ 11. Evaluate the limit. Select the correct answer.
a. –2
b. 1
c. 0
d. 2
12. Sketch the graph of the function using the curve-sketching guidelines.Stewart - Calculus 8e ET Chapter 4 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 13. What is the shortest possible length of the line segment that is cut off by the first quadrant and is
tangent to the curve at some point? Select the correct answer.
a. 2 6
b.
c.
d.
e. None of these
14. Find two positive numbers whose product is and whose sum is a minimum.
15. A rectangular storage container with an open top is to have a volume of 10 . The length of its
base is twice the width. Material for the base costs $12 per square meter. Material for the sides
costs $5 per square meter. Find the cost of materials for the cheapest such container.
____ 16. A production editor decided that a promotional flyer should have a 1-in. margin at the top and the
bottom, and a -in. margin on each side. The editor further stipulated that the flyer should have
an area of 72 . Determine the dimensions of the flyer that will result in the maximum printed
area on the flyer. Select the correct answer.
a. 6 in. 12 in.
b. 3 in. 24 in.
c. in. in.
d. in. in.
17. Consider the following problem: A farmer with ft of fencing wants to enclose a rectangular
area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the
largest possible total area of the four pens?Stewart - Calculus 8e ET Chapter 4 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
18. Use Newton’s method to approximate the zero of between and
using . Continue until two successive approximations differ by less than 0.00001.
____ 19. Find f. Select the correct answer.
,
a.
b.
c.
d.
e. None of these
20. A particle is moving with the given data. Find the position of the particle.
,Stewart - Calculus 8e ET Chapter 4 Form E
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. D
2. B
3. Abs. max.
abs. min.
4.
5. B
6. 1
7. (a) Increasing on and , decreasing on and
(b) Rel. max. ,
rel. min.
8.
9. C
10. 0
11. C
12.
–5 –4 –3 –2 –1 1 2 3 4 5 x
6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
y
13. A
14.
15. $153.9
16. A
17.
18. 0.16904
19. B
20.Stewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Find the critical number(s), if any, of the function . Select the correct answer.
a. 10
b. 5
c. 10 and 10
d. 5 and 5
2. Find the absolute maximum and absolute minimum values, if any, of the function
on [0, 9]
3. You are given the graph of the derivative of a function f.
(a) Determine the intervals on which f is increasing or decreasing.
(b) Find the x-coordinate of the relative maxima and relative minima of f.
–10 –8 –6 –4 –2 2 4 6 8 10 x
y
4. Given .
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.Stewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 5. Given .
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the relative maxima and relative minima of f.
a. (a) Increasing on , decreasing on and
(b) Rel. max. ,
rel. min.
b. (a) Increasing on and decreasing on and
(b) Rel. max. ,
rel. min.
c. (a) Increasing on and , decreasing on and
(b) Rel. max. ,
rel. min.
d. (a) Increasing on and , decreasing on
(b) Rel. max. ,
rel. min.
6. A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and
before deploying her parachute is given by
where is measured in meters per second. What is her terminal velocity? Hint: Evaluate
7. Evaluate the limit using l’Hôpital’s Rule.
8. Find the limit.
9. Sketch the curve.Stewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 10. Find the slant asymptote of the function . Select the correct answer.
a.
b.
c.
d.
e.
11. Sketch the graph of the function using the curve-sketching guidelines.
____ 12. Sketch the graph of the function using the curve-sketching guidelines.
a.
1 2 3 4 5 6 7 8 x
1
–1
–2
–3
–4
y
c.
–6 –5 –4 –3 –2 –1 x
4 3 2 1
–1
–2
–3
–4
y
b.
–6 –5 –4 –3 –2 –1 x
4 3 2 1
–1
–2
–3
–4
y
d.
1 2 3 4 5 6 7 8 x
5 4 3 2 1
–1
yStewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
13. What can you say about point of inflation for ?
14. a) Graph the funtion .
b) Use l’Hospitals’ rule to explain the behavior as
____ 15. Find two positive numbers whose product is and whose sum is a minimum.
Select the correct answer.
a.
b. 3, 48
c. 6, 24
d.
e. 2, 72
16. A rectangular storage container with an open top is to have a volume of 10 . The length of its
base is twice the width. Material for the base costs $12 per square meter. Material for the sides
costs $5 per square meter. Find the cost of materials for the cheapest such container.
17. Find the point on the line that is closest to the origin.
18. A rectangular beam will be cut from a cylindrical log of radius inches. Suppose that the
strength of a rectangular beam is proportional to the product of its width and the square of its
depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.Stewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
19. Find the dimensions of the rectangle enclosed in the semicircle with the largest
possible area.
–12 –8 –4 4 8 12 x
8 4
12
–4
–8
–12
y
____ 20. A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey
indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase by 60
per week. Find the demand function. Select the correct answer.
a.
b.
c.
d.
e.Stewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. B
2. Abs. max.
abs. min.
3. (a) Increasing and
decreasing on
(b) Rel. max at , rel. min. at
4. (a) Increasing on and decreasing on
(b) Rel. max. ,
rel. min.
5. C
6. 47 m/sec
7. 0
8.
9.
10. BStewart - Calculus 8e ET Chapter 4 Form F
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
11.
–5 –4 –3 –2 –1 1 2 3 4 5 x
3 2 1
–1
–2
–3
y
12. D
13. point of inflation gets closer to the origin
14. a)
–5 –4 –3 –2 –1 1 2 3 4 5 x
5 4 3 2 1
–1
–2
–3
–4
–5
y
b)
There is a hole at (0, 0)
15. A
16. $153.9
17.
18. in, in
19. in. in.
20. BStewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 1. Find the critical number(s), if any, of the function . Select the correct answer.
a. 10
b. 5
c. 10 and 10
d. 5 and 5
____ 2. Find the critical number(s) of the function. Select the correct answer.
a.
b.
c.
d.
e.
3. For what values of c does the curve have maximum and minimum points?
4. Find the point(s) of inflection of the graph of
____ 5. Find the inflection points for the function given. Select the correct answer.
,
a. ,
b. ,
c. ,
d. ,
e. ,Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
6. The graph of the derivative of a continuous function f is shown. On what intervals is f
decreasing?
.
____ 7. Find the limit. Select the correct answer.
a. 0
b.
c.
d.
e. 1
8. Find the limit.Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
9. What is the function of the graph?
10. Sketch the graph of the function using the curve-sketching guidelines.
11. Sketch the graph of the function using the curve-sketching guidelines.Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 12. What is the function of the graph? Select the correct answer.
a.
b.
c.
d.
e.
13. Find two positive numbers whose product is and whose sum is a minimum.
____ 14. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius
. Select the correct answer.
a.
b.
c.
d.
e.Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
15. Find the dimensions of the rectangle enclosed in the semicircle with the largest
possible area.
–12 –8 –4 4 8 12 x
8 4
12
–4
–8
–12
y
____ 16. An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per
acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many
trees per acre should be planted to maximize yield? Select the correct answer.
a. 27 trees/acre
b. 30 trees/acre
c. 37 trees/acre
d. 23 trees/acre
____ 17. Use Newton’s method to obtain an approximation to the root of to within 0.00001.
Select the correct answer.
a. 0.62821
b. 0.48918
c. 0.56127
d. 0.51493
18. Find f.
,Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 19. Given that the graph of f passes through the point (4, 69) and that the slope of its tangent line at
is , find f (1) . Select the correct answer.
a. –4.5
b. 8
c. 12
d. 11
e. 1
20. Find the critical number(s), if any, of the function .Stewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1. B
2. E
3.
4.
5. B
6.
7. D
8.
9.
10.
–7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x
7 6 5 4 3 2 1
–1
–2
–3
–4
–5
–6
–7
y
11.
1 2 3 4 5 6 7 8 x
5 4 3 2 1
–1
yStewart - Calculus 8e ET Chapter 4 Form G
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
12. D
13.
14. C
15. in. in.
16. D
17. D
18.
19. A
20.Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
1. Estimate the extreme values of the function.
____ 2. Find the critical number(s), if any, of the function Select the correct answer.
a.
b. –3
c.
d.
3. Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval.
Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
4. Determine where the graph of is concave upward and where it is concave
downward. Also, find all inflection points of the function.
5. The graph of the derivative of a continuous function f is shown. On what intervals is f
decreasing?
.Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
6. A skydiver leaps from a helicopter hovering high above the ground. Her velocity t sec later and
before deploying her parachute is given by
where is measured in meters per second. What is her terminal velocity? Hint: Evaluate
____ 7. Evaluate the limit using l’Hôpital’s Rule. Select the correct answer.
a.
1
42
b. 13
42
c. 1
42
d. 1
2
8. Evaluate the limit.
9. Use l'Hospital's Rule to calculate the exact value of the limit as
10. Find the limit.Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
11. Sketch the graph of the function using the curve-sketching guidelines.
12. Sketch the graph of the function using the curve-sketching guidelines.Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 13. Select the correct graph for the given function .
Select the correct answer.
a.
–4 –3 –2 –1 1 2 3 4
8 6 4 2
10
–2
–4
–6
–8
–10
d.
–4 –3 –2 –1 1 2 3 4
8 6 4 2
10
–2
–4
–6
–8
–10
b.
–4 –3 –2 –1 1 2 3 4
8 6 4 2
10
–2
–4
–6
–8
–10
e.
–4 –3 –2 –1 1 2 3 4
8 6 4 2
10
–2
–4
–6
–8
–10
c.
–4 –3 –2 –1 1 2 3 4
8 6 4 2
10
–2
–4
–6
–8
–10Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. What is the shortest possible length of the line segment that is cut off by the first quadrant and is
tangent to the curve at some point? Select the correct answer.
a.
b.
c.
d. 4 2
e. None of these
15. A rectangular storage container with an open top is to have a volume of 10 . The length of its
base is twice the width. Material for the base costs $12 per square meter. Material for the sides
costs $5 per square meter. Find the cost of materials for the cheapest such container.
16. The size of the monthly repayment k that amortizes a loan of A dollars in N years at an interest rate
of r per year, compounded monthly, on the unpaid balance is given by
The value of r can be found by performing the iteration
A family secured a loan of $360,000 from a bank to finance the purchase of a house. They have
agreed to repay the loan in equal monthly installments of $2342 over 25 years. Find the interest
rate on this loan. Round the rate to one decimal place.Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 17. Use Newton’s method to approximate the zero of between and
using . Continue until two successive approximations differ by less than 0.00001.
Select the correct answer.
a. 0.55341
b. 0.52221
c. 0.48751
d. 0.45061
18. Evaluate , and tell whether its antiderivative F is increasing or decreasing at the
point radians.
____ 19. Find f.
Select the correct answer.
a.
b.
c.
d.
e.
20. Given that the graph of f passes through the point (4, 69) and that the slope of its tangent line at
is , find f (1) .Stewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Answer Key
1.
2. D
3.
4. concave downward: and , concave upward: nowhere; inflection points: none
5.
6. 48 m/sec
7. D
8.
9.
10.
11.
–5 –4 –3 –2 –1 1 2 3 4 5 x
5 4 3 2 1
–1
–2
–3
–4
–5
yStewart - Calculus 8e ET Chapter 4 Form H
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
12.
1 2 3 4 5 6 7 8 x
5 4 3 2 1
–1
y
13. E
14. D
15. $153.9
16. 6.1%
17. B
18. , decreasing
19. B
20. 12
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