Deck 4: Applications of Differentiation

Question 1

A car braked with a constant deceleration of producing skid marks measuring 60 ft before coming to a stop.How fast was the car traveling when the brakes were first applied?

Answer 1

69.28 mi/h

Question 2

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.

Answer 2

CU on CD on IP

Question 3

Find the position function of a particle moving along a coordinate line that satisfies the given conditions.

Answer 3

Question 4

Estimate the extreme values of the function.Round theAnswers to the nearest hundredth.

Question 5

Find all the critical numbers of the function.

Question 6

Sketch the graph of the function on using the curve-sketching guidelines.

Question 7

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?

Question 8

Approximate the zero of to within 0.00001.

Question 9

Find the local and absolute extreme values of the function on the given interval.

Question 10

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?

Question 11

Use Newton's method to approximate the zero of between and using Continue until two successive approximations differ by less than 0.00001.

Question 12

A city's population (in thousands) years from now is estimated by What will the population of the city be in the long run? Hint: Find

Question 13

A tall runs parallel to a tall building at a distance of 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.

Question 14

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.

Question 15

Use Newton's method to find the zero of to within 0.00001 by solving the equation using

Question 16

Find two numbers whose difference is 170 and whose product is a minimum.

Question 17

(a) Find the intervals on which is increasing or decreasing.(b) Find the relative maxima and relative minima of

Question 18

The graph of the first derivative of a function is shown below.At what values of does have a local maximum or minimum?

Question 19

Find the maximum and minimum points of the function.

Question 20

Find the maximum and minimum points of the function.

Question 21

Find all the critical numbers of the function.

Question 22

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.

Question 23

Estimate the extreme values of the function.Round theAnswers to the nearest hundredth.

Question 24

The graph of the second derivative of a function is shown.State the x-coordinates of the inflection points of

Question 25

Sketch the curve.Find the equation of the slant asymptote.

Question 26

A woman is on a lake in a rowboat located one mile form the closest point of a straight shoreline (see the figure).She wishes to get to point 8 miles along the shore from by rowing to a point between and 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 to get to as quickly as possible? How much time does she require?

Question 27

Consider the function .(a) Find the intervals on which is increasing or decreasing.(b) Find the relative maxima and relative minima of

Question 28

What is the minimum vertical distance between the parabolas and

Question 29

Estimate the absolute maximum value of the function to two decimal places on the interval

Question 30

For what values of and is (2, 2.5) is an inflection point of the curve What additional inflection points does the curve have?

Question 31

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.

Question 32

Find the maximum and minimum points of the function.

Question 33

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 is the manufacturing cost, in millions of dollars.The company estimates that if it charges a price (in millions of dollars) for each plane, it will be able to sell Find the cost function.

Question 34

The manager of a 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?

Question 35

Sketch the graph of the function using the curve-sketching guidelines.

Question 36

Use Newton's method to approximate the indicated root of in the interval correct to six decimal places.Use as the initial approximation.

Question 37

The quantity demanded per month of an item is related to the unit price by the demand equation where 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

Question 38

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?

Question 39

Find the critical numbers of the function.

Question 40

Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length and width W

Question 41

Select the correct Answer for each question.
Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval.

A)
B)
C)
D)
E)None of these

Question 42

Select the correct Answer for each question.
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

Question 43

Select the correct Answer for each question.
A production editor decided that a promotional flyer should have a 1-in.margin at the top and the bottom, and a margin on each side.The editor further stipulated that the flyer should have an area of Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

A)
B)
C)
D)

Question 44

Select the correct Answer for each question.
A piece of wire 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 yourAnswer to the nearest hundredth.

A)
B)
C)
D)
E)

Question 45

Select the correct Answer for each question.
The function satisfies the hypotheses of Rolle's Theorem on the interval Find all values of that satisfy the conclusion of the theorem.

A)2, 4
B)2
C)3, 4
D)3

Question 46

Select the correct Answer for each question.
Find

A)
B)
C)
D)
E)None of these

Question 47

Select the correct Answer for each question.
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

Question 48

Select the correct Answer for each question.
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)

Question 49

Select the correct Answer for each question.
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)

Question 50

Select the correct Answer for each question.
Sketch the graph of the function using the curve-sketching guidelines.

A)
B)
C)
D)

Question 51

Select the correct Answer for each question.
A skydiver leaps from a helicopter hovering high above the ground.Her velocity 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

Question 52

Select the correct Answer for each question.
Find all the critical numbers of the function.

A)
B)
C)
D)
E)None of these

Question 53

Select the correct Answer for each question.
Evaluate the limit using l'Hôpital's Rule.

A)0
B)36
C)18
D)108

Question 54

Select the correct Answer for each question.
The function satisfies the hypotheses of Rolle's Theorem on the interval Find all values of that satisfy the conclusion of the theorem.

A)
B)
C)
D)

Question 55

Select the correct Answer for each question.
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)

Question 56

Select the correct Answer for each question.
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 roots

Question 57

Select the correct Answer for each question.
Given (a) Find the intervals on which is increasing or decreasing.(b) Find the relative maxima and relative minima of

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.

Question 58

Select the correct Answer for each question.
Find the absolute maximum value of on the interval

A)7
B)5
C)8
D)6
E)4

Question 59

Select the correct Answer for each question.
The size of the monthly repayment that amortizes a loan of dollars in years at an interest rate of per year, compounded monthly, on the unpaid balance is given by The value of 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%

Question 60

Select the correct Answer for each question.
Find the value of the limit.

A)
B)
C)
D)
E)

Question 61

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)
B)
C)
D)
E)

Question 62

Find the slant asymptote of the graph of using the curve-sketching guidelines.

A)
B)
C)
D)

Question 63

Find the dimensions of the rectangle enclosed in the semicircle with the largest possible area.

A)
B)5 in. 7 in.
C)
D)

Question 64

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)
D)0
E)4

Question 65

Find two positive numbers whose product is and whose sum is a minimum.

A)6, 24
B)2, 72
C)
D)3, 48
E)

Question 66

Identify any transitional values of at which the basic shape of the curve changes.

A)
B)
C)
D)

Question 67

Find

A)
B)
C)
D)
E)None of these

Question 68

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)

Question 69

The graph of the derivative of a continuous function f is shown.On what intervals is decreasing? .

A)
B)
C)
D)
E)None of these

Question 70

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 to

Question 71

Find the limit.

A)
B)
C)
D)

Question 72

A skydiver leaps from a helicopter hovering high above the ground.Her velocity 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

Question 73

Find the intervals where is increasing and where it is decreasing.

A)increasing: decreasing:
B) increasing: decreasing:
C)increasing: decreasing:
D) increasing: decreasing:

Question 74

The size of the monthly repayment that amortizes a loan of dollars in years at an interest rate of per year, compounded monthly, on the unpaid balance is given by The value of 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%

Question 75

Evaluate the limit using Rule.

A)6
B)
C)12
D)27

Question 76

Find the critical number(s), if any, of the function

A)3
B)
C)
D)

Question 77

Evaluate the limit.

A)-5
B)
C)
D)1
E)0

Question 78

Evaluate the limit.

A)9
B)0
C)
D)

Question 79

Evaluate the limit.

A)2
B)
C)0
D)
E)

Question 80

Find the limit.

A)
B)
C)
D)5
E)0