Deck 4: Using the Derivative

Question 1

If for , what is ?

A)
B)
C)
D)

Answer 1

Question 2

Estimate the inflection points of if the following graph is the graph of
A. B. C. List inflection points in increasing order of x-coordinates; separate each point with a comma.

Answer 2

A. (2.5,-1)
B. (1,1), (4,-3)
C. (0,0), (2,0), (6,0)

Question 3

Given the curve , with , find the relation between the parameters a, b, and c that will ensure that the curve has no turning points.

Answer 3

< 0

Question 4

Find all of the critical points of . List them from smallest to largest, separated by commas.

Question 5

For and , find the value(s) of x for which
A has a local minimum.
B has a local maximum.
C. has a global minimum.
D. has a global maximum.
List the value(s) from smallest to largest, separated by commas.

Question 6

The following figure is a graph of a derivative function, . Indicate on the graph the x-values that are critical points and label each as a local maximum, a local minimum, or neither.

Question 7

The point x=1 on the following closed graph corresponds to

A)a local minimum
B)a local maximum
C)neither a maximum nor a minimum
D)a local and global minimum
E)a local and global maximum

Question 8

For and , which of the following statements are true?

A) has global maxima at and

B) has a global minima at and

C) has a global maximum at
D) has a global minimum at

Question 9

Assume that the polynomial f has exactly one local maximum, two local minima, and two inflection points. What is the largets number of zeros f could have?

Question 10

Sketch a graph of a function such that at x = -1, > 0 when x< -1, > 0 when x > -1.

Question 11

The following figure represents the function f, with , and A. Determine the first two positive x-intercepts of f. Separate them by a comma.
B. What must the value of a be so that ?
C. If , calculate .

Question 12

The graph of the function is:
A. increasing and concave up on what interval?
B. increasing and concave down on what interval?
C. decreasing and concave upon what interval?
D. decreasing and concave down on what interval?

Question 13

Find a value of a such that the function has a critical point at .

Question 14

A brick is heated in an oven and taken out to cool off after a certain time. The temperature T of the brick at any time t is given by for , with T in degrees Celsius and t in minutes. What is the temperature of the brick when it is placed in the oven (to the nearest degree)?

Question 15

For which interval(s) is the function decreasing?

A) -9 or 9
B)-9 9
C)-9 0
D)0 9
E)-10 < x < 10

Question 16

A stone is thrown vertically upward so that its height, measured in feet, after t seconds is given by . What is the maximum height of the stone (in feet)?

Question 17

The distance, s, traveled by a runner in a 20 mile race is given in the following figure, where time, t, is in hours. At which of the following values of t is the runner's speed is the slowest?

A)4
B)3
C)1
D)2

Question 18

Sketch the curve .

Question 19

The following is a graph of . Which of the following statements about are true?

A) changes sign at ,
, and

B) changes sign at and

C) has a local maximum or minimum at ,
, and

D) has a local maximum or minimum at and

Question 20

Suppose f has a continuous derivative whose values are given in the following table. Is x=-2.5 a potential local minimum, local maximum, or neither?

A)local minimum
B)neither
C)local maximum

Question 21

The number of plants in a terrarium is given by the function , where c is the number of mg of plant food added to the terrarium. The amount of plant food that produces the highest number of plants is _____ mg (round to the nearest hundredth).

Question 22

If the total revenue and total cost (in dollars) are given by ?
What quantity of gadgets, to the nearest whole number should be produced to maximize profit? What is the maximum profit?

Question 23

The function gives the population of a town (in 1000's of people) at time x where x is the number of years since 1980. The population was a minimum in the year _____.

Question 24

The revenue for selling q items is and the total cost is .
A. Write a function that gives total profit earned.
B. Find the quantity that maximizes profit.

Question 25

Total cost and revenue are approximated by the functions and , both in dollars.
A. What is the fixed cost?
B. What is the profit function?

Question 26

A normal distribution in statistics is modeled by the function . Determine where the maximum value of the function would occur.

A)-3
B)-2
C)0
D)
E)e^-3

Question 27

The quantity of a medication in the bloodstream t hours after it is ingested is given, in mg, by . What is the maximum quantity of the medication in the bloodstream?

A)110 mg
B)300 mg
C)815 mg
D)150 mg

Question 28

What quantity of gadgets should be produced to maximize profit if they sell for $500 per unit and the total cost (in dollars) of producing x units is given by ? What is that profit?

Question 29

Given the following table of production quantities with their corresponding marginal revenue and marginal cost, estimate the production level that maximizes profit.

Question 30

Let C(q) represent the cost, R(q), the revenue and π(q) the profit, in dollars of producing q items. If and approximately, how much profit is earned by the item?

Question 31

The following table shows cost and revenue for a product (in dollars).
A. What is the price of the product?
B. At what value of q is profit is maximized?

Question 32

Daily production levels in a plant can be modeled by the function , which gives units produced at t, the number of hours since the factory opened at 8 am. Factory productivity is at a maximum at _____ am.

Question 33

In the function y = 3sin (x) + 5, in the interval from 0 , at which value(s) of x does the function contain a global maximum?

A) and
B)0 and
C) only
D)4
E)2

Question 34

The function gives cost in dollars of producing r items. What is the marginal cost of increasing r by 1 item from the current production level of r=3?

Question 35

Consider the following graph of a function. Assume the entire graph is shown. How many local minima does the function have?

Question 36

With x people aboard, a South African airline makes a profit of (1200-4x) rands per person for a specific flight. What is the maximum number of passengers that can board such that the airline still profits?

Question 37

A. Find the marginal cost for q=110 when the fixed costs in dollars are 2,000, the variable costs are 100 per item, and each sells for $400.
B. Find the marginal revenue under the same conditions.

Question 38

The total cost, C, in dollars,when producing q items is . Calculate and interpret the marginal cost if q= 6.

Question 39

The total revenue, R, in dollars,when selling q items is . Calculate and interpret the marginal revenue if q= 10.

Question 40

Write a formula for total cost, C, as a function of quantity r when fixed costs are $45,000 and and variable costs are $2,000 per item.

Question 41

The cost function, in dollars, is . The average cost of producing 130 units is .

Question 42

The elasticity for a good is E=1.3. What is the effect on demand of a 7% price increase?

A)9.1% decrease
B)9.1% increase
C)5.4% decrease
D)5.4% increase

Question 43

The average cost per item to produce q items is given by .
A. What is the total cost, , of producing q items?
B. What is the marginal cost, MC, of producing q items?
C. At what production level does marginal cost equal average cost?

Question 44

You sell hot dogs at a baseball game for $2.75 each. You have 100 hot dogs to sell at an average cost to you of $1.75 each. The marginal cost at q=100 is $1.90. Assume you can always sell out of hot dogs. Will increasing the number of hot dogs you have to sell increase or decrease the average cost?

A)increase
B)decrease

Question 45

There is only one barber in a small town. Would you expect the elasticity of demand for the first barber to increase or decrease if a second barber moved into town?

Question 46

The graph of a cost function is given in the following figure. Estimate the value of q at which average cost is minimized.

Question 47

The demand curve for a product is .
A. Find the elasticity of demand (to three decimal places) at a price of p=5.
B. Is demand elastic or inelastic at this price?

Question 48

The cost of producing q items is dollars. What is the average cost of producing the 20^th item?

Question 49

A factory produces a product that sells for $12. They currently produce 2200 items per month, at an average cost of $3 per item. The marginal cost at this level is $2. Assume that the factory can sell all the items that it produces.
A. What is the profit at this production level?
B. Would increasing production increase or decrease profit?

Question 50

The demand equation for a product is , where q is the number of units produced, p is the price of each unit, and a and b are positive constants. Find the critical point(s) of the revenue function.

Question 51

The Revenue is given by and the Cost is given by . What value of q will maximize profit? What is the profit at that value?
q=________________________, P(q)= ___________________________

Question 52

An amusement park finds that when it charges $24 for an all-day pass, attendance is about 3700 per day. When it charges $27 , attendance is about 3400 per day.
A. Estimate the elasticity for the amusement park to two decimal places.
B. Is demand elastic or inelastic?

Question 53

The demand for doughnuts at a bakery is given by , where q is the number of doughnuts sold at a price of p dollars each.
A. Find the elasticity of demand to two decimal places if the price is $0.80.
B. Will revenue be increased by raising or lowering the price?

Question 54

An amusement park finds that when it charges $15 for an all-day pass, attendance is about 2400 per day. When it charges $19 , attendance is about 2200 per day. Is daily revenue higher at a price of $15 or a price of $19?

Question 55

Given the cost function and the demand function , find the value of q (to the nearest whole number) for which average cost is a minimum.

Question 56

The demand equation for a product is , where q is the number of units produced, p is the price of each unit, and a and b are positive constants. Find a formula, in terms of p, for the elasticity of demand.

Question 57

Which average cost function corresponds to the total cost function shown in the following figure?

A)
B)
C)
D)

Question 58

If the cost function is . For the 120^th unit, find the marginal cost and average cost, identify the units.

Question 59

You sell hot dogs at a baseball game for $3.25 each. You have 100 hot dogs to sell at an average cost to you of $2.00 each. The marginal cost at q=100 is $1.95. Assume you can always sell out of hot dogs. Will increasing the number of hotdogs you have to sell increase or decrease your profit?

A)increase
B)decrease

Question 60

A youth group wishes to hold a car wash as a fundraiser. Through past experience with car washes, they have constructed the following table, which shows the price, p, charged for a car wash and the quantity, q, of cars washed at that price. A. At what price is revenue maximized?
B. What is the elasticity at that price?

Question 61

The drug concentration curve for a drug after t hours is given by ng/ml. The minimum effective concentration is 10 ng/ml. Is the drug effective at t = 10 hours?

Question 62

Does the maximum value of the surge function increase or decrease when a is increased and b is held constant?

Question 63

The following table shows the total sales, in thousands, since a new DVD was released.
A. Estimate the point of diminishing returns.
B. Using your answer from part (A), predict the total possible sales for the DVD.

Question 64

A disease is released into a town. The number of people, in thousands infected is modeled by the equation . How many people are infected after hours?

Question 65

The following table gives the number of students who have joined a new school club t days after it was formed.
A. Estimate the value of t where concavity changes in this function.
B. Use your answer from part (a) to estimate the maximum membership in the club.

Question 66

A disease is released into a small town. The number of people infected is modeled by the equation . What is the population at t=0 of this disease?

Question 67

If time, t, is in hours and concentration, C, is in ng/ml, the drug concentration curve for a drug is given by . About how many hours does it take for the drug to reach peak concentration? .

A)2.5 hours
B)8.5 hours
C)0.4 hours
D)3.4 hours

Question 68

The following three equations are graphed in the figure. Which graph corresponds to equation C?
A. B. C.

Question 69

The following table gives the concentration C, of a drug in ng/ml, at time t, in hours, after it is administered to 2 different people.
A. If the concentration for Person A is given by and the concentration for Person B is given by , would you expect b_A to be larger or smaller than b_B?
B. If the minimum effective concentration is 5 ng/ml, until what time is the drug effective for person A?
C. If the minimum effective concentration is 5 ng/ml, until what time is the drug effective for person B?

Question 70

If time, t, is in hours and concentration, C, is in ng/ml, the drug concentration curve for a drug is given by . Suppose scientists wish to alter the drug so that it is effective for more hours. Would the coefficient of 9 in its concentration curve equation increase or decrease?

A)increase
B)decrease

Question 71

The demand curve for a product is given by .
A. Write revenue as a function of price and take its derivative to find the price (to the nearest cent) that will maximize revenue.
B. Find the elasticity (to two decimal places) at the price you found in part A.

Question 72

Raising the average price of an entree at a restaurant from $13 to $15 reduces the number of customers per day from 225 to 175.
A. What is the elasticity of demand to two decimal places for entrees at a price of $13?
B. Would raising the price from $13 to $15 increase or decrease the profit?

Question 73

In Wilson corners, population 2000, a rumor spreads according to the logistic model. If 5 people know the rumor at 4 PM and 110 people have heard it by 5 PM, how many people will have heard the rumor by 6 PM (to the nearest person)?

Question 74

The dose response curve in the following figure is given by , where R is percent of maximum response and x is the dose of the drug in mg. The inflection point is at (10,50) and . Would be greater or less than 7 for values of x less than 10?

Question 75

A biologist found that the number of Drosophila fruit flies, N(t), assumes the following growth pattern if the food source is limited: .
A. How many fruit flies were there in the beginning (to the nearest fly)?
B. At what time was the population increasing most rapidly (to the nearest day)?
C. At what rate does the number of fruit flies increase after 5 days (to the nearest fly per day)?

Question 76

The rabbit population, P, in a wilderness area is approximated by the function ,
where t is the number of weeks since the rabbits were introduced into the area.
A. How many rabbits were initially introduced into the area (to the nearest rabbit)?
B. How many rabbits were in the area after 5 weeks (to the nearest rabbit)?
C. What is the carrying capacity of rabbits in the area?

Question 77

A flu epidemic spreads amongst a group of people according to the formula ,
where represents the number of people that are infected by the end of day t.
A. How many people are infected by the end of the fifth day (to the nearest person)?
B. At what rate do the people become infected on day 5 (to the nearest person per day)?

Question 78

A disease is released into a town. The number of people infected each day is modeled by the equation . Estimate when Estimate the value of n at this time.

Question 79

The peak concentration of 9 ng/ml for a drug occurs 1.5 hours after a 7 mg dose is administered. Sketch a graph that represents the concentration, C, as a function of time, t.

Question 80

If time, t, is in hours and concentration, C, is in ng/ml, the drug concentration curve for a drug is given by . What is the peak concentration of the drug?

A)6.0 mg
B)6.5 mg
C)2.6 mg
D)16.3 mg