Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences
30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40REIn Problems 40 and 41, decide whether the statements are true or false.
41. gives the equation of the tangent line to .
42RE43RE44RE45RE46RE47RE48RE49RE50RE51. If
52RE53RE54RE55RE56. If
57RE58RE59RE60. Write the equation of the line tangent to the curve x at the point where .
61RE62RE63. If
65RE66RE67RE68RE
70RE71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84RE85RE86RE87RE88RE89RE90RE91RE92RE93RE94RE95RE96RE97RE98RE99REIn Problems 100-107, cost, revenue, and profit are in dollars and x is the number of units.
100. Cost If the cost function for a particular good is , what is the
(a) marginal cost function?
(b) marginal cost if 30 units are produced?
(c) interpretation of your answer in part (b)?
101RE
In Problems 100-107, cost, revenue, and profit are in dollars and x is the number of units.
102. Revenue The total revenue function for a commodity is , with x representing the number of units.
(a) Find the marginal revenue function.
(b) At what level of production will marginal revenue be 0?
103RE104RE105RE106RE107RE108RE1T2T3T4T5T6T7T8T9T10T11T12T13T14T15T16T18T19T20T21T22T23T1. Can exist if is undefined?
2. Does exist if ?
3. Does
4. If , does exist?
5. Let
(a) Does
(b) Find .
6. Evaluate the following limits (if they exist).
(a)
(b)
(c)
7CP8CP9CP10CPIn Problems 1 -6, a graph of is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
1.
In Problems 1 -6, a graph of is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
2.
In Problems 1 -6, a graph of is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
3.
In Problems 1 -6, a graph of is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
4.
In Problems 1 -6, a graph of is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
5.
In Problems 1-6, a graph of y = f(x) is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.
(a) and (b)
6.
In Problems 7-10, use the graph of and the given c-value to find the following, whenever they exist.
(a)
(b)
(c)
(d)
7.
In Problems 7-10, use the graph of and the given c-value to find the following, whenever they exist.
(a)
(b)
(c)
(d)
8.
9EIn Problems 7-10, use the graph of y = and the given c-value to find the following, whenever they exist.
(a) (b)
(c) (d)
10.
In Problems 11-14, complete each table and predict the limit, if it exists.
11.
X f(x)
0.9
0.99
0.999
1.001
1.01
1.1
12E13EIn Problems 11-14, complete each table and predict the limit, if it exists.
14.
x
-2.1
-2.01
-2.001
-1.999
-1.99
15EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
16.
17E18EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
19.
20EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
21.
22E23EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
24.
25E26E27E28EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
29.
30E31E32EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
33.
In Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
34.
35EIn Problems 15-40, use properties of limits and algebraic methods to find the limits, if they exist.
36.
37E38E39E40E41E42E43E44E45E46E47E48E49. Use values 0.1, 0.01, 0.001, 0.0001, and 0.00001 to approximate
to three decimal places. This limit equals the special number e that is discussed in Section 5.1, "Exponential Functions,” and Section 6.2, “Compound Interest; Geometric Sequences."
50. (a) If if it exists. Explain your conclusions.
(b) If if it exists. Explain your conclusions.
51E52. If and find
(a)
(b)
(c)
(d)
53E54E55. Revenue The total revenue for a product is given by
where x is the number of units sold. What is
56. Profit If the profit function for a product is given by
Find .
57E58. Sales and training During the first 4 months of employment, the monthly sales S (in thousands of dollars) for a new salesperson depend on the number of hours x of training as follows:
(a) Find .
(b) Find .
59. Advertising and sales Suppose that the daily sales S (in dollars) t days after the end of an advertising campaign are
(a) Find S(0).
(b) Find .
(c) Find .
60E61. Productivity During an 8-hour shift, the rate of change of productivity (in units per hour) of infant activity centers assembled after t hours on the job is
(a) Find .
(b) Find .
(c) Is the rate of productivity higher near the lunch break (at t = 4) or near quitting time (at t = 8)?
62E63. Cost-benefit Suppose that the cost C of obtaining water that contains p percent impurities is given by
(a) Find , if it exists. Interpret this result.
(b) Find , if it exists.
(c) Is complete purity possible? Explain.
64. Cost-benefit Suppose that the cost C of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by
(a) Find .
(b) Find , if it exists.
(c) Can 100% of the particulate pollution be removed? Explain.
65. Federal income tax The following table shows part of a tax rate schedule for single filers. Use this schedule and create a table of values that could be used to find the following limits, if they exist. Let x represent the amount of taxable income and let T(x) represent the tax due.
(a) (b) (c)
Single Filers
If taxable income is over— But not over— The tax is:
$0 $9075 of the amount over $0
$9075 $36,900 $907.50 plus of the amount over 9075
$36,900 $89,350 $5081.25 plus of the amount over 36,900
$89,350 $186,350 $18,193.75 plus of the amount over 89,350
Source: Internal Revenue Service
66E67E68. Airport parking The Hourly Parking Garage at BWI International Airport in Baltimore costs $2 per half hour during the first hour and $4 for each hour or part of an hour for more than 1 hour and up to 5 hours. If is the total charge for t hours in BWI’s Hourly Parking Garage, write a piecewise function that describes for . Use to find the following limits, if they exist.
(a)
(b)
(c)
(d)
Dow Jones Industrial Average The graph in the figure shows the Dow Jones Industrial Average (DJIA) on a particularly tumultuous day. Use the graph for Problems 69 and 70, with t as the time of day and D(t) as the DJIA at time t.
69. Estimate , if it exists. Explain what this limit corresponds to.
Dow Jones Industrial Average The graph in the figure shows the Dow Jones Industrial Average (DJIA) on a particularly tumultuous day. Use the graph for Problems 69 and 70, with t as the time of day and D(t) as the DJIA at time t.
70. Estimate , if it exists. Explain what this limit corresponds to.
71E72E1. Find any x-values where the following functions are discontinuous.
(a) (b)
2CP3CPIn Problems 1 and 2, refer to the figure. For each given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. If the function is discontinuous, state which of the three conditions that define continuity is not satisfied.
1. (a) (b) (c) (d)
In Problems 1 and 2, refer to the figure. For each given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. If the function is discontinuous, state which of the three conditions that define continuity is not satisfied.
2. (a) (b) (c) (d)
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
3.
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
4.
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
5.
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
6.
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
7.
In Problems 3-8, determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
8.
9E10EIn Problems 9-16, determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing each function with a graphing utility.
11.
12EIn Problems 9-16, determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing each function with a graphing utility.
13.
14E15EIn Problems 9-16, determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing each function with a graphing utility.
16.
17E18E19E20E
Each of Problems 21-24 contains a function and its graph. For each problem, answer parts (a) and (b).
(a) Use the graph to determine, as well as you can,
(i) vertical asymptotes, (ii) lim .
(iii) . (iv) horizontal asymptotes.
(b) Check your conclusions in (a) by using the functions to determine items (i)-(iv) analytically
21.
Each of Problems 21-24 contains a function and its graph. For each problem, answer parts (a) and (b).
(a) Use the graph to determine, as well as you can,
(i) vertical asymptotes, (ii) lim .
(iii) . (iv) horizontal asymptotes.
(b) Check your conclusions in (a) by using the functions to determine items (i)-(iv) analytically
22.
23E24E25E26E27E28E29E30E31E32E33E34E35EIn Problems 35 and 36, complete (a)-(c). Use analytic methods to find (a) any points of discontinuity and (b) limits as and . (c) Then explain why, for these functions, a graphing calculator is better as a support tool for the analytic methods than as the primary tool for investigation.
36.
37EFor Problems 37 and 38, let
be a rational function.
38. (a) If m > n, show that and hence that
y = 0 is a horizontal asymptote.
(b) If m < n, find What does this say about horizontal asymptotes?
39. Sales volume Suppose that the weekly sales volume (in thousands of units) for a product is given by
where p is the price in dollars per unit. Is this function continuous
(a) for all values of ? (b) at p = 24?
(c) for all ?
(d) What is the domain for this application?
40. Worker productivity Suppose that the average number of minutes M that it takes a new employee to assemble one unit of a product is given by
where t is the number of days on the job. Is this function continuous
(a) for all values of t? (b) at t = 14?
(c) for all ?
(d) What is the domain for this application?
41. Demand Suppose that the demand for a product is defined by the equation
where p is the price and q is the quantity demanded.
(a) Is this function discontinuous at any value of q? What value?
(b) Because q represents quantity, we know that . Is this function continuous for ?
42. Advertising and sales The sales volume y (in thousands of dollars) is related to advertising expenditures x (in thousands of dollars) according to
(a) Is this function discontinuous at any points?
(b) Advertising expenditures x must be nonnegative. Is this function continuous for these values of x?
43. Annuities If an annuity can make an unending number of equal payments at the end of the interest periods, it is called a perpetuity. If a lump sum investment of
is needed to result in n periodic payments of R when the interest rate per period is i, then
(a) Evaluate to find a formula for the lump sum payment for a perpetuity.
(b) Find the lump sum investment needed to make payments of $100 per month in perpetuity if interest is 12% compounded monthly.
44. Response to adrenalin Experimental evidence suggests that the response y of the body to the concentration x of injected adrenalin is given by
where a and b are experimental constants.
(a) Is this function continuous for all x?
(b) On the basis of your conclusion in part (a) and the fact that in the context of the application , must a and b both be positive, both be negative, or have opposite signs?
45. Cost-benefit Suppose that the cost C of removing p percent of the impurities from the wastewater in a manufacturing process is given by
Is this function continuous for all those p
values for which the problem makes sense?
46. Pollution Suppose that the cost C of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by
Describe any discontinuities for C(p). Explain what each discontinuity means.
47. Pollution The percent p of particulate pollution that can be removed from the smokestacks of an industrial plant by spending C dollars is given by
Find the percent of the pollution that could be removed if spending C were allowed to increase without bound. Can 100% of the pollution be removed? Explain.
48E49E50. Calories and temperature Suppose that the number of calories of heat required to raise 1 g of water (or ice) from is given by
(a) What can be said about the continuity of the function f(x)?
(b) What happens to water at that accounts for the behavior of the function at ?
52E1. Find the average rate of change of over [1, 4].
2. For the function , find each of the following.
(a)
(b)
(c)
(d)
3. Which of the following are given by ?
(a) The slope of the tangent when
(b) The y-coordinate of the point where
(c) The instantaneous rate of change of
(d) The marginal revenue at is the revenue function
4. Must a graph that has no discontinuity, corner, or cusp at x = c be differentiable at x = c?
In Problems 1-4, for each given function, find the average rate of change over each specified interval.
In Problems 1-4, for each given function, find the average rate of change over each specified interval.
2. over (a) [-1, 2] and (b) [1, 10]
In Problems 1-4, for each given function, find the average rate of change over each specified interval.
3. For f(x) given by the table, over (a) [2, 5] and (b) [3.8, 4]
X 0 2 2.5 3 3.8 4 5
f(x) 14 20 22 19 17 16 30
In Problems 1-4, for each given function, find the average rate of change over each specified interval.
4. For f(x) given in the table, over (a) [3, 3.5] and (b) [2, 6]
X 1 2 3 3.5 3.7 6
f(x) 40 25 18 15 18 38
5. Given , find the average rate of change of f(x) over each of the following pairs of intervals.
(a) [2.9,3] and [2.99,3] (b) [3, 3.1] and [3, 3.01]
(c) What do the calculations in parts (a) and (b) suggest the instantaneous rate of change of f(x) at x = 3 might be?
6. Given , find the average rate of change of f(x) over each of the following pairs of intervals.
(a) [1.9, 2] and [1.99, 2]
(b) [2, 2.1] and [2, 2.01]
(c) What do the calculations in parts (a) and (b) suggest the instantaneous rate of change of f(x) at x = 2 might be?
7E8. In Example 6 in this section, we were given
. Find
(a) the instantaneous rate of change of f(x) at x = 6.
(b) the slope of the tangent to the graph of y = f(x) at x = 6.
(c) the point on the graph of y = f(x) at x = 6.
EXAMPLE 6 Tangent Line
Given , find
(a) the derivative of f(x) at any point (x, f(x)).
(b) the slope of the tangent to the curve at (1, 16).
(c) the equation of the line tangent to at (1.16).
Solution
(a) The derivative of f(x) at any value x is denoted by f'(x) and is
(b) The derivative is , so the slope of the tangent to the curve at (1,16) is .
(c) The equation of the tangent line uses the given point (1, 16) and the slope m = 8. Using gives .
The discussion in this section indicates that the derivative of a function has several interpretations.
9. Let .
(a) Use the definition of derivative and the Procedure/Example box on page 562 to verify that .
(b) Find the instantaneous rate of change of .
(c) Find the slope of the tangent to the graph of .
(d) Find the point on the graph of .
10. Let .
(a) Use the definition of derivative and the
Procedure/Example box on page 562 to verify that .
(b) Find the instantaneous rate of change of .
(c) Find the slope of the tangent to the graph of .
(d) Find the point on the graph of .
In Problems 11-14, the tangent line to the graph of f(x) at = 1 is shown. On the tangent line, P is the point of tangency and A is another point on the line.
(a) Find the coordinates of the points P and A.
(b) Use the coordinates of P and A to find the slope of the tangent line.
(c) Find f'(l).
(d) Find the instantaneous rate of change of at P.
12EIn Problems 11-14, the tangent line to the graph of f(x) at = 1 is shown. On the tangent line, P is the point of tangency and A is another point on the line.
(a) Find the coordinates of the points P and A.
(b) Use the coordinates of P and A to find the slope of the tangent line.
(c) Find f'(l).
(d) Find the instantaneous rate of change of at P.
14EFor each function in Problems 15-18, find
(a) the derivative using the definition.
(b) the instantaneous rate of change of the function any value and at the given value.
(c) the slope of the tangent at the given value.
15.
For each function in Problems 15-18, find
(a) the derivative using the definition.
(b) the instantaneous rate of change of the function any value and at the given value.
(c) the slope of the tangent at the given value.
16.
17E18E For each function in Problems 19-22, approximate f'(a) in the following ways.
(b) Graph the function on a graphing calculator. Then zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see.
19.
20E21E For each function in Problems 19-22, approximate f'(a) in the following ways.
(b) Graph the function on a graphing calculator. Then zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see.
23E24EIn the figures given in Problems 25 and 26, at each point A and B, draw an approximate tangent line and then use it to complete parts (a) and (b).
(a) Is greater at point A or at point B? Explain.
(b) Estimate at point B.
26E27E28E29E30. If the instantaneous rate of change of g(x) at (- 1, -2) is 1/2, write the equation of the line tangent to the graph of g(x) at x = -1.
Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In Problems 31 and 32, use the graph of the function given in Figure 9.26.
31. (a) Over what interval(s) (a) through (d) is the rate of change of f(x) positive?
(b) Over what interval(s) (a) through (d) is the rate of change of f(x) negative?
(c) At what point(s) A through E is the rate of change of f(x) equal to zero?
Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In Problems 31 and 32, use the graph of the function given in Figure 9.26.
32. (a) At what point(s) A through E does the rate of change of f(x) change from positive to negative? (b) At what point(s) A through E does the rate of change of f(x) change from negative to positive?
33. Given the graph of ) in Figure 9.27, determine for which x-values A, B, C, D, or E the function is (a) continuous. (b) differentiable.
34E35E36E37EIn Problems 35-38, (a) find the slope of the tangent to the graph of at any point, (b) find the slope of the tangent at the given point, (c) write the equation of the line tangent to the graph of at the given point, and (d) graph both and its tangent line (use a graphing utility).
38.
39. Total cost Suppose total cost in dollars from the production of x printers is given by
Find the average rate of change of total cost when production changes
(a) from 100 to 300 printers.
(b) from 300 to 600 printers.
(c) Interpret the results from parts (a) and (b).
40. Average velocity If an object is thrown upward at 64 feet per second from a height of 20 feet, its height S after t seconds is given by
What is the average velocity in the
(a) first 2 seconds after it is thrown?
(b) next 2 seconds?
41. Demand If the demand for a product is given by
what is the average rate of change of demand when p increases from
(a) 1 to 25?
(b) 25 to 100?
42. Revenue If the total revenue function for a blender is
where x is the number of units sold, what is the average rate of change in revenue as x increases from 10 to 20 units?
43E44. b The figure shows the percent of the U.S. population that was foreign-born for selected years from 1910 and projected to 2020.
(a) Use the figure to find the average rate of change in the percent of the U.S. population that was foreign- horn from 1960 to 2020. Interpret your result.
(b) From the figure, determine for which two consecutive data points the average rate of change in the percent of foreign-born was (i) closest to zero and (ii) furthest from zero.
Foreign-Born Population Percent:
1910-2020
45. Marginal revenue The revenue function for a home theater system is
dollars
where x denotes the number of units sold.
(a) What is the function that gives marginal revenue?
(b) What is the marginal revenue if 50 units are sold, and what does it mean?
(c) What is the marginal revenue if 140 units are sold, and what does it mean?
(d) What is the marginal revenue if 150 units are sold?
(e) As the number of units sold passes through 150, what happens to revenue?
46. Marginal revenue Suppose the total revenue function
for a blender is
dollars
where x is the number of units sold.
(a) What function gives the marginal revenue?
(b) What is the marginal revenue when 600 units are sold, and what does it mean?
(c) What is the marginal revenue when 2000 units are sold, and what does it mean?
(d) What is the marginal revenue when 1800 units are sold, and what does it mean?
47. Labor force and output The monthly output at the Olek Carpet Mill is
where x is the number of workers employed at the mill. If there are currently 50 workers, find the instantaneous rate of change of monthly output with respect to the number of workers. That is, find Q'(50).
48. Consumer expenditure Suppose that the demand for x units of a product is
where p dollars is the price per unit. Then the consumer expenditure for the product is
What is the instantaneous rate of change of consumer expenditure with respect to price at
(a) any price p? (b) (c)
49. Profit Suppose that the profit function for the monthly sales of a car by a dealership is
where x is the number of cars sold. What is the instantaneous rate of change of profit when
(a) 200 cars are sold? Explain its meaning.
(b) 300 cars are sold? Explain its meaning.
50. Profit It the total revenue function and the total cost function for a toy are
what is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning.
51. Heat index The highest recorded temperature in the state of Alaska was , occurring on June 27,1915, at Fort Yukon. The heat index is the apparent temperature of the air at a given temperature and humidity level. If x denotes the relative humidity (in percent), then the heat index (in degrees Fahrenheit) for an air temperature of can be approximated by the function
(a) At what rate is the heat index changing when the humidity is ?
(b) Write a sentence that explains the meaning of your answer in part (a).
52. Receptivity In learning theory, receptivity is defined as the ability of students to understand a complex concept. Receptivity is highest when the topic is introduced and tends to decrease as time passes in a lecture. Suppose that the receptivity of a group of students in a mathematics class is given by
where t is minutes after the lecture begins.
(a) At what rate is receptivity changing 10 minutes after the lecture begins?
(b) Write a sentence that explains the meaning of your answer in part (a).
53E54E1. True or false: The derivative of a constant times a function is equal to the constant times the derivative of the function.
2. True or false: The derivative of the sum of two functions is equal to the sum of the derivatives of the two functions.
3. True or false: The derivative of the difference of two functions is equal to the difference of the derivatives of the two functions.
4. Does the Coefficient Rule apply to , where c is a constant? Explain.
5CP