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All Textbook Solutions for Calculus and Its Applications (11th Edition)
Find .
47.
Find .
48.
Find dydu,dudx,anddydx. y=(u+1)(u1)andu=x3+1Find dydu,dudx,anddydx. y=u(u+1)andu=x32xFind dydx for each pair of functions. y=5u2+3u,whereu=x3+1Find for each pair of functions.
52.
Find dydx for each pair of functions. y=73u,whereu=x29Find dydx for each pair of functions. y=2u+53,whereu=x2xFind dydx for each pair of functions. Find dydxify=1u2+uandu=5+3t.Find dydx for each pair of functions. Find dydtify=1u2+uandu=5+3t.57. Find an equation for the tangent line to the graph of at the point .
Find an equation for the tangent line to the graph of y=x2+3x at the point (1,2).59. Find an equation for the tangent line to the graph of at the point .
60. Find an equation for the tangent line to the graph of at the point .
Consider g(x)=(6x+12x5)2. a. Find g(x) using the Extended power Rule. b. note that g(x)=36x2+12x+14x220x+25. Find g(x) using the Quotient Rule. c. c) Compare your answers to part (a) and (b). which approach was easier, and why?62. Consider
.
a. Find using the Quotient and the Extended Power Rule.
b. Note that . Find using the product Rule and the Extended Power Rule.
c. Compare your answers to part (a) and (b).
63. Let .
Find .
Let f(u)=u+1u1andg(x)=u=x. Find (fg)(4).Let f(u)=u3andg(x)=u=1+3x2. Find (fg)(2).66. Let .
Find .
For Exercises 67-70, Use the chain Rule to differentiate each function. You may need to apply the more than once.
67.
For Exercises 67-70, Use the chain Rule to differentiate each function. You may need to apply the more than once.
68.
For Exercises 67-70, Use the chain Rule to differentiate each function. You may need to apply the more than once.
69.
For Exercises 67-70, Use the chain Rule to differentiate each function. You may need to apply the more than once.
70.
Total revenue. A total-revenue function is given by R(x)=1000x20.1x, where R(x) is the total revenue, in thousands of dollars, from the sale of x airplanes. Find the rate at which total revenue is changing when 20 airplanes have been sold.Total cost. A total-cost function is given by C(x)=2000(x2+2)1/3+700, where C(x) is the total cost, in thousands of dollars, of producing x airplanes, Find the rate at which total cost is changing when 20 airplanes have been produced.73. Total profit. Use the total-cost and total revenue functions in Exercises 71 and 72 to find the rate at which total profit is changing when x airplanes have been produced and sold.
74. Total cost. A company determine that its total cost, in thousands of dollars, for producing, for x chairs is
,
and it plan to boost production t months from now according to the function
.
How fast will costs be rising 4 months from now?
Consumer credit. The total outstanding consumer credit of the united states (in billions of dollars) can be modeled by the function C(x)=9.26x485.27x3+287.24x2309.12x+2651.4, where x is the number of your since 2008. (Source; Based on data from Federalreserve.gov.) a. Find dC/dx. b. Interpret the meaning of dC/dx. c. Using this model, estimate how quickly outstanding consumer credit was rising in 2014.Utility. Utility is a type of function that occurs in economics. When a consumer receives x units of a product, a certain amount of pleasure, or utility, U, is derived. Suppose the utility related to the number of ticket x sold for a ride at a country fair is U(x)=802x+13x+4. dUdx=200(2x+1)1/2(3x+4)3/2 Find the rate at which utility at changes with respect to number of tickets bought.Compound interest. If 1000 is invested at interest rate r, compounded quarterly, 5 yr it will grow to an amount, A, given by (See Section R.1) A=1000(1+r4)20. a. Find the rate of change, dA/dr. b. Interpret the meaning of dA/dr.Compound interest. If 1000 is invested at interest rate r compared annually, in 3 yr it will grow to an amount A given by (see Section R.1) A=1000(1+r)3. a. Find the rate of change, dA/dr. b. b) Interpret the meaning of dA/dr.79. Business profit. French’s Electronics is selling laptop computers. It determines that its total profit, in dollars, is given by
,
where x is the number of units produced and sold. Suppose that x is a function of time, in months, where .
a. Find the total profit as a function of time t.
b. Find the rate of change of total profit when .
Consumer demand. Suppose the demand function for a new autobiography is given by D(p)=80,000p, and that price p is a function of time, given by p=1.6t+9, where t is in days. a. Find the demand as a function of time t. b. Find the rate of change of the quantify demand when t=100days.Chemotherapy. The dosage for Carboplatin chemotherapy drugs depends on several parameters for the particular drug as well as the age, weight, and sex of the patient. For female, the formulas giving the dosage for such drugs are D=0.85A(c+25)andc=(140y)w72x, where A and x depend on which drug is used, D is the dosage in milligrams (mg), c is called the clearance, y is the patients age in years, and w is the patients weight in kilograms (kg). (Source; U.S. Oncology.) a. Suppose a patient is a 45-year-old woman and the drug has parameters A=5 and x=0.6. Use this information to write formulas for D and c that give D as a function of c and c as a function of w. b. Use your formulas from part (a) to compute dD/dc. c. Use your formulas from part (a) to compute dc/dw. d. Compute dD/dw. e. e) Interpret the meaning of the derivative dD/dw.If f(x) is a function, then (f)(x)=f(f(x)) is the composition of f with itself. This is called an iterated function, and the composition can be repeated many times. For example, (fff)(x)=f(f(f(x))). Iterated functions are very useful in many areas, including finance (compound interest is a simple case) and the science (in weather forecasting, for example). For each function, use the Chain Rule to find the derivative. If f(x)=x2+1,findddx[(ff)(x)].If f(x) is a function, then (f)(x)=f(f(x)) is the composition of f with itself. This is called an iterated function, and the composition can be repeated many times. For example, (fff)(x)=f(f(f(x))). Iterated functions are very useful in many areas, including finance (compound interest is a simple case) and the science (in weather forecasting, for example). For each function, use the Chain Rule to find the derivative. If f(x)=x+x,findddx[(ff)(x)].If is a function, then is the composition of with itself. This is called an iterated function, and the composition can be repeated many times. For example, . Iterated functions are very useful in many areas, including finance (compound interest is a simple case) and the science (in weather forecasting, for example). For each function, use the Chain Rule to find the derivative.
84. If .
If f(x) is a function, then (f)(x)=f(f(x)) is the composition of f with itself. This is called an iterated function, and the composition can be repeated many times. For example, (fff)(x)=f(f(f(x))). Iterated functions are very useful in many areas, including finance (compound interest is a simple case) and the science (in weather forecasting, for example). For each function, use the Chain Rule to find the derivative. If f(x)=x+x,findddx[(fff)(x)].Differentiate. y=(2x3)3+1Differentiate.
87.
Differentiate. y=(xx1)389EDifferentiate. y=1x21xDifferentiate. y=(x2x1x2+1)3Differentiate.
92.
93E94E95. The Extended Power Rule (for positive integer powers can be verified using the Product Rule. For example, If then the Product Rule is applied by recognizing that . Therefore,
.
a. Use the Product Rule to show that .
b. Use the Product Rule to show that .
96. The following is the beginning of an alternative proof of the Quotient Rule that uses the Product Rule and the Power Rule. Complete the proof, giving reasons for each step.
Proof: Let
.
Then
.
Therefore, …
For the function in each of Exercises 97 and 98, graph and over the given interval. Then estimate points at which the line tangent to f is horizontal.
97.
For the function in each of Exercises 97 and 98, graph f and f over the given interval. Then estimate points at which the line tangent to f is horizontal. f(x)=6x33x248x+45;[5,5]99EFind the derivative of each of the following function functions. Then use a calculate to check the results. f(x)=(2x1+x3)5Find .
1.
Find d2y/dx2. y=x5+9Find .
3.
Find .
4.
Find .
5.
Find d2y/dx2. y=4x2+3x1Find d2y/dx2. y=7x+2Find d2y/dx2. y=6x3Find .
9.
Find .
10.
Find .
11.
Find d2y/dx2. y=x4Find f(x). f(x)=x35xFind f(x). f(x)=x4+3xFind .
15.
Find .
16.
Find .
17.
Find f(x). f(x)=4x3Find .
19.
Find f(x). f(x)=(x3+2x)6Find .
21.
Find f(x). f(x)=(2x23x+1)10Find .
23.
Find f(x). f(x)=(x21)23Find y. y=x3/25xFind y. y=x2/3+4xFind y. y=(x3x)3/4Find y. y=(x4+x)2/3Find .
29.
Find y. y=2x5/4+x1/2Find y. y=2x3+1x2Find y. y=3x41xFind y. y=(x2+3)(4x1)Find y. y=(x2+3)(4x1)Find y. y=3x+12x3Find y. y=2x+35x1For y=x5, find d4y/dx4.38. For , find .
39. For , find .
40. For , find .
41. For , find .
For f(x)=x2x1/2, find f(4)(x).For g(x)=x43x37x26x+9, find g(6)(x).44. For , find .
Given s(t)=10t2+2t+5, where s(t) is in meters and t is in seconds, find each of the following. a. a) v(t) b. b) a(t) c. c) The velocity and acceleration when t=1 secGiven s(t)=t3+t where s(t) is in feet and t is in seconds, find each of the following. a. v(t) b. a(t) c. The velocity and acceleration when t=4sec47. Given
,
where is in miles and t is in hours, find each of the following.
a.
b.
c. The velocity and acceleration when
d. When the distance function is given by a linear function, we have uniform motion. What does uniform motion mean in term of velocity and acceleration?
48. Given
,
where is in meters and t is in seconds, find each of the following.
a.
b.
c. The velocity and acceleration when
Free fall. When an object is dropped the distance it falls in t seconds, assuming negligible air resistance, is given by s(t)=16t2 where s(t) is in feet. Suppose a medics reflex hammer falls from a hovering helicopter. Find (a) how far the hammer falls in 3 sec, (b) how fast the hammer is traveling 3 sec after being dropped, and (c) the hammers acceleration after it has been falling for 3 sec.50. Free fall. (See Exercises 49.) Suppose a worker drops a bolt from a scoffold high above a work site. Assuming negligible air resistance, find (a) how far the bolt falls in 2 sec. (b) how fast the bolt is traveling 2 secs after being dropped, an (c) the bolt’s acceleration after it has been falling for 2 sec.
Free fall. Find the velocity and acceleration of the stone in Exercises 3 after it has been falling for 2 sec.52. Free fall. Find the velocity and acceleration of the stone in Exercises 3 after it has been falling for 3 sec.
53. The following graph describes a bicycle racer’s distance from a roadside television camera.
a. When is the bicyclist’s velocity the greatest? How can you-tell?
b. Is the bicyclist’s acceleration positive or negative? How can you-tell?
The following graph describes an airplanes distance from its last point of rest. a. Is the planes velocity greater at t=6sec or t=20sec? How-can you tell? b. Is the planes acceleration positive or negative? How can you tell?Sales. The following graph represents the sales, y, of a new video game after t weeks on the market. a. In what open interval(s) f(t)=0? b. In what open interval(s) f(t)=0? c. In what open interval(s) f(t)0? d. In what open interval(s) f(t)0? e. Describe in works and graphically the difference in meaning between sales are increasing and the rate of sales is increasing.Velocity and acceleration. The following graph describe the distance of Jesses car from home as a function of time t. a. In what open interval(s) is Jesses car accelerating? b. In what open interval(s) is Jesses car decelerating? c. In what open interval(s) is Jesses car maintaining a constant velocity?57. Sales. A company determine that monthly sales , in thousands of dollars, after t months of marketing a product are given by
.
a. Find .
b. Find .
c. c) Interval the meaning of your answers to parts (a) and (b).
Sales. Nadias fashions discovers that the number of items. Sold N(t)=2t33t2+2t. days after launching a new sales promotion is given by N(t)=2t33t2+2t. a. Find N(1),N(2),andN(4). b. Find N(1),N(2),andN(4). c. c) Interpret the meaning of your answers to part (a) and (b).Population. The function P(t)=2000t4t+75 gives the population of deer in an area after t months. a. Find p(10),p(50),andp(100). b. Find p(10),p(50),andp(100). c. c) Interpret the meaning of your answers to parts (a) and (b). What is happening to this population of deer in the long term?60. Medicine. A medication is injected into the bloodstream, where it is quickly metabolized. The percent concentration p of the medication after t minutes in the bloodstream is modeled by the function .
a. Find .
b. Find .
c. c) Interpret the meaning of your answers to part (a) and (b). What is happening to the concentration of medication in the bloodstream in the long term?
61EFind y for each function. y=12x+1Find y for each function. y=x+1x1Find y for each function. y=xx1For y=xk, find d5y/dx5.66E67E68E69. Free fall. On Earth, all free-fall distance functions are of the form where t is in seconds and is in meters. The second derivative always has the same value. What does that value represent?
the value of the gravitational constant for Earth
Free fall. On the moon, all free-fall distance functions are of the form s(t)=0.81t2, where t is in seconds and s(t) is in meters. An object is dropped from a height of 200 meters above the moon. After t=2sec, a. How far has the object fallen? b. How fast is it traveling? c. What is its acceleration? d. Explain the meaning of the second derivative of this free-fall function.71. Hang time. On Earth, an object travels after of free fall. Thus, by symmetry, an athlete would require 1 sec to jump high, and another second to come back down. Is it possible for a person to stay in the air (have a “hang time” of) 2 sec? can a person have a hang time of ? What do you think is the longest possible hang time achievable by humans jumping from level ground?
Free fall. Skateboarder Danny way free-fell 28 ft from the fender Stratocaster Guitar atop the Hard Rock Hotel Casino in Las Vegas onto a ramp below. The distance s(t) in feet, traveled by a body falling freely from rest t seconds is approximated by s(t)=16t2. Estimate Ways velocity at the moment he touched down onto the ramp. (Note: Use the result from Exercises 26 in section R.1.)An object rolls 1 m in 1 min. Below are four possible graphs showing the objects distance traveled as a function of time. Match each graph with one of the following descriptions: a. The object rolled unimpeded along a level table. b. The object rolled along a level concrete walkway, then rolled along a level grassy lawn. c. The object rolled down an incline. d. the object rolled along a level table, then down an incline.A bicyclists distance from her starting point is given by y=s(t). Suppose the graph of s has a corner. Give three situations in which a corner, rather than a smooth curve, can occur. What can you conclude about s(t) and s(t) at the corner of the graph of s?75E76E77EIndeterminate Forms and IHopitals Rule, Let f and g be differentiable over an open interval containing x=a If limxa(f(x)g(x))=00orlimxa(f(x)g(x))= and if limxa(f(x)g(x)) exists, then limxa(f(x)g(x))=limxa(f(x)g(x)). The forms 0/0 and / are said to be indeterminate. In such cases, the limit may exist, and IHopitals Rule a way to find the limit using differentiation. For example, in Example 1 of section 1.1, we showed that limx1(x21x1)=2. Since, for x=1 we have (x21)(x1)=0/0, we differentiate the numerator and denominator separately. And reevaluate the limit: limx1(x21x1)=limx1(2x1)=2. Use this method to find the following limits. Be sure to check that the initial substitution results in an indeterminate from. limx3(x3x24x29)79E80E81E82EIndeterminate Forms and IHopitals Rule, Let f and g be differentiable over an open interval containing x=a If limxa(f(x)g(x))=00orlimxa(f(x)g(x))= and if limxa(f(x)g(x)) exists, then limxa(f(x)g(x))=limxa(f(x)g(x)). The forms 0/0 and / are said to be indeterminate. In such cases, the limit may exist, and IHopitals Rule a way to find the limit using differentiation. For example, in Example 1 of section 1.1, we showed that limx1(x21x1)=2. Since, for x=1 we have (x21)(x1)=0/0, we differentiate the numerator and denominator separately. And reevaluate the limit: limx1(x21x1)=limx1(2x1)=2. Use this method to find the following limits. Be sure to check that the initial substitution results in an indeterminate from. limx(4x2+x32x2+1)84E85E86E87EFor the distance function in each of Exercises 85-88, graph s, v, and a over the given interval. Then use the graphs to determine the point(s) at which the velocity switches from increasing to decreasing or from decreasing to increasing.
88.
For the distance function in each of Exercises 85-88, graph s, v, and a over the given interval. Then use the graphs to determine the point(s) at which the velocity switches from increasing to decreasing or from decreasing to increasing.
89.
These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each description in column A with the most appropriate graph in column B. Column A Column B A function with a relative maximum but no absolute extremaThese review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect.
Concept Reinforcement
Match each description in column A with the most appropriate graph in column B.
2.
Column A Column B
A function with both a vertical asymptote and a horizontal asymptote
These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each description in column A with the most appropriate graph in column B. Column A Column B A function that is concave up and decreasingThese review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect. Concept Reinforcement Match each description in column A with the most appropriate graph in column B. Column A Column B A function that is concave up and increasingThese review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect.
Concept Reinforcement
Match each description in column A with the most appropriate graph in column B.
5.
Column A Column B
A function with three critical values
These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect.
Concept Reinforcement
Match each description in column A with the most appropriate graph in column B.
6.
Column A Column B
A function with one critical value and a second derivative that is always positive
These review exercises are for test preparation. They can also be used as a practice test. Answers are at the back of the book. The red bracketed section references tell you what part(s) of the chapter to restudy if your answer is incorrect.
Concept Reinforcement
Match each description in column A with the most appropriate graph in column B.
7.
Column A Column B
A function with a first derivative that is always positive
In Exercises 8-16, classify each statement as either true or false.
8. Every continuous function has at least one critical value.
In Exercises 8-16, classify each statement as either true or false.
9. If a continuous function has extrema, they will always occur where.
In Exercises 8-16, classify each statement as either true or false.
10. If , Then is a relative minimum.
In Exercises 8-16, classify each statement as either true or false.
11. If , then cannot be a relative minimum.
In Exercises 8-16, classify each statement as either true or false.
12. If the graph of has a horizontal asymptote, then the degree of the polynomial P (x) is less than or equal to that of the polynomial.
In Exercises 8-16, classify each statement as either true or false.
13. Absolute extrema of a continuous function f always occur at the endpoints of a closed interval.
In Exercises 8-16, classify each statement as either true or false.
14. In general, if f is a continuous function, then.
In Exercises 8-16, classify each statement as either true or false. If the elasticity of demand at a certain price x is 1, then revenue is maximizes at this price. [2.7]In Exercises 8-16, classify each statement as either true or false.
16. In a related-rate problem, the variables are commonly differentiated implicitly with respect to time, t.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function. [2.1] f(x)=43xx2For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
18.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
19.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function. [2.1] f(x)=4+(x1)3For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
21.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
22.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
23.
For each function given, find any extrema, along with the x-value at which they occur. Then sketch a graph of the function.
24.
Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down.
25.
Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down.
26.
Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down.
27.
Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down.
28.
Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down. [2.2] f(x)=3x4+2x33x2+1 (Round to three decimal places where appropriate.)Sketch the graph of each function. List any minimum or maximum values, where they occur, and any points of inflection. State where the function is increasing or decreasing and where it is concave up or concave down. [2.2] f(x)=15x5+34x443x3+8 (Round to three decimal places where appropriate.)Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3] f(x)=2x+5x+1Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3]
32.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3] f(x)=5x216Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3]
34.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3]
35.
Sketch the graph of each function. Indicate where each function is increasing or decreasing, the coordinates at which any relative extrema occur, where any asymptotes occur, where the graph is concave up or concave down, and where any intercepts occur. [2.3]
36.
For each function, find any absolute extrema over the indicated interval. Indicate the x-value at which each extremum occurs. Where no interval is specified, use the real line. [2.4] f(x)=x42x2+3;[0,3]For each function, find any absolute extrema over the indicated interval. Indicate the x-value at which each extremum occurs. Where no interval is specified, use the real line. [2.4]
38.
For each function, find any absolute extrema over the indicated interval. Indicate the x-value at which each extremum occurs. Where no interval is specified, use the real line. [2.4] f(x)=x+50x;(0,)For each function, find any absolute extrema over the indicated interval. Indicate the x-value at which each extremum occurs. Where no interval is specified, use the real line. [2.4] f(x)=x42x2+1MaximizeQ=xy,where3x+4y=24. [2.5]42. Find the minimum value of.
Business: maximizing profit. If R(x)=52x0.5x2andC(x)=22x1, find the maximum profit and the number of units that must be produced and sold in order to yield this maximum profit Assume that R(x)andC(x) are in dollars. [2.5]44. Business: minimizing cost. A rectangular box with a square base and a cover is to have a volume of .If the cost of material is for the sides, what should the dimensions of the box be in order to minimize the cost? [2.5]
Business: minimizing inventory cost. Venice Cyclery sells 360 hybrid bicycles per year. It costs $8 to store one bicycle for a year. To reorder, there is a fixed cost of $10, plus $2 for each bicycle. How many times per year should the shop order bicycles, and in what lot size, in order to minimize inventory costs? [2.5]46REFor Exercises 47 and 48, . [2.6]
47. Find
.
For Exercises 47 and 48, y=f(x)=2x3+x. [2.6] a. Find dy. b. Find dywhenx=2anddx=0.01Approximate 83usingyf(x)x. [2.6]50. Physical science: waste storage. The Waste Isolation pilot plant (WIPP) in New Mexico consists of large rooms carved into a salt deposit and is used for long-term storage of radioactive waste. (Source: www.wipp.energy.gov.) A new storage room in the shape of a cube with an edge length of 200 ft is to be carved into the salt. Use a differential to estimate the potential difference in the volume of this room if the edge measurements have a tolerance of. [2.6]
51. Economics: elasticity of demand. Consider the demand function given by
.
[2.7]
a. Find the elasticity.
b. Find the elasticity at x = 10, and state whether the demand is elastic or inelastic.
c. Find the elasticity at x = 50, and state whether the demand is elastic or inelastic.
d. At a price of $25, will a small increase in price cause total revenue to increase or
decrease?
e. Find the price at which total revenue is a maximum.
52. Differentiate the following implicitly to find. Then find the slope of the curve at the given point.
[2.8]
53. A ladder 25 ft long leans against a vertical wall. If the bottom moves away from the wall at the rate of 6 ft/sec, how fast is the height of the top changing when the lower end is 7 ft from the wall? [2.8]
54. Business: total revenue, cost, and profit. Find the rates of change, with respect to time, of total revenue, cost, and profit for.
. [2.8]
55. Find the absolute maximum and minimum values, if they exist, over the indicated interval.
[2.4]
57RE58. Find any relative maxima and minima of
.
59. Determine a rational function f whose graph has a vertical asymptote at and horizontal asymptote at and include the point. [2.3]
Use a calculator to estimate the relative extrema of each function. [2.1, 2.2] f(x)=3.8x518.6x3Use a calculator to estimate the relative extrema of each function. [2.1, 2.2]
61.
62REFind all relative minimum or maximum values as well as the x-values at which they occur. State each function is increasing or decreasing. Then sketch a graph of the function f(x)=x24x5Find all relative minimum or maximum values as well as the x-values at which they occur. State each function is increasing or decreasing. Then sketch a graph of the function
2.
Find all relative minimum or maximum values as well as the x-values at which they occur. State each function is increasing or decreasing. Then sketch a graph of the function
3.
Find all relative minimum or maximum values as well as the x-values at which they occur. State each function is increasing or decreasing. Then sketch a graph of the function
4.
Sketch a graph of each function. List any extrema, and indicate any asymptotes or points of inflection.
5.
Sketch a graph of each function. List any extrema, and indicate any asymptotes or points of inflection.
6.
7TSketch a graph of each function. List any extrema, and indicate any asymptotes or points of inflection. f(x)=x9x29TSketch a graph of each function. List any extrema, and indicate any asymptotes or points of inflection. f(x)=8x2411T12T13TFind the absolute maximum and minimum and minimum values, if they exist, of each function over the indicated interval. Where no interval is specified, use the real line. f(x)=x3+x2x+1;[2,12]Find the absolute maximum and minimum and minimum values, if they exist, of each function over the indicated interval. Where no interval is specified, use the real line. f(x)=x2+8.6x+10Find the absolute maximum and minimum and minimum values, if they exist, of each function over the indicated interval. Where no interval is specified, use the real line.
16.
Find the absolute maximum and minimum and minimum values, if they exist, of each function over the indicated interval. Where no interval is specified, use the real line. f(x)=2x+5Find the absolute maximum and minimum and minimum values, if they exist, of each function over the indicated interval. Where no interval is specified, use the real line.
18.
19T20T21TBusiness: maximum profit. Find the maximum profit and the number of units, x, of a leather briefcase that must be produced and sold in order to yield the maximum profit. Assume that R(x)andC(x) are the revenue and cost, in dollars, when x units are produced: R(x)=x2+110x+6, C(x)=1.1x2+10x+80.23. Business: minimizing cost. From a thin piece of cardboard 60 in. by 60 in., square corners are cut out so that the sides can be folded up to make an open box. What dimensions will yield a box of maximum volume? What is the maximum volume?
24. Business: minimizing inventory costs. Ironside Sports sells 1225 tennis rackets per year. It costs $2 to store one tennis racket for a year. To reorder, there is a fixed cost of $1, plus $0.50 for each tennis racket. How many times per year should Ironside order tennis rackets, and in what lot size, in order to minimize inventory costs?
25. .
Approximate50usingyf(x)x.27.
a. Find dy.
b. .
28. Economics: elasticity of demand. Consider the demand function given by
.
a. Find the elasticity.
b. Find the elasticity at x = 1, stating whether demand is elastic or inelastic.
c. Find the elasticity at x = 12, stating whether demand is elastic or inelastic.
d. At a price of $12, will a small increase in price cause total revenue to increase or decrease?
e. Find the price at which total revenue is a maximum.
Differentiate the following implicitly to find dy/dx . Then find the slope of the curve at (1,2): x3+y3=9.30. A spherical balloon has a radius of 15 cm. Use a differential to find the approximate change in the volume of the balloon if the radius is increased or decreased by 0.5 cm. (The volume of a sphere is)
31. A pole 13 ft long leans against a vertical wall. If the lower end moves away from the wall at the rate of 0.4 ft/sec how fast is the upper end sliding down when the lower end is 12 ft from the wall?
32T33TEstimate any extrema of the function given by f(x)=5x330x2+5x.35. Estimate any extrema of the function given by
.
36T37. Business: advertising. The business of manufacturing and selling bowling balls in one of frequent changes. Companies introduce new models to market about every 3 to 4 months. Typically, a new model is created because of advances in technology such as new surface stock or a new way to place weight blocks in a ball. To decide how to best use advertising dollars, companies track sales in relation to amount spent on advertising. Suppose a company has the following data from past sales.
Amount Spent on Advertising (in thousands) Number of Bowling Balls Sold, N
0 $ 8
50 13,115
100 19,780
150 22,612
200 20,083
250 12,430
300 4
a. Use regression to fit linear, quadratic, cubic, and quartic functions to the data.
b. Determine the domain of the function in part (a) that fits the data and the problem situation. How did you arrive at your answer?
c. Determine the maximum value of the function over the domain. How much should the company spend on advertising a new model in order to maximize the number of bowling balls sold?
For Exercises 1–3, do the following.
Graph the reproduction curve, the line, and the harvest function using the same viewing window.
Find the population at which the maximum sustainable harvest occurs. Use both a graphical solution and a calculus solution.
Find the maximum sustainable harvest.
1. , where P is measured in thousands.
For Exercises 13, do the following. Graph the reproduction curve, the line y=P, and the harvest function using the same viewing window. Find the population at which the maximum sustainable harvest occurs. Use both a graphical solution and a calculus solution. Find the maximum sustainable harvest. f(P)=0.025P2+4P, where P is measured in thousands. This is the reproduction curve in the Hudson Bay area for the snowshoe hare.For Exercises 13, do the following. Graph the reproduction curve, the line y=P, and the harvest function using the same viewing window. Find the population at which the maximum sustainable harvest occurs. Use both a graphical solution and a calculus solution. Find the maximum sustainable harvest. f(P)=0.01P2+2P, where P is measured in thousands. This is the reproduction curve in the Hudson Bay area for the lynx.4ETE5ETE6. The table below lists data regarding the reproduction of a certain animal.
a. Use regression to fit a cubic polynomial to these data.
b. Graph the reproduction curve, the line , and the harvest function using the same
viewing window.
c. Graphically determine the population at which the maximum sustainable harvest occurs.
Population, P (in thousands) Population, 1 Year Later
10 9.7
20 23.1
30 37.4
40 46.2
50 42.6
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
1.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
2.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=23x2x2Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
4.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. F(x)=0.5x2+2x11Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. g(x)=1+6x+3x2Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
7.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
8.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
9.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=x33x+6Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=x3+3xFind any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
12.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. F(x)=1x3Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
14.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
15.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
16.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
17.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
18.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=13x32x2+4x1Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
20.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=3x415x2+12Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. g(x)=2x420x2+18Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. G(x)=x+23Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
24.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
25.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function. f(x)=(x+3)2/35Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
27.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
28.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
29.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
30.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
31.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
32.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
33.
Find any relative extrema of each function. List each extremum along with the x-value at which it occurs. Then sketch a graph of the function.
34.
35–68. Check the results of Exercises 1–34 using a calculator.
35.
35–68. Check the results of Exercises 1–34 using a calculator.
36.
Check the results of Exercises 134 using a calculator. f(x)=23x2x235–68. Check the results of Exercises 1–34 using a calculator.
38.
35–68. Check the results of Exercises 1–34 using a calculator.
39.
Check the results of Exercises 134 using a calculator. g(x)=1+6x+3x235–68. Check the results of Exercises 1–34 using a calculator.
41.
Check the results of Exercises 134 using a calculator. G(x)=x3x2x+2Check the results of Exercises 134 using a calculator. f(x)=x33x235–68. Check the results of Exercises 1–34 using a calculator.
44.
35–68. Check the results of Exercises 1–34 using a calculator.
45.
Check the results of Exercises 134 using a calculator. f(x)=3x2+2x335–68. Check the results of Exercises 1–34 using a calculator.
47.
Check the results of Exercises 134 using a calculator. g(x)=2x316Check the results of Exercises 134 using a calculator. G(x)=x36x2+1035–68. Check the results of Exercises 1–34 using a calculator.
50.
35–68. Check the results of Exercises 1–34 using a calculator.
51.
Check the results of Exercises 134 using a calculator. f(x)=x42x335–68. Check the results of Exercises 1–34 using a calculator.
53.