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All Textbook Solutions for Calculus and Its Applications (11th Edition)
44EClassify each statement as either true or false.
1. Riemann sums are a way of approximating the area under a curve by using rectangles.
Classify each statement as either true or false. If a and b are both negative, then abf(x)dx is negative.Classify each statement as either true or false. For any continuous function f defined over [1,7], it follows that 12f(x)dx+27f(x)dx=17f(x)dxClassify each statement as either true or false.
4. Every integral can be evaluated using integration by parts.
Classify each statement as either true or false. x2.exdx=(13x3)ex+C6REMatch each integral in column A with the corresponding anti-derivative in column B. Column A Column B (1+2x) 2 dx a) lnx+C b) x1 +C c) (1+x2 )1 +C d) 12 (1+2x)1 +C e) 2x1/2 +C f) ln(1+x2 )+C g) ln(1+x2 )+C8RE9RE10RE11REBusiness: total cost. The marginal cost, in dollars, of producing the xth car stereo is given by C(x)=0.004x22x+500 Approximate the total cost of producing 300 car stereos by computing the sum i=16C(xi)x with x=50. Find each antiderivative.Find each antiderivative. 20x4dxFind each antiderivative.
14.
15REFind the area under each curve over the Indicated Interval
16.
Find the area under each curve over the Indicated Interval
17.
18REIn each case, give an interpretation of what the shaded region represents.
19.
20RE21REEvaluate.
22.
23REEvaluate.
24. , where
Evaluate.
25. , for g as shown in the graph at right
26REDecide whether abf(x)dx is positive, negative, or zeroDecide whether is positive, negative, or zero
28.
Find the area of the region bounded by y=x2+3x+1 and y=6x1.Find each antiderivative using substitution. Do not use Table 1
30.
Find each antiderivative using substitution. Do not use Table 1 24t54t6+3dt32REFind each antiderivative using substitution. Do not use Table 1 e2xe2x+2dxFind each antiderivative using integration by parts. Do not use Table 1. 3xe4xdxFind each antiderivative using integration by parts. Do not use Table 1. lnx23dx36REFind each antiderivative using integration by parts. Do not use Table 1. x4e3xdx38RE39RE40RE41RE42RE43. Business: total cost. Refer to Exercise 12. Calculate the total cost of producing 300 car stereos.
44. Find the average value of over. .
A particle starts out from the origin. Its velocity in mph after t hours is v(t)=3t2+2t. Find the distance the particle travels during the first 4 hr (from t=0 to t=4 ).46. Business: total revenue. A company estimates that its revenue grows continuously at a rate given by, where t is the number of days since an innovation is introduced. Find the accumulated revenue for the first 4 days.
47RE48RE49RE50REIntegrate using any method. t7(t8+3)11dtIntegrate using any method. ln(7x)dxIntegrate using any method. xln(8x)dx54REFind each antiderivative.
55.
56REFind each antiderivative. x91ln|x|dxFind each antiderivative. ln|x3x4|dxFind each antiderivative. dxx(ln|x|)4Find each antiderivative. xx+33dxFind each antiderivative.
61.
Use a graphing calculator to approximate the area between tile following curves: y=2x22x,y=12x212x3.1. Approximate
by computing the area of each rectangle and adding.
Find each antiderivative.
2.
Find each antiderivative.
3.
Find each antiderivative. (e5x+1x+x3/8)dx(assumex0)Find the area under the curve over the indicated interval. y=xx2;[0,1]Find the area under the curve over the indicated interval. y=4x;[1,3]Give an interpretation of the shaded area.Evaluate.
8.
9T10T11TFind 37f(x)dx, for f as shown in the graph.13TFind each antiderivative using substitution. Assume when appears. Do not consult Table 1.
14.
Find each antiderivative using substitution. Assume u0 when lnu appears. Do not consult Table 1. e0.5xdx16TFind each antiderivative using integration by parts. Do not consult Table 1. xe5xdx18T19T20T21T22T23T24T25. A robot leaving a spacecraft has velocity given by where is in kilometers per hour and t is the number of hours since the robot left the spacecraft. Find the total distance traveled during the first 4 hr.
26TFind each antiderivative using any method. Assume u0 when lnu appears. x5exdx28T29T30TFind each antiderivative using any method. Assume u0 when lnu appears. x4e0.1xdxFind each antiderivative using any method. Assume u0 when lnu appears. xln(13x)dx33TFind each antiderivative using any method. [(lnx)34(lnx)2+5]xdx,x035T36T37T38T39T40TFind each integral.
1.
Find each integral.
2.
Find each integral. 2dxFind each integral. 4dxFind each integral. x1/4dxFind each integral. x1/3dxFind each integral.
7.
Find each integral.
8.
Find each integral. (2t2+5t3)dtFind each integral.
10.
Find each integral.
11.
Find each integral.
12.
Find each integral. x3dxFind each integral. xdxFind each integral. x5dxFind each integral. x23dxFind each integral. dxx4Find each integral. dxx2Find each integral.
19.
Find each integral.
20.
Find each integral.
21.
Find each integral.
22.
Find each integral. 7x23dxFind each integral.
24.
Find each integral. 2e2xdxFind each integral.
26.
Find each integral. e3xdxFind each integral. e5xdxFind each integral.
29.
Find each integral. e6xdxFind each integral. 5e3xdxFind each integral. 2e5xdxFind each integral.
33.
Find each integral.
34.
Find each integral. 23e9xdxFind each integral. 45e10xdxFind each integral.
37.
Find each integral. (2x54e3x)dxFind each integral.
39.
Find each integral. (x4+18x45x2/5)dxFind each integral. (3x+2)2dx (Hint: Expand first.)Find each integral.
42.
Find each integral. (3x5e2e+x7)dx,x0Find each integral. (2e6x3x+x43)dx,x0Find each integral. (7x23e5x8x)dx,x0Find each integral. (4x5+34e6x7x)dx,x0Find f such that: f(x)=x3,f(2)=9Find such that:
48.
Find such that:
49.
Find such that:
50.
Find f such that: f(x)=5x2+3x7,f(0)=9Find f such that: f(x)=8x2+4x2,f(0)=6Find f such that: f(x)=3x25x+1,f(1)=72Find f such that: f(x)=6x24x+2,f(1)=9Find f such that: f(x)=5e2x,f(0)=12Find f such that: f(x)=3e4x,f(0)=74Find such that:
57.
58ECredit Market Debt.
Since 2009, the annual rate of change in the national credit market debt can be modeled by the
function
,
Where is in billions of dollars per year and t is the number of years since 2009. (Source: Based on data from federalreserve.gov/release/g19/current.) Use the preceding information for Exercises 59 and 60.
59. Find the national credit market debt, , since 2009, given that
Credit Market Debt.
Since 2009, the annual rate of change in the national credit market debt can be modeled by the
function
,
Where is in billions of dollars per year and t is the number of years since 2009. (Source: Based on data from federalreserve.gov/release/g19/current.) Use the preceding information for Exercises 59 and 60.
60. What was the national credit market debt in 2013?
Total cost from marginal cost. Belvedere, Inc., determines that the marginal cost, C of producing the thermos is given by C(x)=x32x. Find the total-cost function, C, assuming that C(x) is in dollars and that fixed costs are 7000.62. Total cost from marginal cost. Solid Rock industries determines that the marginal cost, C’ of producing the climbing harness is given by
.
Find the total-cost function, C, assuming that is in dollars and that fixed costs are .
63. Total revenue from marginal revenue. Eloy Chutes determines that the marginal revenue, , in dollars per unit, from selling the parachute is given by
a. Find the total-revenue function, R, assuming that
b. Why is a reasonable assumption?
Total revenue from marginal revenue. Taylor Ceramics determines that the marginal revenue, R, in dollars per unit, from selling the xth vase is given by R(x)=x21R(x)=x33x. a. Find the total-revenue function R, assuming that R(0)=0 b. Why is R(0)=0 a reasonable assuming?Demand from marginal demand. Lessard Company finds that the rate at which the quantify of flameless candles candles that consumer demand changes with respect to price is given by the marginal-demand function D(x)=4000x2 Where x is the price per candle, in dollars. Find the demand function if 1003 candles are demanded by consumers when the price is $4 per candle.66E67. Efficiency of a machine operator. The rate at which a machine operator’s efficiency, E (expressed as a percentage), changes with respect to time t is given by
Where t is the number of hour the operator has been at work.
A machine operator’s efficiency changes with respect to time.
Find , given that the operator’s efficiency after working 2 hr is that is,
Use the answer to part (a) to find the operator’s efficiency after 3 hr; after 5 hr.
With respect to time t is given by
Efficiency of a machine operator. The rate at which a machine operators efficiency, E (expressed as a percentage), changes with respect to time t is given by dEdt=4010t Where t is the number of hours the operator has been at work. Find E(t), given that the operators efficiency after working 3 hr is 56%; that is, E(3)=56. b. use the answer to part (a) to find the operators efficiency after 4 hr; after 7 hr.69. Heart rate. The rate of change in Trisha’s pulse (in beats per minute) t minutes after she stops exercising is given by
a. Find , if Trisha’s pulse is 78 beats per minute 2 min after she stopped exercising.
b. Find Trisha’s pulse rate 4 min after she stops exercising.
c. Find the rate of change in Trisha’s pulse after 4 min.
d. How can and be used to find Trisha’s resting pulse rate?
70. Memory. In a memory experiment, the rate at which students memorize Spanish vocabulary is found to be given by
,
Where is the number of word memorized in t minutes.
Find is the know that
How many words are memorized in 8 min?
71. Physics: height of a thrown baseball. A baseball is thrown directly upward with an initial velocity of 75ft/sec from an initial height
Where v is in feet per second.
a. Find the function h that gives the height (in feet) of the baseball after t seconds.
b. What are the height and the velocity of the baseball after 2 sec of flight?
c. After how many seconds does the ball reach its highest points? (Hint: The ball “stops” for a moment before staring its downward fall.)
d. How high is the ball at its highest point?
e. After how many seconds will the ball hit the ground?
f. What is the ball velocity at the moment it hits the ground?
Physics: height of an object. A football player punts a football, which leaves his foot at a height of 3 ft above the ground and with an initial upward of 70ft/sec. the vertical velocity of the football t seconds after it is punted is given by v(t)=32t+70 Where v is in feet per second. Find the function h that gives the height (in feet) of the football after t seconds. What are the height and the velocity of the football after 1.5 sec? After how many seconds does the ball reach its highest point, and how high is the ball at this point? d. The punt returner catches the football 5 ft above the ground. What is the vertical of the ball immediately before it is caught?73. Population growth. The rates of changes in population for two cities are as followers:
,
,
Where is the number of years since 2000, and both and are measured in people year. In 2000, Alphaville has a population of 5000, and Betaburgh had a population of 3500.
Determine the population models for both cities.
What were the populations of Alphaville and Betaburgh, to the nearest, in 2010?
c. Sketch the graph of each city’s population model, and estimate the year in which the two cities have the same population.
Comparing rates of change. Jim is offered a job that will pay him $50 in the first day, $100 on the second day, $150 on the third day, and so on; thus, the rate of change of his pay t days after starting the job is given by J(t)=50 Larry is offered the same job, but the rate of change of this his pay is given by L(t)=e0.1t, Both J(t) and L(t) are measured in dollars per day. a. Determine the total pay model for Jim and for Larry. b. After 30 day, what is Jims total pay and Larrys total pay? c. On what day does Larrys dally pay first exceed Jims dally pay? d. In general, how does exponential growth compare to linear growth? Explain.75E76ESolve each integral. Each can be found using rules developed in this section, but some algebra may be required. (5t+4)2t4dtSolve each integral. Each can be found using rules developed in this section, but some algebra
may be required.
78.
Solve each integral. Each can be found using rules developed in this section, but some algebra
may be required.
79.
Solve each integral. Each can be found using rules developed in this section, but some algebra may be required. x4+6x27x3dxSolve each integral. Each can be found using rules developed in this section, but some algebra may be required. (t+1)3dt82E83E84E85E86E87EIn Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit.
Total profit from marginal profit. A concert promoter sells x tickets and has a marginal-profit function given by
,
where is in dollars per ticket. This means that the rate of changes of total profit with respect to the number of ticket sold, x, is . Find the total profit from the sale of the first 300 tickets.
In Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit.
Total profit from marginal profit. Poyse Inc. has a marginal-profit function given by
,
where Is in dollars per units. This means that the rate of change of total profit with respect to the number of units produced, x, is . Find the profit from the production and sale of the first 40 units.
In Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit. Total cost from marginal cost. Sylvies Old Worlds Cheeses has found that its marginal cost, in dollars per kilogram, is C(x)=0.003x+4.25,forx500, where x is the number of kilograms of cheese produced. Find the total cost of producing 400 kg of cheese.In Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit. Total cost from marginal cost. Redline Roasting has found that its marginal cost, in dollars per pound, is C(x)=0.012x+6.50,forx300, where x is the number of pounds of coffee roasted. Find the total cost of roasting 200 lb of coffee.In Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit. Total cost from marginal cost. Cleos Custom Fabrics has found that its marginal cost, in dollars per yard, is C(x)=0.007x+12,forx350, where x is the number of yards of fabric produced. Find the total cost of producing 200 yd of this fabric.In Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit.
Total cost from marginal cost. Photos from nature has found that its marginal cost, in cents per card, is
for ,
where x is the number of cards produced. Find the total cost of producing 650 cards.
7EIn Exercises 1-8, calculate total cost (disregarding any fixed costs) or total profit.
Total cost from marginal cost. Using the information and answer from Exercises 7, find the cost of producing the 201st dress through the 400th dress.
9E10E11. Express without using summation notation.
12. Express without using summation notation.
Express i=010i2 without using summation notation.14E15E16. Express without using summation notation.
a. Approximate the area under the following graph of f(x)=1x2 over the internal [1,7] by computing the area of each rectangle to four decimal places and then adding. b. Approximate the area under the graph of f(x)=1x2 over the internal [1,7] by computing the area of each rectangle to four decimal places and then adding. Compare your answer to that for part (a).18. a. Approximate the area under the following graph of over the internal by computing the area of each rectangle and then adding.
b. Approximate the area under the graph of over the internal by computing the area of each rectangle and then adding. Compare your answer to that for part (a).
19E20E21E22. Approximate the area under the graph of
over the interval using 4 subintervals.
Approximate the area under the graph of F(x)=0.2x3+2x20.2x2 Over the interval [8,3] using 5 subintervals.24E25E26E27E28EIn exercises 29-36, use geometry to evaluate each definite integral.
29.
In exercises 29-36, use geometry to evaluate each definite integral.
30.
In exercises 29-36, use geometry to evaluate each definite integral. 263dxIn exercises 29-36, use geometry to evaluate each definite integral. 144dxIn exercises 29-36, use geometry to evaluate each definite integral. 03xdxIn exercises 29-36, use geometry to evaluate each definite integral.
34.
In exercises 29-36, use geometry to evaluate each definite integral. 01012xdxIn exercises 29-36, use geometry to evaluate each definite integral. 05(2x+5)dx37E38EUse geometry and the following graph of f(x)=12x to evaluate each definite integral. a. 01f(x)dx b. 13f(x)dx c. Find c such that 0cf(x)dx=4 d. Find c such that 0cf(x)dx=3.The Trapezoidal Rule
We can approximate an integral by replacing each rectangle in a Riemann sun with a trapezoid. The area of a trapezoid is, where and are the lengths of the parallel sides and h is the distance between the sides. Thus,
Area under f over
.
If is subdivided into n equal subintervals of length, we have
Area under over
,
Where and .
40. Use the Trapezoidal Rule and the interval subdivision of Exercises 17 (a) to approximate the area under the graph of
.
The Trapezoidal Rule We can approximate an integral by replacing each rectangle in a Riemann sun with a trapezoid. The area of a trapezoid is h(c1+c2)/2, where c1 and c2 are the lengths of the parallel sides and h is the distance between the sides. Thus, Area under f over [a,b] xf(a)=f(m)2+xf(m)+f(b)2 x[f(a)2+f(m)+f(b)2]. If [a,b] is subdivided into n equal subintervals of length x=(ba)/n, we have Area under f over [a,b] x[f(x1)2+f(x2)+f(x3)++f(xn)+f(b)2], Where x1=a and xn=xn1+xorxn=a+(n1)x. Use the Trapezoidal Rule and the interval subdivision of Exercise 18 (a) to approximate the area under the graph of f(x)=x2+1over[0.5]Simpson’s Rule.
To use Simpson’s Rule to approximate the area under a graph of a function f, interval is subdivided into n equal subintervals, where n is an even number. As show in the following graph a parabola is fitted to the first, second, and third points, another parabola is fitted to the third, fourth, and fifth points, and so on.
The area under the graph of f over is approximated by
where .
42. Use Simpson’s Rule and the interval subdivision of Exercises 17 (a) to approximate the area under the graph of over.
Simpson’s Rule.
To use Simpson’s Rule to approximate the area under a graph of a function f, interval is subdivided into n equal subintervals, where n is an even number. As show in the following graph a parabola is fitted to the first, second, and third points, another parabola is fitted to the third, fourth, and fifth points, and so on.
The area under the graph of f over is approximated by
where .
43. Use Simpson’s Rule and 4 subintervals to approximate the area under the graph of .
44E45E46. When using Riemann summation to approximate the area under the graph of a function, is it necessary to divide the interval into subintervals of equal width? Why or why not?
The area, A, of a semicircle of radius r is given by. Using this equation, compare the answer to Exercises 47 and 48 with the exact area.
47. Approximated the area under the graph of using 10 rectangles.
The area, A, of a semicircle of radius r is given by. Using this equation, compare the answer to Exercises 47 and 48 with the exact area.
48. Approximated the area under the graph of using 14 rectangles.
Find the area under the given curve over the indicated interval.
1.
Find the area under the given curve over the indicated interval.
2.
Find the area under the given curve over the indicated interval. y=2x;[1,3]Find the area under the given curve over the indicated interval. y=x2;[0,3]Find the area under the given curve over the indicated interval. y=x2;[0,5]Find the area under the given curve over the indicated interval. y=x3;[0,2]Find the area under the given curve over the indicated interval. y=x3;[0,1]Find the area under the given curve over the indicated interval.
8.
Find the area under the given curve over the indicated interval.
9.
Find the area under the given curve over the indicated interval. y=ex;[0,2]Find the area under the given curve over the indicated interval. y=ex;[0,3]Find the area under the given curve over the indicated interval. y=2x;[1,4]Find the area under the given curve over the indicated interval. y=3x;[6,1]Find the area under the given curve over the indicated interval.
14.
In each of Exercises 15-24, explain what the shaded area represents.
15.
In each of Exercises 15-24, explain what the shaded area represents.
16.
In each of Exercises 15-24, explain what the shaded area represents.In each of Exercises 15-24, explain what the shaded area represents.
18.
In each of Exercises 15-24, explain what the shaded area represents.
19.
In each of Exercises 15-24, explain what the shaded area represents.In each of Exercises 15-24, explain what the shaded area represents.In each of Exercises 15-24, explain what the shaded area represents.In each of Exercises 15-24, explain what the shaded area represents.In each of Exercises 15-24, explain what the shaded area represents.
24.
25E26E27E28E29E30E31EFind the area under the graph of each function over the given interval.
32.
In Exercises 33 and 34, determine visually whether is positive, negative, or zero, and express in terms of area A.
33.
a.
b.
In Exercises 33 and 34, determine visually whether is positive, negative, or zero, and express in terms of area A.
34.
a.
b.
Evaluate each integral. Then state whether the result indicates that there is more area above or below the x-axis or that areas above and below the axis are equal. 01.5(xx2)dxEvaluate each integral. Then state whether the result indicates that there is more area above or below the x-axis or that areas above and below the axis are equal.
36.
Evaluate each integral. Then state whether the result indicates that there is more area above or below the x-axis or that areas above and below the axis are equal. 11(x33x)dxEvaluate each integral. Then state whether the result indicates that there is more area above or below the x-axis or that areas above and below the axis are equal.
38.
39E40E41E42EEvaluate. 13(3t2+7)dtEvaluate. 12(4t3+1)dtEvaluate. 14(x1)dxEvaluate. 18(x32)dxEvaluate. 25(2x23x+7)dx48E49E50E51E52E53E54EEvaluate. 1e(x+1x)dxEvaluate.
56.
Evaluate. 022xdx(Hint:simplifyfirst.)58EBusiness: total revenue. Sallys Sweets finds that the marginal revenue, in dollars, from the sale of x pounds of maple-coated pecans is given by R(x)=2x1/6. Find the revenue when 300 lb of maple-coated pecans are produced.Business: total profit. Pure Water Enterprises finds that the marginal profit, in dollars, from drilling a well that is x feet deep is given by P(x)=x5. Find the profit when a well 250 ft deep is dripped.61E62. Business: increasing total cost. Kitchens-to-Please Contracting determine that the marginal cost, in dollars per square foot, of installing x square feet of kitchen countertop is given by
.
a. Find the cost of installing of countertop
Find the cost of installing an extra of countertop after have already been installed.