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Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

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BuyFindarrow_forward

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
Chapter 4, Problem 11P
Textbook Problem
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Suppose the coefficients of the cubic polynomial P(x) = a + bx + cx2 + dx3 satisfy the equation

a + b 2 + c 3 + d 4 = 0

Show that the equation P(x) = 0 has a root between 0 and 1. Can you generalize this result for an nth-degree polynomial?

To determine

To show: The equation P(x)=0 has a root between 0 and 1 and then generalize this result for an nth degree polynomial.

Explanation of Solution

Given:

Suppose the coefficients of the cubic polynomial P(x)=a+bx+cx2+dx3 satisfy the relation a+b2+c3+d4=0.

Theorem used:

Rolle`s Theorem: A function f which is differentiable with respect to an open interval (a,b) and continuous with respect to a closed interval [a,b], such that: f(a)=f(b), then there must exist at least a point “ c” belonging to open interval (a,b), so that f(c)=0.

Calculation:

The equation, P(x)=0 has a root between 0 and 1.

Consider the a fourth degree polynomial of the form Q(x)=ax+b2x2+c3x3+d4x4.

Differentiate Q(x) with respect to x once gives Q(x)=a+bx+cx2+dx3.

Hence, Q(x)=P(x).                                                                                              (1)

Substitute x=0 in Q. Then,

Q(0)=a(0)+b2(0)2+c3(0)3+d4(0)4=0

Again substitute x=1 in Q gives as follows.

Q(1)=a(1)+b2(1)2+c3(1)3+d4(1)4=a+b2+c3+d4=0

Observe the function Q(x) which is differentiable with respect to an open interval (0,1)  and continuous with respect to a closed interval [0,1], such that: f(0)=f(1), then there must exist at least a point “c” belonging to open interval (0,1), so that Q(c)=0.

Substitute x=0 in equation (1) as follows.

Q(0)=P(0)P(0)=0[Q(0)=0]

That is P(0)=0 for at least a c in open interval (0,1).

Hence, the equation P(x)=0 has a root between 0 and 1.

Generalization of the above result for any nth degree polynomial is as follows.

Suppose the coefficients of the nth degree polynomial P(x)=a0+a1x+a2x2+a3x3++anxn satisfy the relation a0+a12+a23+a34++ann+1=0

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Chapter 4 Solutions

Single Variable Calculus
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Ch. 4.1 - Some computer algebra systems have commands that...Ch. 4.1 - (a) If f(x)=x/(x+2), 1 x 4, use the commands...Ch. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - The table shows speedometer readings at 10-second...Ch. 4.1 - Oil leaked from a tank at a rate of r(t) liters...Ch. 4.1 - When we estimate distances from velocity data, it...Ch. 4.1 - The velocity graph of a braking car is shown. Use...Ch. 4.1 - The velocity graph of a car accelerating from rest...Ch. 4.1 - In someone infected with measles, the virus level...Ch. 4.1 - The table shows the number of people per day who...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Determine a region whose area is equal to the...Ch. 4.1 - Determine a region whose area is equal to the...Ch. 4.1 - (a) Use Definition 2 to find an expression for the...Ch. 4.1 - Let A be the area under the graph of an increasing...Ch. 4.1 - If A is the area under the curve y = sin x from 0...Ch. 4.1 - (a) Express the area under the curve y = x5 from 0...Ch. 4.1 - (a) Express the area under the curve y = x4 + 5x2...Ch. 4.1 - Find the exact area under the cosine curve y = cos...Ch. 4.1 - (a) Let An be the area of a polygon with n equal...Ch. 4.2 - Evaluate the Riemann sum for f(x)=x1,6x4, with...Ch. 4.2 - If f(x)=cosx0x3/4 evaluate the Riemann sum with n...Ch. 4.2 - If f(x)=x24,0x3, find the Riemann sum with n = 6,...Ch. 4.2 - (a) Find the Riemann sum for f(x)=1/x,1x2, with...Ch. 4.2 - The graph of a function f is given. Estimate...Ch. 4.2 - The graph of g is shown. Estimate 24g(x)dx with...Ch. 4.2 - A table of values of an increasing function f is...Ch. 4.2 - The table gives the values of a function obtained...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - Use the Midpoint Rule with the given value of n to...Ch. 4.2 - If you have a CAS that evaluates midpoint...Ch. 4.2 - With a programmable calculator or computer (see...Ch. 4.2 - Use a calculator or computer to make a table of...Ch. 4.2 - Use a calculator or computer to make a table of...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - (a) Find an approximation to the integral...Ch. 4.2 - Prove that abxdx=b2a22.Ch. 4.2 - Prove that abx2dx=b3a33.Ch. 4.2 - Express the integral as a limit of Riemann sums....Ch. 4.2 - Express the integral as a limit of Riemann sums....Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - The graph of f is shown. Evaluate each integral by...Ch. 4.2 - The graph of g consists of two straight lines and...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate 111+x4dx.Ch. 4.2 - Give that 0sin4xdx=38, what is 0sin4d?Ch. 4.2 - In Example 4.1.2 we showed that 01x2dx=13. Use...Ch. 4.2 - Use the properties of integrals and the result of...Ch. 4.2 - Use the results of Exercises 27 and 28 and the...Ch. 4.2 - Use the result of Exercise 27 and the fact that...Ch. 4.2 - Write as a single integral in the form abf(x)dx...Ch. 4.2 - If 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.Ch. 4.2 - If 09f(x)dx=37 and 09g(x)dx=16, find...Ch. 4.2 - Find 05f(x)dx if f(x)={3forx3xforx3Ch. 4.2 - For the function f whose graph is shown, list the...Ch. 4.2 - If F(x)=2f(t)dt, where f is the function whose...Ch. 4.2 - Each of the regions A, B, and C bounded by the...Ch. 4.2 - Suppose f has absolute minimum value m and...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use Property 8 of integrals to estimate the value...Ch. 4.2 - Use properties of integrals, together with...Ch. 4.2 - Use properties of integrals, together with...Ch. 4.2 - Which of the integrals 12xdx,121/xdx and 12xdx has...Ch. 4.2 - Which of the integrals 00.5cos(x2)dx,00.5cosxdx is...Ch. 4.2 - Prove Property 3 of integrals.Ch. 4.2 - (a) If f is continuous on [a, b], show that...Ch. 4.2 - Let f(x) = 0 if x is any rational number and f(x)...Ch. 4.2 - Let f(0) = 0 and f(x) = 1/x if 0 x 1. Show that...Ch. 4.2 - Express the limit as a definite integral. 73....Ch. 4.2 - Express the limit as a definite integral. 74....Ch. 4.2 - Find12x2dx. Hint: Choose xi to be the geometric...Ch. 4.3 - Explain exactly what is meant by the statement...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Evaluate the integral. 19. 13(x2+2x4)dxCh. 4.3 - Evaluate the integral. 20. 11x100dxCh. 4.3 - Evaluate the integral. 21. 02(45t334t2+25t)dtCh. 4.3 - Evaluate the integral. 22. 01(18v3+16v2)dvCh. 4.3 - Evaluate the integral. 23. 19xdxCh. 4.3 - Evaluate the integral. 24. 18x-2/3dxCh. 4.3 - Evaluate the integral. 25. /6sindCh. 4.3 - Evaluate the integral. 26. 55dxCh. 4.3 - Evaluate the integral. 27. 01(u+2)(u3)duCh. 4.3 - Evaluate the integral. 28. 04(4t)tdtCh. 4.3 - Evaluate the integral. 29. 142+x2xdxCh. 4.3 - Evaluate the integral. 30. 12(3u2)(u+1)duCh. 4.3 - Evaluate the integral. 31. /6/2csctcottdtCh. 4.3 - Evaluate the integral. 32. /4/3csc2dCh. 4.3 - Evaluate the integral. 33. 00(1+r)3drCh. 4.3 - Evaluate the integral. 34. 12s4+1s2dsCh. 4.3 - Evaluate the integral. 35. 12v5+3v6v4dvCh. 4.3 - Evaluate the integral. 36. 1183zdzCh. 4.3 - Evaluate the integral. 37....Ch. 4.3 - Evaluate the integral. 38....Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Sketch the region enclosed by the given curves and...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Evaluate the integral and interpret it as a...Ch. 4.3 - Evaluate the integral and interpret it as a...Ch. 4.3 - What is wrong with the equation? 49....Ch. 4.3 - What is wrong with the equation? 50....Ch. 4.3 - What is wrong with the equation? 51....Ch. 4.3 - What is wrong with the equation? 52....Ch. 4.3 - Find the derivative of the function. 53....Ch. 4.3 - Find the derivative of the function. 54....Ch. 4.3 - Find the derivative of the function. 55....Ch. 4.3 - Find the derivative of the function. 56....Ch. 4.3 - Let F(x)=xcosttdt. Find an equation of the tangent...Ch. 4.3 - If f(x)=0x(1t2)cos2tdt, on what interval is f...Ch. 4.3 - On what interval is the curve y=0xt2t2+t+2dt...Ch. 4.3 - Let F(x)=1xf(t)dt, where f is the function whose...Ch. 4.3 - If f(1) = 12, f is continuous, and 14f(x)dx=17,...Ch. 4.3 - If f(x)=0sinx1+t2dt and g(y)=3yf(x)dx, find g(/6).Ch. 4.3 - The Fresnel function S was defined in Example 3...Ch. 4.3 - The sine integral function Si(x)=0xsinttdt is...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Evaluate the limit by first recognizing the sum as...Ch. 4.3 - Justify (3) for the case h 0.Ch. 4.3 - If f is continuous and g and h are differentiable...Ch. 4.3 - (a) Show that 11+x31+x3 for x 0. (b) Show that...Ch. 4.3 - (a) Show that cos(x2) cos x for 0 x 1. (b)...Ch. 4.3 - Show that 0510x2x4+x2+1dx0.1 by comparing the...Ch. 4.3 - Let f(x)={0ifx0xif0x12xif1x20ifx2 and...Ch. 4.3 - Find a function f and a number a such that...Ch. 4.3 - Suppose h is a function such that h(1) = 2, h(1) =...Ch. 4.3 - A manufacturing company owns a major piece of...Ch. 4.3 - A high-tech company purchases a new computing...Ch. 4.3 - Evaluate the integral. 79. 1912xdxCh. 4.3 - Evaluate the integral 80. 0110xdxCh. 4.3 - The following exercises are intended only for...Ch. 4.3 - Evaluate the integral 82. 014t2+1dtCh. 4.3 - Evaluate the integral 83. 11eu+1duCh. 4.3 - The following exercises are intended only for...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Find the general indefinite integral. 5....Ch. 4.4 - Find the general indefinite integral. 6. x54dxCh. 4.4 - Find the general indefinite integral. 7....Ch. 4.4 - Find the general indefinite integral. 8....Ch. 4.4 - Find the general indefinite integral. 9....Ch. 4.4 - Find the general indefinite integral. 10....Ch. 4.4 - Find the general indefinite integral. 11. 1+x+xxdxCh. 4.4 - Find the general indefinite integral. 12....Ch. 4.4 - Find the general indefinite integral. 13....Ch. 4.4 - Find the general indefinite integral. 14....Ch. 4.4 - Find the general indefinite integral. 15....Ch. 4.4 - Find the general indefinite integral. 16....Ch. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Evaluate the integral. 19. 23(x23)dxCh. 4.4 - Evaluate the integral. 20. 12(4x33x2+2x)dxCh. 4.4 - Evaluate the integral. 21. 20(12t4+14t3t)dtCh. 4.4 - Evaluate the integral. 22. 03(1+6w210w4)dwCh. 4.4 - Evaluate the integral. 23. 02(2x3)(4x2+1)dxCh. 4.4 - Evaluate the integral. 24. 11t(1t)2dtCh. 4.4 - Evaluate the integral. 25. 0(4sin3cos)dCh. 4.4 - Evaluate the integral. 26. 12(1x24x3)dxCh. 4.4 - Evaluate the integral. 27. 14(4+6uu)duCh. 4.4 - Evaluate the integral. 28. 12(21p2)2dpCh. 4.4 - Evaluate the integral. 29. 145xdxCh. 4.4 - Evaluate the integral. 30. 18(2w3w3)dwCh. 4.4 - Evaluate the integral. 31. 14t(1+t)dtCh. 4.4 - Evaluate the integral. 32. 0/4sectandCh. 4.4 - Evaluate the integral. 33. 0/41+cos2cos2dCh. 4.4 - Evaluate the integral. 34. 0/3sin+sintan2sec2dCh. 4.4 - Evaluate the integral. 35. 182+tt23dtCh. 4.4 - Evaluate the integral. 36. 064u(uu3)duCh. 4.4 - Evaluate the integral. 37. 01(x54+x45)dxCh. 4.4 - Evaluate the integral. 38. 01(1+x2)3dxCh. 4.4 - Evaluate the integral. 39. 25|x3|dxCh. 4.4 - Evaluate the integral. 40. 02|2x1|dxCh. 4.4 - Evaluate the integral. 41. 12(x2|x|)dxCh. 4.4 - Evaluate the integral. 42. 03/2|sinx|dxCh. 4.4 - Use a graph to estimate the x-intercepts of the...Ch. 4.4 - Repeat Exercise 43 for the curve y = 2x + 3x4 ...Ch. 4.4 - The area of the region that lies to the right of...Ch. 4.4 - The boundaries of the shaded region in the figure...Ch. 4.4 - If w'(t) is the rate of growth of a child in...Ch. 4.4 - The current in a wire is defined as the derivative...Ch. 4.4 - If oil leaks from a tank at a rate of r(t) gallons...Ch. 4.4 - A honeybee population starts with 100 bees and...Ch. 4.4 - In Section 3.7 we defined the marginal revenue...Ch. 4.4 - If f(x) is the slope of a trail at a distance of x...Ch. 4.4 - If x is measured in meters and f(x) is measured in...Ch. 4.4 - If the units for x are feet and the units for a(x)...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - The acceleration function (in m/s2) and the...Ch. 4.4 - The acceleration function (in m/s2) and the...Ch. 4.4 - The linear density of a rod of length 4 m is given...Ch. 4.4 - Water flows from the bottom of a storage tank at a...Ch. 4.4 - The velocity of a car was read from its...Ch. 4.4 - Suppose that a volcano is erupting and readings of...Ch. 4.4 - Lake Lanier in Georgia, USA, is a reservoir...Ch. 4.4 - Water flows into and out of a storage tank. A...Ch. 4.4 - The graph of the acceleration a(t) of a car...Ch. 4.4 - Shown is the graph of traffic on an Internet...Ch. 4.4 - The following graph shows the power consumption in...Ch. 4.4 - On May 7, 1992, the space shuttle Endeavour was...Ch. 4.4 - Evaluate the integral. 69. (sinx+sinhx)dxCh. 4.4 - The following exercises are intended only for...Ch. 4.4 - Evaluate the integral. 71. (x2+1+1x2+1)dxCh. 4.4 - The following exercises are intended only for...Ch. 4.4 - The following exercises are intended only for...Ch. 4.4 - The area labeled B is three times the area labeled...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Evaluate the indefinite integral. 7. x1x2dxCh. 4.5 - Evaluate the indefinite integral. 8. x2sin(x3)dxCh. 4.5 - Evaluate the indefinite integral. 9. (12x)9dxCh. 4.5 - Evaluate the indefinite integral. 10. sint1+costdtCh. 4.5 - Evaluate the indefinite integral. 11. sin(2/3)dCh. 4.5 - Evaluate the indefinite integral. 12. sec22dCh. 4.5 - Evaluate the indefinite integral. 13. sec3ttan3tdtCh. 4.5 - Evaluate the indefinite integral. 14. y2(4y3)2/3dyCh. 4.5 - Evaluate the indefinite integral. 15. cos(1+5t)dtCh. 4.5 - Evaluate the indefinite integral. 16. sinxxdxCh. 4.5 - Evaluate the indefinite integral. 17. sec2tan3dCh. 4.5 - Evaluate the indefinite integral. 18....Ch. 4.5 - Evaluate the indefinite integral. 19....Ch. 4.5 - Evaluate the indefinite integral. 20. xx+2dxCh. 4.5 - Evaluate the indefinite integral. 21....Ch. 4.5 - Evaluate the indefinite integral. 22. cos(/x)x2dxCh. 4.5 - Evaluate the indefinite integral. 23. z21+z33dzCh. 4.5 - Evaluate the indefinite integral. 24....Ch. 4.5 - Evaluate the indefinite integral. 25. cotxcsc2xdxCh. 4.5 - Evaluate the indefinite integral. 26. sec2xtan2xdxCh. 4.5 - Evaluate the indefinite integral. 27. sec3xtanxdxCh. 4.5 - Evaluate the indefinite integral. 28. x22+xdxCh. 4.5 - Evaluate the indefinite integral. 29. x(2x+5)8dxCh. 4.5 - Evaluate the indefinite integral. 30. x3x2+1dxCh. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the indefinite integral. Illustrate and...Ch. 4.5 - Evaluate the definite integral. 35. 01cos(t/2)dtCh. 4.5 - Evaluate the definite integral. 36. 01(3t1)50dtCh. 4.5 - Evaluate the definite integral. 37. 011+7x3dxCh. 4.5 - Evaluate the definite integral. 38. 0xcos(x2)dxCh. 4.5 - Evaluate the definite integral. 39. 0/6sintcos2tdtCh. 4.5 - Evaluate the definite integral. 40....Ch. 4.5 - Evaluate the definite integral. 41....Ch. 4.5 - Evaluate the definite integral. 42....Ch. 4.5 - Evaluate the definite integral. 43. 013dx(1+2x)23Ch. 4.5 - Evaluate the definite integral. 44. 0axa2x2dxCh. 4.5 - Evaluate the definite integral. 45. 0axx2+a2dx(a0)Ch. 4.5 - Evaluate the definite integral. 46. /3/3x4sinxdxCh. 4.5 - Evaluate the definite integral. 47. 12xx1dxCh. 4.5 - Evaluate the definite integral. 48. 04x1+2xdxCh. 4.5 - Evaluate the definite integral. 49....Ch. 4.5 - Evaluate the definite integral. 50....Ch. 4.5 - Evaluate the definite integral. 51. 01dx(1+x)4Ch. 4.5 - Verify that f(x)=sinx3 is an odd function and use...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Evaluate 22(x+3)4x2dx by writing it as a sum of...Ch. 4.5 - Evaluate 01x1x4dx by making a substitution and...Ch. 4.5 - Breathing is cyclic and a full respiratory cycle...Ch. 4.5 - A model for the basal metabolism rate, in kcal/h,...Ch. 4.5 - If f is continuous and 04f(x)dx=10, find...Ch. 4.5 - If f is continuous and 09f(x)dx=4, find...Ch. 4.5 - If f is continuous on , prove that...Ch. 4.5 - If f is continuous on , prove that...Ch. 4.5 - If a and b are positive numbers, show that...Ch. 4.5 - If f is continuous on [0, ], use the substitution...Ch. 4.5 - If f is continuous, prove that...Ch. 4.5 - Use Exercise 65 to evaluate 0/2cos2xdx and...Ch. 4.5 - Evaluate the integral. 67. dx53xCh. 4.5 - Evaluate the integral. 68. e5rdrCh. 4.5 - Evaluate the integral. 69. (lnx)2xdxCh. 4.5 - Evaluate the integral. 70. dxax+b(a0)Ch. 4.5 - Evaluate the integral. 71. ex1+exdxCh. 4.5 - Evaluate the integral. 72. ecostsintdtCh. 4.5 - Evaluate the integral. 73. (arctanx2)x2+1dxCh. 4.5 - Evaluate the integral. 74. xx2+4dxCh. 4.5 - Evaluate the integral. 75. 1+x1+x2dxCh. 4.5 - Evaluate the integral. 76. sin(lnx)xdxCh. 4.5 - Evaluate the integral. 77. sin2x1+cos2xdxCh. 4.5 - Evaluate the integral. 78. sinx1+cos2xdxCh. 4.5 - Evaluate the integral. 79. cotxdxCh. 4.5 - Evaluate the integral. 80. x1+x4dxCh. 4.5 - Evaluate the integral. 81. ee4dxxlnxCh. 4.5 - Evaluate the integral. 82. 01xex2dxCh. 4.5 - Evaluate the integral. 83. 01ez+1ez+zdzCh. 4.5 - Evaluate the integral. 84. 01(x1)e(x1)2dxCh. 4.5 - Use Exercise 64 to evaluate the integral...Ch. 4 - (a) Write an expression for a Riemann sum of a...Ch. 4 - (a) Write the definition of the definite integral...Ch. 4 - State the Midpoint Rule.Ch. 4 - State both parts of the Fundamental Theorem of...Ch. 4 - (a) State the Net Change Theorem. (b) If r(t) is...Ch. 4 - Suppose a particle moves back and forth along a...Ch. 4 - (a) Explain the meaning of the indefinite integral...Ch. 4 - Explain exactly what is meant by the statement...Ch. 4 - State the Substitution Rule. In practice, how do...Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Use the given graph of f to find the Riemann sum...Ch. 4 - (a) Evaluate the Riemann sum for f(x)=x2x0x2 with...Ch. 4 - Evaluate 01(x+1x2)dx by interpreting it in terms...Ch. 4 - Express limni=1nsinxix as a definite integral on...Ch. 4 - If 06f(x)dx=10 and04f(x)dx=7, find46f(x)dx.Ch. 4 - (a) Write 15(x+2x5)dx as a limit of Riemann sums,...Ch. 4 - The figure shows the graphs of f,f, and 0xf(t)dt....Ch. 4 - Evaluate: (a) 0/2ddx(sinx2cosx3)dx (b)...Ch. 4 - The graph of f consists of the three line segments...Ch. 4 - If f is the function in Exercise 9, find g(4).Ch. 4 - Evaluate the integral, if it exists. 11....Ch. 4 - Evaluate the integral, if it exists. 12....Ch. 4 - Evaluate the integral, if it exists. 13. 01(1x9)dxCh. 4 - Evaluate the integral, if it exists. 14. 01(1x)9dxCh. 4 - Evaluate the integral, if it exists. 15. 19u2u2uduCh. 4 - Evaluate the integral, if it exists. 16....Ch. 4 - Evaluate the integral, if it exists. 17....Ch. 4 - Evaluate the integral, if it exists. 18....Ch. 4 - Evaluate the integral, if it exists. 19. 15dt(t4)2Ch. 4 - Evaluate the integral, if it exists. 20....Ch. 4 - Evaluate the integral, if it exists. 21....Ch. 4 - Evaluate the integral, if it exists. 22....Ch. 4 - Evaluate the integral, if it exists. 23....Ch. 4 - Evaluate the integral, if it exists. 24....Ch. 4 - Evaluate the integral, if it exists. 25....Ch. 4 - Evaluate the integral, if it exists. 26....Ch. 4 - Evaluate the integral, if it exists. 27....Ch. 4 - Evaluate the integral, if it exists. 28....Ch. 4 - Evaluate the integral, if it exists. 29. 03|x24|dxCh. 4 - Evaluate the integral, if it exists. 30. 04|x1|dxCh. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Evaluate the indefinite integral. Illustrate and...Ch. 4 - Use a graph to give a rough estimate of the area...Ch. 4 - Graph the function f(x) = cos2x sin x and use the...Ch. 4 - Find the derivative of the function. 35....Ch. 4 - Find the derivative of the function. 36....Ch. 4 - Find the derivative of the function. 37....Ch. 4 - Find the derivative of the function. 38....Ch. 4 - Find the derivative of the function. 39.y=xxcosdCh. 4 - Find the derivative of the function. 40....Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use Property 8 of integrals to estimate the value...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the properties of integrals to verify the...Ch. 4 - Use the Midpoint Rule with n = 6 to approximate...Ch. 4 - A particle moves along a line with velocity...Ch. 4 - Let r(t) be the rate at which the worlds oil is...Ch. 4 - A radar gun was used to record the speed of a...Ch. 4 - A population of honeybees increased at a rate of...Ch. 4 - Let f(x)={x1if3x01x2if0x1 Evaluate 31f(x)dx by...Ch. 4 - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4 - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4 - If f is a continuous function such that...Ch. 4 - Find a function f and a value of the constant a...Ch. 4 - If f' is continuous on [a, b], show that...Ch. 4 - Find limh01h22+h1+t3dtCh. 4 - If f is continuous on [0, 1], prove that...Ch. 4 - Evaluate limn1n[(1n)9+(2n)9+(3n)9+...+(nn)9]Ch. 4 - If xsinx=0x2f(t)dt, where f is a continuous...Ch. 4 - Find the minimum value of the area of the region...Ch. 4 - If f is a differentiable function such that f(x)...Ch. 4 - (a) Graph several members of the family of...Ch. 4 - If f(x)0g(x)11+t3dt, where...Ch. 4 - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4 - Find the interval [a, b] for which the value of...Ch. 4 - Use an integrai to estimate th sum i=110000i.Ch. 4 - (a) Evaluate 0nxdx, where n is a positive integer....Ch. 4 - Find d2dx20x(1sint1+u4du)dt.Ch. 4 - Suppose the coefficients of the cubic polynomial...Ch. 4 - A circular disk of radius r is used in an...Ch. 4 - Prove that if f is continuous, then...Ch. 4 - The figure shows a parabolic segment, that is. a...Ch. 4 - Given the point (a, b) in the first quadrant, find...Ch. 4 - The figure shows a region consisting of all points...Ch. 4 - Evaluate limn(1nn+1+1nn+2++1nn+n).Ch. 4 - For any number c, we let fc(x) be the smaller of...

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