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All Textbook Solutions for Calculus Volume 1

For the following exercises, find the antiderivative of the function, assuming F(0)=0 . 519. [T] f(x)=1( x+1)2 For the following exercises, find the antiderivative of the function, assuming F(0)=0 . 520. [T] f(x)=e2x+3x2 For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false. 521. If f(x) is the antiderivative of v(x), then 2f(x) is the antiderivative of 2v(x).For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false. 522. If f(x) is the antiderivative of v(x), then f(2x) is the antiderivative of v(2x).For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false. 523. If f(x) is the antiderivative of v(x), then f(x) + 1 is the antiderivative of v(x) + 1.For the following exercises, determine whether the statement is true or false. Either prove it is true or find a counterexample if it is false. 524. If f(x) is the antiderivative of v(x), then (f(x))2 is the antiderivative of (v(x))2 .True or False? Justify your answer with a proof or a counterexample. Assume that f(x) is continuous and differentiable unless stated otherwise. 525. If f(1)=6 and f(1)=2 , then there exists at least one point x[1,1] such that f(x)=4 .True or False? Justify your answer with a proof or a counterexample. Assume that f(x) is continuous and differentiable unless stated otherwise. 526. If f' (c) = 0. there is a maximum or minimum at x = c.True or False? Justify your answer with a proof or a counterexample. Assume that f(x) is continuous and differentiable unless stated otherwise. 527. There is a function such that f(x) < 0, f (x) > 0, and f"(x) < 0. (A graphical “proof” is acceptable for this answer.)True or False? Justify your answer with a proof or a counterexample. Assume that f(x) is continuous and differentiable unless stated otherwise. 528* There is a function such that there is both an inflection point and a critical point for some value x = a.Given the graph of f’, determine where f increasing or decreasing.The graph of f is given below. Draw f’.Find the linear approximation L(x) to y=x2+tan(x) near x=14 .Find the differential of y=x25x6 and evaluate for x = 2 with dx = 0.1.Find the critical points and the local and absolute extrema of the following functions on the given interval. 533. f(x)=x+sin2(x)over[0,]Find the critical points and the local and absolute extrema of the following functions on the given interval. 534. f(x)=3x44x312x2+6over[3,3]Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down. 535. f(x)=3t48t318t2 Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down. 536. y=x+sin(x)Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down. 537. g(x)=xx Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down. 538. f()=sin(3)Evaluate the following limits. 539. limx3xx2+1x41 Evaluate the following limits 540. limxcos(1x)Evaluate the following limits. 541. limx1x1sin(x)Evaluate the following limits. 542. limx(3x)1/x Use Newton’s method to find the first two iterations, given the starting point. 543. y=x3+1,x0=0.5 Use Newton’s method to find the first two iterations, given the starting point. 544. 1x+1=12,x0=0 Find the antiderivatives F(x) of the following functions 545. g(x)=x1x2 Find the antiderivatives F(x) of the following functions 546. f(x)=2x+6cosx,F(x)=2+2 Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes. 547. y=1x( x+1)2 Graph the following functions by hand. Make sure to label the inflection points, critical points, zeros, and asymptotes. 548. y=x4x2 A car is being compacted into a rectangular solid. The volume is decreasing at a rate of 2 m3/sec. The length and width of the compactor are square, but the height is not the same length as the length and width. If the length and width walls move toward each other at a rate of 0.25 m/sec, find the rate at which the height is changing when the length and width are 2 m and the height is 1.5 m.A rocket is launched into space; its kinetic energy is given by K(t)=(12)m(t)v(t)2 , where K is the kinetic energy in joules, m is the mass of the rocket in kilograms, and v is the velocity of the rocket in meters/second. Assume the velocity is increasing at a rate of 15 m/sec2and the mass is decreasing at a rate of 10 kg/sec because the fuel is being burned. At what rate is the rocket’s kinetic energy changing when the mass is 2000 kg and the velocity is 5000 m/sec? Give your answer in mega-Joules (MJ), which is equivalent to 106 J.The famous Regiomontanus' problem for angle maximization was proposed during the 15 th century. A painting hangs on a wall with the bottom of the painting a distance a feet above eye level, and the top b feet above eye level. What distance x (in feet) from the wall should the viewer stand to maximize the angle subtended by the painting, ?An airline sells tickets from Tokyo to Detroit for $1200. There are 500 seats available and a typical flight books 350 seats. For every $10 decrease in price, the airline observes an additional five seats sold. What should the fare be to maximize profit? How many passengers would be onboard?State whether the given sums are equal or unequal. i=110i and k=110k i=110i and i=615(i5) i=110i(i1) and j=09(j+1)j i=110i(i1) and k=110(k2k)In the following exercises, use the rules for sums of powers of integers to compute the sums. 2. i=510i In the following exercises, use the rules for sums of powers of integers to compute the sums. 3. i=510i2 Suppose that i=1100ai=15 and i=1100bi=12 . In the following exercises, compute the sums. 4. i=1100(ai+bi)Suppose that i=1100ai=15 and i=1100bi=12 . In the following exercises, compute the sums. 5. i=1100(aibi)Suppose that i=1100ai=15 and i=1100bi=12 . In the following exercises, compute the sums. 6. i=1100(3ai4bi)Suppose that i=1100ai=15 and i=1100bi=12 . In the following exercises, compute the sums. 7. i=1100(5ai+4bi)In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 8. k=120100(k25k+1)In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 9. j=150(j22j)In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 10. j=1120(j210j)In the following exercises, use summation properties and formulas to rewrite and evaluate the sums. 11. k=125[( 2k)2100k]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 12. L4 for f(x)=1x1 on [2, 3]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 13. R4 for g(x)=cos(x) on [0, 1]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 14. L6 for f(x)=1x(x1) on [2, 5]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 15. R6 for f(x)=1x(x1) on [2, 5]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 16. R4 for 1x2+1 on [-2, 2]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 17. L4 for 1x2+1 on [-2, 2]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 18. R4 for x22x+1 on [0, 2]Let Ln denote the left-endpoint sum using n sub intervals and let Rn denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and light sums for the given functions on the indicated interval. 19. L8 for x22x+1 on [0, 2]Compute the left and tight Riemann sums— L4 and R4 . espectively—for f(x)=(2|x|) on [-2, 2]. Compute their average value and compare it with the area under the graph of f.Compute the left and light Riemann sums— L6 and R6 , respectively—for f(x)=(3|3x|) on [0, 6]. Compute their average value and compare it with the area under the graph of f.Compute the left and light Riemann sums— L4 and R4 , respectively—for f(x)=4x2 on [-2, 2] and compare their values.Compute the left and right Riemann sums— L6 and R6 , respectively—for f(x)=9(x3)2 on [0, 6] and compare their values.Express the following endpoint sums in sigma notation but do not evaluate them. 24. L30 for f(x)=x2 on [1, 2]Express the following endpoint sums in sigma notation but do not evaluate them. 25. L10 for f(x)=4x2 on [-2, 2]Express the following endpoint sums in sigma notation but do not evaluate them. 26. R20 for f(x)=sinx on [0,]Express the following endpoint sums in sigma notation but do not evaluate them. 27. R100 for Inx on [1, e]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 28. [T] L100 and R100 for y=x23x+1 on the interval [-1, 1]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 29. [T] L100 and R100 for y=x2 on the interval [0, 1]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 30. [T] L50 and R50 for y=x+1x21 on the interval [2, 4]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 31. [T] L100and R100 for y=x3 on the interval [-1, 1]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 32. [T] L50 and R50 for y=tan(x) on the interval [0,4]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 33. [T] L100 and R100 for y=e2x on the interval [-1, 1]In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 34. Let tj denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014. If there were a total of 21 stage, interpret j=121tj .In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 35. Let rj denote the total rainfall in Portland on the jth 31 day of the year in 2009. Interpret j=131rj In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums? 36. Let dj denote the hours of daylight and j denote the increase in the hours of daylight from day j - 1 to day j in Fargo, Noith Dakota, on thejth day of the year. Interpret d1+j=2365j To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds 110mi to his daily routine each week, what is the total mileage on Joe's shoes after 25 weeks?The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide (CO2) each decade since 1960, in parts per million (ppm). Estimate the total increase in atmospheric (CO2) between 1964 and 2013. Decade Ppm/y 1964-1973 1.07 1974-1983 1.34 1984-1993 1.40 1994-2003 1.87 2004-2013 2.07 Table 5.2 Average Annual Atmospheric (CO2)Increase, 1964-2013 Source: http://www.esrl.noaa.gov/ gmd/ccgg/trends/.The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to 2010. Starting Year 20-Year Change 1870 0.3 1890 1.5 1910 0.2 1930 2.8 1950 0.7 1970 1.1 1990 1.5 Table 5.3 Approximate 20-Year Sea Level Increases, 1870-1990 Source: http://link.springer.com/article/10.1007%2Fs10712-011-9119-1 The following table gives the approximate increase in dollars in the average price of a gallon of gas per decade since 1950. If the average price of a gallon of gas in 2010 was $2.60, what was the average price of a gallon of gas in 1950? Starting Year 10-Year Change 1950 0.03 1960 0.05 1970 0.86 1980 -0.03 1990 0.29 2000 1.12 Table 5.4 Approximate 10-Year Gas Price Increases, 1950-2000 Source: http://epb.lbl.gov/homepages/ Riick_Diamond/docs/ lbnl55011-trends.pdf.The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010. Year % Change/Year 2000 1.12 2001 0.99 2002 0.93 2003 0.86 2004 0.93 2005 0.93 2006 0.97 2007 0.96 2008 0.95 2009 0.88 Table 5.5 Annual Percentage Growth of U.S. Population, 2000-2009 Source:http://www.census.gov/popest/data. (Hint: To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.)In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8 . 42.In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8 . 43.In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8 . 44.In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8 . 45.[T] Use a computer algebra system to compute the Riemann sum, LN , for N = 10, 30, 50 for f(x)=1x2 on [-1, 1].[T] Use a computer algebra system to compute the Riemann sum, LN, for N = 10, 30, 50 for f(x)=11+x2 on [-1, 1].[T] Use a computer algebra system to compute the Riemann sum, LN , for N = 10, 30, 50 for f(x)=sin2x on [0, 2 ]. Compare these estimates with .In the following exercises, use a calculator or a computer program to evaluate the endpoint sums RN and LN for N = 1,10,100. How do these estimates compare with the exact answers, which you can find via geometry? 49. [T] y=cos(x) on the interval [0, 1]In the following exercises, use a calculator or a computer program to evaluate the endpoint sums RN and LN for N = 1,10,100. How do these estimates compare with the exact answers, which you can find via geometry? 50. [T] y=3x+2 on the interval [3, 5]In the following exercises, use a calculator or a computer program to evaluate the endpoint sums RN and LN for N = 1,10,100. 51. [T] y=x45x2+4 on the interval [-2, 2], which has an exact area of 3215 In the following exercises, use a calculator or a computer program to evaluate the endpoint sums RN and LN for N = 1,10,100. 52. [T] y=Inx on the interval [1,2], which has an exact area of 2In(2)1 In the following exercises, use a calculator or a computer program to evaluate the endpoint sums RN and LN for N = 1,10,100. 53. Explain why, if f(a)0 and f is increasing on [a, b], that the left endpoint estimate is a lower bound for the area below the graph of f on [a, b].Explain why, if f(b)0 and f is decreasing on [a, b], that the left endpoint estimate is an upper bound for the area below the graph of f on [a, b].Show that, in general, RNLN=(ba)f(b)f(a)N .Explain why, if f is increasing on |a,b| , the error between either LN or RN and the area A below the graph of f is at most (ba)f(b)f(a)N .For each of the three graphs: Obtain a lower bound L(A) for the area enclosed by the curve by adding the areas of the squares enclosed completely by the curve. Obtain an tipper bound U(A) for the area by adding to L(A) the areas B{A) of the squares enclosed partially by the curve.In the previous exercise, explain why L(A) gets no smaller while U(A) gets no larger as the squares are subdivided into four boxes of equal area.A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is sin(n) . The base of the outer triangle is B=cos(n)+sin(n)tan(n) and the height is H=Bsin(2n) . Use this information to argue that the area of a unit circle is equal to .In the following exercises, express the limits as integrals. 60. limni=1n(xi)x over [1, 3]In the following exercises, express the limits as integrals. 61. limni=1n(5( x i )23( x i )3)x over [0, 2]In the following exercises, express the limits as integrals. 62. limni=1nsin2(2xi)x over [0, 1]In the following exercises, express the limits as integrals. 63. limni=1ncos2(2xi)x over [0, 1]In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 64. Ln=1ni=1ni1n In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 65. Rn=1ni=1nin In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 66. Ln=2ni=1n(1+2i1n)In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 67. Rn=3ni=1n(3+3in)In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 68. Ln=2ni=1n2i1ncos(2i1n)In the following exercises, given Ln or Rn as indicated, express their limits as n as definite integrals, identifying the correct intervals. 69. Rn=1ni=1n(1+in)log(( 1+ i n)2)In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 70.In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 71.In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 72.In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 73.In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 74.In the following exercises, evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis. 75.In the following exercises, evaluate the integral using area formulas. 76. 03(3x)dx In the following exercises, evaluate the integral using area formulas. 77. 23(3x)dx In the following exercises, evaluate the integral using area formulas. 78. 33(3|x|)dx In the following exercises, evaluate the integral using area formulas. 79. 06(3|x3|)dx In the following exercises, evaluate the integral using area formulas. 80. 224x2dx In the following exercises, evaluate the integral using area formulas. 81. 154 ( x3 )2dx In the following exercises, evaluate the integral using area formulas. 82. 01236 ( x6 )2dx In the following exercises, evaluate the integral using area formulas. 83. 23(3|x|)dx In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. 84. {(0, 0), (2, 1), (4, 3), (5, 0), (6, 0), (8, 3)} over [0, 8]In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. 85. {(0, 2), (1, 0), (3, 5), (5, 5), (6, 2), (8, 0)} over [0, 8]In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. 86. {(-4, -4), (-2, 0), (0, -2), (3, 3), (4, 3)} over [-4, 4]In the following exercises, use averages of values at the left (L) and right (R) endpoints to compute the integrals of the piecewise linear functions with graphs that pass through the given list of points over the indicated intervals. 87. {(-4, 0), (-2, 2),(0, 0), (1, 2), (3, 2), (4, 0)} over [-4, 4]Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 88. 04(f(x)+g(x))dx Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 89. 24(f(x)+g(x))dx Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 90. 02(f(x)g(x))dx Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 91. 24(f(x)g(x))dx Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 92. 02(3f(x)4g(x))dx Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises, compute the intervals. 93. 24(4f(x)3g(x))dx In the following exercises, use the identity AAf(x)dx= A0f( x)dx+ 0 A f( x )dx to compute the integrals. 94. sint1+t2dt(Hint:sin(t)=sin(t))In the following exercises, use the identity AAf(x)dx= A0f( x)dx+ 0 A f( x )dx to compute the integrals. 95. t1+costdt In the following exercises, use the identity AAf(x)dx= A0f( x)dx+ 0 A f( x )dx to compute the integrals. 96. 13(2x)dx (Hint: Look at the graph of f.)In the following exercises, use the identity AAf(x)dx= A0f( x)dx+ 0 A f( x )dx to compute the integrals. 97. 24( x3)3dx (Hint: Look at the graph of f.)In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 98. 01(1+x+x2+x3)dx In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 99. 01(1x+x2x3)dx In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 100. 01(1x2)dx In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 101. 01(12x3)dx In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 102. 01(6x43x2)dx In the following exercises, given that 01xdx=12,01x2dx=13 and 01x3dx=14 compute the integrals. 103. 01(75x3)dx In the following exercises, use the comparison theorem. 104. Show that 03(x26x+9)dx0 In the following exercises, use the comparison theorem. 105. Show that 23(x3)(x+2)dx0 .In the following exercises, use the comparison theorem. 106. Show that 011+x3dx011+x2dx..In the following exercises, use the comparison theorem. 107. Show that 121+xdx121+x2dx .In the following exercises, use the comparison theorem. 108. Show that 0/2sintdt4. (Hint: sintet over [0,2] )In the following exercises, use the comparison theorem. 109. Show that /4/4costdt2/4 .In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 110. f(x)=x2 , a = -1, b = 1 In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 111. f(x)=x5 , a = -1, b = 1 In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 112. f(x)=4x2a = 0, b = 2 In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 113. f(x)=(3|x|) , a = -3, b = 3 In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 114. f(x)=sinx , a = 0, b=2 In the following exercises, find the average value fave of f between a and b, and find a point c, where f(c)=fave . 115. f(x)=cosx , a = 0, b=2 In the following exercises, approximate the average value using Riemann sums L100 and R100 . How does your answer compare with the exact given answer? 116. [T] y=In(x) over the interval [1,4]; the exact solution is In(256)31 .In the following exercises, approximate the average value using Riemann sums L100 and R100 . How does your answer compare with the exact given answer? 117. [T] y=ex/2 over the interval [0, 1]; the exact Solution is 2(e1) .the following exercises, approximate the average value using Riemann sums L100 and R100 . How does your answer compare with the exact given answer? 118. [T] y=tanx over the interval [0,4] , the exact solution is 2In(2) .In the following exercises, approximate the average value using Riemann sums L100 and R100 . How does your answer compare with the exact given answer? 119. [T] y=x+14x2 over the interval [-1, 1]; the exact solution is 6 .In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10, 100. How does the accuracy compare with the given exact value? 120. [T] y=x24 over the interval [0, 2]; the exact solution is 83 .In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10, 100. How does the accuracy compare with the given exact value? 121. [T] y=xex2 over the interval [0, 2]; the exact solution is 14(e41) .In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10, 100. How does the accuracy compare with the given exact value? 122. [T] y=(12)x over the interval [0, 4]; the exact solution is 1564In(2) .[T] y=xsin(x2) over the interval [,0] ; the exact solution is cos(2)12 .Suppose that A=02sin2tdt and B=02cos2tdt . Show that A+B=2 and A = B.Suppose that A=/4/4sec2tdt= and B=/4/4tan2tdt= . Show that AB=2 .Show that the average value of sin2t over [0,2] is equal to 1/2 Without further calculation, determine whether the average value of sin2t over [0,] is also equal to 1/2.Show that the average value of cos2t over [0,2] is equal to 1/2. Without further calculation, determine whether the average value of cos2t over [0,] is also equal to 1/2.Explain why the graphs of a quadratic function (parabola) p(x) and a linear function l(x) can intersect in at most two points. Suppose that p(a)=l(a) and p(b)=l(b) , and that abp(t)dtabl( t)dt . Explain why cdp(t)cdl( t)dt whenever acdb .Suppose that parabola p(x)=ax2+bx+c opens downward 3 (a0) and has a vertex of y=b2a0 . For which interval [A, B] is AB(ax2+bx+c)dx as large as possible?Suppose [a, b] can be subdivided into subintervals a=a0a1a2...aN=b such that either f0 over [ai1,ai] or f0 over [ai1,ai] . Set Ai=ai1aif(t)dt . a. Explain why abf(t)dt=A1+A2+...+AN . b. Then, explain why |abf(t)dt|ab|f(t)|dt .Suppose f and g are continuous functions such that cdf(t)dtcdg( t)dt for every subinterval [c, d] of [a, b]. Explain why f(x)g(x) for all values of x.Suppose the average value of f over [a, b] is 1 and the average value of f over [b, c] is 1 where a < c < b. Show that the average value of f over [a, c] is also 1.Suppose that [a, b] can be partitioned, taking a=a0a1...aN=b such that the average value of f over each subinterval [ai1,ai] is equal to 1 for each i = 1,…, N. Explain why the average value of f over [a, b] is also equal to 1.Suppose that for each i such that 1iN one has i1if(t)dt=i . Show that 0Nf(t)dt=N(N+1)2 .Suppose that for each i such that 1iN one has i1if(t)dt=i2 . Show that 0Nf(t)dt=N( N+1)( 2N+1)6 .[T] Compute the left and right Riemann sums L10 and R10 and their average L10+R102 for f(t)=t2 over [0, 1]. Given that 01t2dt=0.33 . to how many decimal places is L10+R102 accurate?[T] Compute the left and right Riemann sums, L10 and R10 , and their average L10+R102 for f(t)=(4t2) over [1, 2]. Given that 12(4t2)dt=1.66 , to how many decimal places is L10+R102 accurate?If 151+t4dt=41.7133..., what is 151+u4du ?Estimate 01tdt using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value 01tdt ?Estimate 01tdt by comparison with the area of a single rectangle with height equal to the value of t at the midpoint t=12 . How does this midpoint estimate compare with the actual value 01tdt ?From the graph of sin(2x) shown: a. Explain why 01sin(2t)dt=0 . b. Explain why, in general, aa+1sin(2t)dt=0 for any value of a.If f is 1-periodic (f(t+1)=f(t)) , odd, and integrable over [0, 1], is it always true that 01f(t)dt=0 ?If f is 1-periodic and 01f(t)dt=A , is it necessarily true that a1+af(t)dt=A for all A?How long after she exits the aircraft does Julie reach terminal velocity?Based on your answer to question 1, set up an expression involving one or more integrals that represents the distance Julie falls after 30 sec.If Julie pulls her ripcord at an altitude of 3000 ft, how long does she spend in a free fall?Julie pulls her ripcord at 3000 ft. It takes 5 sec for her parachute to open completely and for her to slow down, during which time she falls another 400 ft. After her canopy is fully open, her speed is reduced to 16 ft/sec. Find the total time Julie spends in the air, from the time she leaves the airplane until the time her feet touch the ground. On Julie’s second jump of the day, she decides she wants to fall a little faster and orients herself in the “head down” position. Her terminal velocity in this position is 220 ft/sec. Answer these questions based on this velocity:How long does it take Julie to reach terminal velocity in this case?Before pulling her ripcord, Julie reorients her body in the “belly down” position so she is not moving quite as fast when her parachute opens. If she begins this maneuver at an altitude of 4000 ft, how long does she spend in a free fall before beginning the reorientation? Some jumpers wear “ wingsuits” (see Figure 5.31). These suits have fabric panels between the arms and legs and allow the wearer to glide around in a free fall, much like a flying squirrel. (Indeed, the suits are sometimes called “flying squirrel suits.”) When wearing these suits, terminal velocity can be reduced to about 30 mph (44 ft/sec), allowing the wearers a much longer time in the air. Wingsuit flyers still use parachutes to land; although the vertical velocities are within the margin of safety, horizontal velocities can exceed 70 mph, much too fast to land safely.Answer the following question based on the velocity in a wingsuit. 7. If Julie does a wingsuit before her third jump of the day, and she pulls her ripcord at an altitude of 3000 ft, how long does she got to spend gliding around in the air?Consider two athletes running at variable speeds v1(t) and v2(t) . The runners start and finish a race at exactly the same time. Explain why the two runners must be going the same speed at some point.Two mountain climbers start their climb at base camp, raking two different routes, one steeper than the other, and arrive at the peak at exactly the same time. Is it necessarily true that, at some point, both climbers increased in altitude at the same rate?To get on a certain toll road a driver has to take a card that lists the mile entrance point. The card also has a timestamp. When going to pay the toll at the exit, the driver is surprised to receive a speeding ticket along with the toll. Explain how this can happen.Set F(x)=1x(1t)dt . Find F(2) and the average value of F over [1, 2].In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 148. ddx1xet2dt the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 149. ddx1xecostdt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 150. ddx3x9y2dy In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 151. ddx4xds 16s2 In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 152. ddxx2xtdt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 153. ddx0xtdt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 154. ddx0sinx1t2dt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 155. ddxcosx11t2dt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 156. ddx1xt21+t4dt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 157. ddx1x2t1+tdt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 158. ddx0Inxetdt In the following exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative. 159. ddx1e2Inu2du The graph of y=0xf(t)dt , where f is a piecewise constant function, is shown here. Over which intervals is f positive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero? What are the maximum and minimum values of f? What is the average value of f?The graph of y=0xf(t)dt , where f is a piecewise constant function, is shown here. Over which intervals is f positive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero? What are the maximum and minimum values of f? What is the average value of f?The graph of y=0xl(t)dt , where l is a piecewise constant function, is shown here. Over which intervals is l positive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero? Over which intervals is l increasing? Over which is it decreasing? Over which, if any, is it constant? What is the average value of l ?The graph of y=0xl(t)dt , where l is a piecewise constant function, is shown here. Over which intervals is lpositive? Over which intervals is it negative? Over which intervals, if any, is it equal to zero? Over which intervals is l increasing? Over which is it decreasing? Over which, if any, is it constant? What is the average value of l?In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 164. [T] y=x2 over [0, 4]In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 165. [T] y=x3+6x2+x5 over [-4, 2]In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 166. [T] y=x3 over [0, 6]In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 167. [T] y=x+x2 over [1, 9]In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 168. [T] (cosxsinx)dx over [0,]In the following exercises, use a calculator to estimate the area under the curve by computing T10 , the average of the left- and right-endpoint Riemann sums using N = 10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area. 169. [T] 4x2dx over [1, 4]In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 170. 12(x23x)dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 171. 23(x2+3x5)dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 172. 23(t+2)(t3)dt In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 173. 23(t29)(4t2)dt In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 174. 12x9dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 175. 01x99dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 176. 48(4t 5/2 3t 3/2 )dt In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 177. 1/44(x21 x2 )dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 178. 122x3dx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 179. 1412xdx In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 180. 142tt2dt In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 181. 116dtt 1/4 In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 182. 02cosd In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 183. 0/2sind In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 184. 0/4sec2d In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 185. 0/4sectan In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 186. /3/4csccotd In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 187. /4/2csc2d In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 188. 12(1 t2 1 t3 )dt In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2. 189. 21(1 t2 1 t3 )dt In the following exercises, use the evaluation theorem to express the integral as a function F(x). 190. axt2dt In the following exercises, use the evaluation theorem to express the integral as a function F(x). 191. 1xetdt In the following exercises, use the evaluation theorem to express the integral as a function F(x). 192. 0xcostdt In the following exercises, use the evaluation theorem to express the integral as a function F(x). 193. xxsintdt In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2. 194. 23|x|dx In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2. 195. 24|t22t3|dt In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2. 196. 0|cost|dt In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2. 197. /2/2|sint|dt Suppose that the number of hours of daylight on a given day in Seattle is modeled by the function 3.75cos(t6)+12.25 , with t given in months and t = 0 corresponding to the winter solstice. What is the average number of daylight hours in a year? At which times t1 and t2 where 0t1t212, do the number of daylight hours equal the average number? Write an integral that expresses the total number of daylight hours in Seattle between t1 and t2 . Compute the mean hours of daylight in Seattle between t1 and t2 where 0t1t212 , and then between t1 and t2 , and show that the average of the two is equal to the average day length.Suppose the rate of gasoline consumption in the United States can be modeled by a sinusoidal function of the form (11.21cos(t6))109gal/mo . What is the average monthly consumption, and for which values of t is the rate at time t equal to the average rate? What is the number of gallons of gasoline consumed in the United States in a year? Write an integral that expresses the average monthly U.S. gas consumption during the part of the year between the beginning of April (t = 3) and the end of September (t = 9).Explain why, if f is continuous over [a, b], there is at least one point c[a,b] such that f(c)=1baabf(t)dt .Explain why, if f is continuous over [a, b] and is not equal to a constant, there is at least one point M[a,b] such that f(M)=1baabf(t)dt and at least one point M[a,b] such that f(m)1baabf(t)dt .Kepler’s first law states that the planets move in elliptical orbits with the Sun at one focus. The closest point of a planetary orbit to the Sun is called the perihelion (for Earth, it currently occurs around January 3) and the farthest point is called the aphelion (for Earth, it currently occurs around July 4). Kepler's second law states that planets sweep out equal areas of their elliptical orbits in equal times. Thus, the two arcs indicated in die following figure are swept out in equal times. At what time of year is Earth moving fastest in its orbit? When is it moving slowest?A point on an ellipse with major axis length 2a and minor axis length 2b has the coordinates (acos,bsin),02. Show that the distance from this point to the focus at (-c, 0) is d=a+ccos , where c=a2b2 . Use these coordinates to show that the average distance d from a point on the ellipse to the focus at (-c, 0), with respect to angle , is a.As implied earlier, according to Kepler's laws, Earth's orbit is an ellipse with the Sun at one focus. The perihelion for Earth’s orbit around the Sun is 147,098,290 km and the aphelion is 152,098,232 km. By placing the major axis along the x-axis, find the average distance from Earth to the Sun. The classic definition of an astronomical unit (AU) is the distance from Earth to the Sun, and its value was computed as the average of the perihelion and aphelion distances. Is this definition justified?The force of gravitational attraction between the Sun and a planet is F()=GmMr2 , where m is the mass of the planet, M is the mass of the Sun, G is a universal constant, and r( ) is the distance between the Sun and the planet when the planet is at an angle with the major axis of its orbit. Assuming that M, m, and the ellipse parameters a and b (half-lengths of the major and minor axes) are given, set up—but do not evaluate—an integral that expresses in terms of G, m, M, a, b the average gravitational force between the Sun and the planet.The displacement from rest of a mass attached to a spring satisfies the simple harmonic motion equation x(t)=Acos(t) , where is a phase constant, is the angular frequency, and A is the amplitude. Find the average velocity, the average speed (magnitude of velocity), the average displacement, and the average distance from rest (magnitude of displacement) of the mass.Use basic integration formulas to compute the following antiderivatives. 207. (x1 x )dx Use basic integration formulas to compute the following antiderivatives. 208. (e 2x12e x/2 )dx Use basic integration formulas to compute the following antiderivatives. 209. dx2x Use basic integration formulas to compute the following antiderivatives. 210. x1x2dx Use basic integration formulas to compute the following antiderivatives. 211. 0(sinxcosx)dx Use basic integration formulas to compute the following antiderivatives. 212. 0/2(xsinx)dx Write an integral that expresses the increase in the perimeter P(s) of a square when its side length s increases from 2 units to 4 units and evaluate the integral.Write an integral that quantifies the change in the area A(s)=s2 of a square when the side length doubles from S units to 2S units and evaluate the integral.A regular N-gon (an N-sided polygon with sides that have equal length s, such as a pentagon or hexagon) has perimeter Ns. Write an integral that expresses the increase in perimeter of a regular N-gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.The area of a regular pentagon with side length a0 is pa2 with p=145+5+25 . The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area.A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2a units?Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2s units and evaluate the integral.Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2s units and evaluate the integral.Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.Suppose that a particle moves along a straight line with velocity v(t)=42t , where 0t2 (in meters per second). Find the displacement at time r and the total distance traveled up to t = 2.Suppose that a particle moves along a straight line with velocity defined by v(t)=t23t18 , where 0t6 (in meters per second). Find the displacement at time t and the total distance traveled up to t = 6.Suppose that a particle moves along a straight line with velocity defined by v(t)=|2t6| , where 0t6 (in meters per second). Find the displacement at time t and the total distance traveled up to t = 6.Suppose that a particle moves along a straight line with acceleration defined by a(t)=t3 , where 0t6 (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to t = 6 if v(0) = 3 and d(0) = 0.A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is -9.8 m/sec2. Neglecting air resistance, solve for the velocity v(t) and the height h(t) of the ball t seconds after it is thrown and before it returns to the ground.A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is -9.8 m/sec2. Neglecting air resistance, solve for the velocity v(t) and die height h(t) of the ball t seconds after it is thrown and before it returns to the ground.The area A(t) of a circular shape is growing at a constant rate. If the area increases from 4 units to 9 units between times t = 2 and t = 3, find the net change in the radius during that time.A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36in.3 to 288in.3 between time t = 30 and t = 60 seconds, find the net change in the radius of the balloon during that time.Water flows into a conical tank with cross-sectional area x2 at height x and volume x33 up to height x. If water flows into the tank at a rate of 1 m3/min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.A horizontal cylindrical tank has cross-sectional area A(x)=4(6xx2)m2 at height x meters above the bottom when x3 . a. The volume V between heights a and b is abA(x)dx . Find the volume at heights between 2 m and 3 m. b. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, dxdt=dxdVdVdt , at how many meters per minute is the height of oil in the tank changing, expressed in terms of x, when the height is at x meters? c. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical 24-hour period, with hour 1 corresponding to midnight to 1 a.m. Hour Power Hour Power 1 28 13 48 2 25 14 49 3 24 15 49 4 23 16 50 5 24 17 50 6 27 18 50 7 29 19 46 8 32 20 43 9 34 21 42 10 39 22 40 11 42 23 37 12 46 24 34 Find the total amount of power in gigawatt-hours (gW-h) consumed by the city in a typical 24-hour period.The average residential electrical power use (in hundreds of watts) per hour is given in the following table. Hour Power Hour Power 1 8 13 12 2 6 14 13 3 5 15 14 4 4 16 15 5 5 17 17 6 6 18 19 7 7 19 18 8 8 20 17 9 9 21 16 10 10 22 16 11 10 23 13 12 11 24 11 Compute the average total energy used in a day in kilowatt-hours(kWh). If a ton of coal generates 1842 kWh, how long does it take for an average residence to burn a ton of coal? Explain why the data might fit a plot of the form p(t)=11.57.5sin(t12) .The data in the following table are use to estimate the average power output produced by Peter Sagan for each of the last 18 sec of Stage 1 of the 2012 Tour de France. Second Watts Second Watts 1 600 10 1200 2 500 11 1170 3 575 12 1125 4 1050 13 1100 5 925 14 1075 6 950 15 1000 7 1050 16 950 8 950 17 900 9 1100 18 780 Table 5.6 Average Power Output Source: Sportsexercisenginnering.com Estimate the net energy used in kilojoules (kJ), noting that 1W=1 J/s, and the average power output by Sagan during this time interval.The data in the following table are used to estimate the average power output produced by Peter Sagan for each Minutes Watts Minutes Watts 15 200 165 170 30 180 180 220 45 190 195 140 60 230 210 225 75 240 225 170 690 210 240 210 105 210 255 200 120 220 270 220 135 210 285 250 130 150 300 400 Table 5.7 Average Power Output Source: sportsexercisengineering.com Estimate the net energy used in kilojoules, noting that 1W = 1 J/s.The distribution of incomes as of 2012 in the United States in $5000 increments is given in the following table. The kth row denotes the percentage of households with incomes between $5000xk and 5000 xk +4999. The row k = 40 contains all households with income between $200,000 and $250,000 and k = 41 accounts for all households with income exceeding $250,000. 0 3.5 21 1.5 1 4.1 22 1.4 2 5.9 23 1.3 3 5.7 24 1.3 4 5.9 25 1.1 5 5.4 26 1.0 6 4.4 27 0.75 7 4.1 28 0.8 8 4.8 29 1.0 9 4.1 30 0.6 10 4.3 31 0.6 11 3.5 32 0.5 12 3.7 33 0.5 13 3.2 34 0.4 14 3.0 35 0.3 15 2.8 36 0.3 16 2.5 37 0.3 17 2.2 38 0.2 18 2.2 39 1.8 19 1.8 40 2.3 20 2.1 41 Table 5.8 Income Distributions Source: http: //www. census.gov/prod/2013 pubs/p60-245.pdf a. Estimate the percentage of U.S households in 2012 with incomes less than $55,000. b. What percentage of households had incomes exceeding $85,000? c. Plot the data and try to fit its shape to that of a graph of the form a(x+c)eb(x+e) for suitable a, b, c.Newton’s law of gravity states that the gravitational force exerted by an object of mass M and one of mass m with centers that are separated by a distance r is F=GmMr2 , with G an empirical constant G=6.671011m3/(kgs2)G=6.671011m3/(kgs2) The work done by a variable force over an interval [a, b] is defined as W=abF(x)dx. If Earth has mass 5.972191024 and radius 6371 km. compute the amount of work to elevate a polar weather satellite of mass 1400 kg to its orbiting altitude of 850 km above Earth.For a given motor vehicle, the maximum achievable deceleration from braking is approximately 7 m/sec on dry concrete. On wet asphalt, it is approximately 2.5 m/sec2. Given that 1 mph corresponds to 0.447 m/sec, find the total distance that a car travels in meters on dry concrete after the brakes are applied until it comes to a complete stop if the initial velocity is 67 mph (30 m/sec) or if die initial braking velocity is 56 mph (25 m/sec). Find the corresponding distances if the surface is slippery wet asphalt.John is a 25-year old man who weighs 160 lb. He burns 500 — 50t calories/hr while riding his bike for t hours. If an oatmeal cookie has 55 cal and John eats 4t cookies during the tth hour, how many net calories has he lost after 3 hours riding his bike?Sandra is a 25-year old woman who weighs 120 lb. She bums 300 — 50t cal/hr while walking on her treadmill. Her caloric intake from drinking Gatorade is 100t calories during the tth hour. What is her net decrease in calories after walking for 3 hours?A motor vehicle has a maximum efficiency of 33 mpg at a cruising speed of 40 mph. The efficiency drops at a rate of 0.1 mpg/mph between 40 mph and 50 mph, and at a rate of 0.4 mpg/mph between 50 mph and 80 mph. What is the efficiency in miles per gallon if the car is cruising at 50 mph? What is the efficiency in miles per gallon if the car is cruising at 80 mph? If gasoline costs $3.50/gal, what is the cost of fuel to drive 50 mi at 40 mph, at 50 mph, and at 80 mph?Although some engines are more efficient at given a horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of 1/25 mpg/horsepower. If a typical 50-horsepower engine has an average fuel efficiency of 32 mpg, what is the average fuel efficiency of an engine with the following horsepower: 150, 300, 450?[T] The following table lists the 2013 schedule of federal income tax versus taxable income. Taxable Income Range The Tax Is… ... Of the Amount Over $0-$3925 10% $0 $8925-$36,250 $892.50 + 15% $8925 $36,250-$87,850 $4,991.25 + 25% $36,250 $87,8 50-$183,250 $17,891.25 + 28% $87,850 $183,250-$398,350 $44,603.25 + 33% $103,250 $398,350-$400,000 $115,586.25 + 35% $398,350 > $400,000 $116,163.75 + 39.6% $400,000 Table 5.9 Federal Income Tax Versus Taxable Income Source: http://www.irs.gov/pub/irs-prior/ i1040tt--2013.pdf.’ Suppose that Steve just received a $10,000 raise. How much of this raise is left after federal taxes if Steve’s salary before receiving tire raise was $40,000? If it was $90,000? If it was $385,000?[T] The following table provides hypothetical data regarding the level of service for a certain highway. Highway Speed Range (mph) Vehicles per Hour per Lane Density Range (vehicles/ mi) >60 < 600 < 10 60-57 600-1000 10-20 57-54 1000-1500 20-30 54-46 1500-1900 30-45 46-30 1900-2100 45-70 <30 Unstable 70-200 Table 5.10 Plot vehicles per hour per lane on the x-axis and highway speed on the y-axis. Compute the average decrease in speed (in miles per hour) per unit increase in congestion (vehicles per hour per lane) as the latter increases from 600 to 1000, from 1000 to 1500, and from 1500 to 2100. Does the decrease in miles per hour depend linearly on the increase in vehicles per hour per lane? Plot minutes per mile (50 times the reciprocal of miles per hour) as a function of vehicles per hour per lane. Is this function linear? For the next two exercises use the data in the following table, which displays bald eagle populations from 1963 to 2000 in the continental United States. Year Population of Breeding Pairs of Bald Eagles 1963 487 1974 791 1981 1188 1986 1875 19921996 3749 5094 2000 6471 Table 5.11 Population of Breeding Bald Eagle Pairs Source: http://www.fws.gov/Midwest/eagle/ population/chtofprs.html.[T] The graph below plots the quadratic p(t)=6.48t280.31t+585.69 against the data in preceding table, normalized so that t = 0 corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over [0, 37].[T] the graph below plots the cubic p(t)=0.07t3+2.42t225.63t+521.23 against the data in the preceding table, normalized so that t = 0 corresponds to 1963.Estimate the average number so that t = 0 corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over [0, 37].[T] Suppose you go on a road trip and record your speed at every half hour, as compiled in the following table. The best quadratic fit to the data is q(t)=5x211x+49 , shown in the accompanying graph. Integrate q to estimate the total distance driven over the 3 hours. Time(hr) Speed (mph) 0 (start) 50 1 40 2 50 3 60 As a car accelerates, it does not accelerate at a constant rate; rather, the acceleration is variable. For the following exercises, use die following table, which contains the acceleration measured at every second as a driver merges onto a freeway. Time(sec) Acceleration (mph/sec) 1 11.2 2 10.6 3 8.1 4 5.4 5 0 [T] The accompanying graph plots the best quadratic fit, a(t)=0.70t2+1.44t+10.44 , to the data from the preceding table. Compute the average value of a(t) to estimate the average acceleration between t = 0 and t = 5.[T] Using your acceleration equation from the previous exercise, find the corresponding velocity equation. Assuming the final velocity is 0 mph, find the velocity at time t = 0.[T] Using your velocity equation from the previous exercise, find the corresponding distance equation, assuming your initial distance is 0 mi. How far did you travel while you accelerated your car? (Hint: You will need to convert time units.)[T] The number of hamburgers sold at a restaurant throughout the day is given in the following table, with the accompanying graph plotting the best cubic fit to the data, b(t)=0.12t32.13t3+12.13t+3.91 , with t = 0 corresponding to 9 a.m. and t= 12 corresponding to 9 p.m. Compute the average value of b(t) to estimate the average number of hamburgers sold per hour. Hours Past Midnight No. of Burgers Sold 9 3 12 28 15 20 18 30 21 45 [T] An athlete runs by a motion detector, which records her speed, as displayed in the following table. The best linear fit to this data, l(t)=0.068t+5.14 , is shown in the accompanying graph. Use the average value of l(t) between t = 0 and t = 40 to estimate the runner’s average speed. Minutes Speed (m/sec) 0 5 10 4.8 20 3.6 30 3.0 40 2.5 Why is u-substitution referred to as change of variable?2. If f=gh, when reversing the chain title, ddx(gh)(x)=g(h(x))hx , should you take u=g(x) or u=h(x) ?In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify / such that the integral takes the form f(u)du. 256. xx+1dx=215( x+1)3/2(3x2)+C;u=x+1 In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify / such that the integral takes the form f(u)du. 257. x2 x1dx(x1)=215x1(3x2+4x+8)+C;u=x1 In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify / such that the integral takes the form f(u)du. 258. x4x2+9dx=112( 4x2 +9)3/2+C;u=4x2+9 In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify / such that the integral takes the form f(u)du. 259. x 4x2 +9dx=144x2+9+C;u=4x2+9

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