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All Textbook Solutions for Calculus (MindTap Course List)

69E 70E a Show that 11+x31+x3forx0 b Show that 1011+x3dx1.25.72E Show that 0510x2x4+x2+1dx0.1 by comparing the integrand to a simpler function.Let f(x)={0ifx0xif0x12xif1x20ifx2 and g(x)=0xf(t)dt a Find an expression for f(x) similar to the one for f(x). b Sketch the graphs of f and g. c Where is f differentiable? Where is g differentiable?75E 76E A manufacturing company owns a major piece of equipment that depreciates at the continuous rate f=f(t), where t is the time measured in months since its last overhaul. Because a fixed cost A is incurred each time the machine is overhauled, the company wants to determine the optimal time T in months between overhauls. a Explain why 0tf(s)ds represents the loss in value of the machine over the period of time t since the last overhaul. b Let C=C(t) be given by C(t)=1t[A+0tf(s)ds] What does C represent and why would the company want to minimize C? c Show that C has a minimum value at the numbers t=T where C(T)=f(T).78E The following exercises are intended only for those who have already covered Chapter 6. Evaluate the integral. 1912xdx The following exercises are intended only for those who have already covered Chapter 6. Evaluate the integral. 0110xdx 81E 82E 83E 84E Verify by differentiation that the formula is correct. 1x21+x2dx=1+x2x+C Verify by differentiation that the formula is correct. cos2xdx=12x+14sin2x+C Verify by differentiation that the formula is correct. tan2xdx=tanxx+C Verify by differentiation that the formula is correct. xa+bxdx=215b2(3bx2a)(a+bx)3/2+C 5E 6E Find the general indefinite integral. (5+23x2+34x3)dx 8E Find the general indefinite integral. (u+4)(2u+1)du 10E 11E 12E 13E 14E Find the general indefinite integral. 1sin3tsin2tdt 16E 17E 18E 19E 20E 21E 22E 23E Evaluate the integral. 11t(1t)2dt 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E Evaluate the integral. 25|x3|dx 40E 41E 42E Use a graph to estimate the x-intercepts of the curve y=12x5x4. Then use this information to estimate the area of the region that lies under the curve and above the x-axis.44E The area of the region that lies to the right of the y-axis and to the left of the parabola x=2yy2 the shaded region in the figure is given by the integral 02(2yy2)dy. Turn your head clockwise and think of the region as lying below the curve x=2yy2 from y=0 to y=2 Find the area of the region.The boundaries of the shaded region in the figure are the y-axis, the line y = 1, and the curve y=x4. Find the area of this region by writing x as a function of y and integrating with respect to y as in Exercise 45.47E 48E 49E A honeybee population starts with 100 bees and increases at a rate of n(t) bees per week. What does 100+015n(t) represent?51E 52E 53E 54E The velocity function in meters per second is given for a particle moving along a line. Find a the displacement and b the distance traveled by the particle during the given time interval. v(t)=3t5,0t3 56E The acceleration function in m/s2 and the initial velocity are given for a particle moving along a line. Find a the velocity at time t and b the distance traveled during the given time interval. a(t)=t+4,v(0)=5,0t10 58E 59E 60E 61E Suppose that a volcano is erupting and readings of the rate rt at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for rt are tonnes metric tons per second. t 0 1 2 3 4 5 6 r(t) 2 10 24 36 46 54 60 a Give upper and lower estimates for the total quantity Q(6) of erupted materials after six seconds. b Use the Midpoint Rule to estimate Q(6).Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 am by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 am to July 26th at 7:30 am. Day Inflow rate (ft3/s) July 18 5275 July 19 6401 July 20 2554 July 21 4249 July 22 3016 July 23 3821 July 24 2462 July 25 2628 July 26 3003 64E The graph of the acceleration a(t) of a car measured in ft/s2 is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval.Shown is the graph of traffic on an Internet service providers T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period.67E 68E 69E 70E 71E 72E 73E The area labeled B is three times the area labeled A. Express b in terms of a.Evaluate the integral by making the given substitution. cos2xdx,u=2x Evaluate the integral by making the given substitution. x(2x2+3)4dx,u=2x2+3 3E Evaluate the integral by making the given substitution. sin2cosd,u=sin 5E Evaluate the integral by making the given substitution. 2t+1dt,u=2t+1 7E Evaluate the indefinite integral. x2sin(x3)dx 9E Evaluate the indefinite integral. sin1+costdt Evaluate the indefinite integral. sin(2/3)d Evaluate the indefinite integral. sec22d 13E Evaluate the indefinite integral. y2(4y3)2/3dy 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E Evaluate the indefinite integral. sec2xtan2xdx Evaluate the indefinite integral. sec3xtanxdx 28E Evaluate the indefinite integral. x(2x+5)8dx 30E Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. take C=0. x(x21)3dx 32E Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. take C=0. sin3xcosxdx Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative. take C=0. sinxcos4xdx 35E 36E Evaluate the definite integral. 011+7x3dx 38E Evaluate the integral. 0/6sintcos2tdt 40E Evaluate the definite integral. /4/4(x3+x4tanx)dx 42E 43E 44E 45E Evaluate the definite integral. /3/3x4sinxdx Evaluate the definite integral. 12xx1dx 48E Evaluate the definite integral. 1/21cos(x2)x3dx 50E Evaluate the definite integral. 01dx(1+x)4 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E If f is continuous function on , prove that abf(x+c)dx=a+cb+cf(x)dx For the case where f(x)0, draw a diagram to intercept this equation geometrically as an equality of areas.If a and b are positive numbers, show that 01xa(1x)bdx=01xb(1x)adx 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E 83E 84E 85E a Write an expression for a Riemann sum of a function f on an interval [a,b]. Explain the meaning of the notation that you use. b If f(x)0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. c If f(x) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.a Write the definition of the definite integral of a continuous function from a to b. b What is the geometric interpretation of abf(x)dx if f(x)0? c What is the geometric interpretation of abf(x)dx if f(x) takes on both positive and negative values? Illustrate with a diagram.3CC 4CC 5CC 6CC a Explain the meaning of the indefinite integral f(x)dx. b What is the connection between the definite integral abf(x)dx and the indefinite integral f(x)dx?Explain exactly what is meant by the statement that differentiation and integration are inverse processes.State the Substitution Rule. In practice, how do you use it?1TFQ 2TFQ 3TFQ 4TFQ 5TFQ 6TFQ 7TFQ 8TFQ 9TFQ 10TFQ 11TFQ 12TFQ 13TFQ 14TFQ 15TFQ 16TFQ 17TFQ 18TFQ Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be a left endpoints and b mid points. In each case draw a diagram and explain what the Riemann sum represents.a Evaluate the Riemann sum for f(x)=x2x0x2 With four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. b Use the definition of a definite integral with right endpoints to calculate the value of the integral 02(x2x)dx c Use the Fundamental Theorem to check your answer to part b. d Draw a diagram to explain the geometric meaning of the integral in part b.3E 4E 5E 6E 7E 8E The graph of f consists of the three line segments shown. If g(x)=0xf(t)dt, find g(4) and g(4).10E 11E 12E 13E 14E 15E 16E 17E 18E Evaluate the integral, if it exists. 15dt(t4)2 20E 21E 22E 23E 24E 25E 26E 27E 28E Evaluate the integral, if it exists. 03|x24|dx 30E 31E 32E 33E 34E 35E 36E 37E Find the derivative of the function. g(x)=1sinx1t21+t4dt Find the derivative of the function. y=xxcosd 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E Let f(x)={x1if3x01x2if0x1 Evaluate 31f(x)dx by interpreting the integral as a difference of areas.51E The Fresnel function S(x)=0xsin(12t2)dt was introduced in Section 4.3. Fresnel also used the function C(x)=0xcos(12t2)dt in his theory of the diffraction of light waves. a On what intervals is C is increasing? b On what intervals is C concave upward? c Use a graph to solve the following equation correct to two decimal places: 0xcos(12t2)dt=0.7 d Plot the graphs of C and S on the same screen. How are these graphs related?53E 54E 55E Find limh01h22+h1+t3dt 57E 58E If xsinxx=0x2f(t)dt, where f is a continuous function, find f4.2P If f is a differentiable function such that f(x) is never 0 and 0xf(t)dt=[f(x)]2 for all x, find f.4P 5P 6P 7P 8P 9P 10P Suppose the coefficients of the cubic polynomial P(x)=a+bx+cx2+dx3 satisfy the equation a+b2+c3+d4=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?12P 13P The figure shows a parabolic segment, that is, a portion of a parabola cut off by a chord AB. It also shows a point C on the parabola with the property that the tangent line at C is parallel to the chord AB. Archimedes proved that the area of the parabolic segment is 43 times the area of the inscribed triangle ABC. Verify Archimedes result for the parabola y=4x2 and the line y=x+2.Given the point a, b in the first quadrant, find the downward-opening parabola that passed through the point a, b and the origin such that the area under the parabola is a minimum.The figure shows a region consisting of all points inside a square that are closer to the center than to the sided of the square. Find the area of the region.17P For any number c, we let fc(x) be the smaller of the two numbers (xc)2 and (xcc)2. Then we define g(c)=01fc(x)dx. Find the maximum and minimum values of g(c)if2c2.Find the area of the shaded region.Find the area of the shaded region.3E Find the area of the shaded region.Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y=x+1,y=9x2,x=1,x=2 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y=sinx,y=x,x=/2,x=Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y=(x2)2,y=x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y=x24x,y=2x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y=x+3,y=(x+3)/2 10E Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x=1y2,x=y21 12E Sketch the region enclosed by the given curves and find its area. y=12x2,y=x26 14E 15E Sketch the region enclosed by the given curves and find its area. y=cosx,y=2cosx,0x2 Sketch the region enclosed by the given curves and find its area. x=2y2,x=4+y2 Sketch the region enclosed by the given curves and find its area. y=x1,xy=1 19E Sketch the region enclosed by the given curves and find its area. x=y4,y=2x,y=0 21E Sketch the region enclosed by the given curves and find its area. y=x3,y=x

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