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All Textbook Solutions for Calculus

47E 48E 49E 54E Finding a Derivative In Exercises 3958, find the derivative of the function. y=ln(t2+4)12arctant2 51E 52E 53E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E Finding Relative Extrema In Exercises 63-66, find any relative extrema of the function. f(x)=arcsecxx Finding Relative Extrema In Exercises 63-66, find any relative extrema of the function. f(x)=arcsinx2x 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 83E 84E 82E 87E 88E 89E True or False? In Exercises 83-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. arcsin2x+arccos2x=1 85E 86E Linear and Quadratic Approximations In Exercises 6568, use a computer algebra system to find the linear approximation P1(x) = f(a) + f'(a)(x-a) and the quadratic approximation P2(x)=f(a)+f(a)(xa)+12f(a)(xa)2 of the function f at x = a. Sketch the graph of the function and its linear and quadratic approximations. f(x)=arctanx,a=0 66E 67E 68E Angular Rate of Change An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider and x as shown in the figure. (a) Write as a function of x. (b) The speed of the plane is 400 miles per hour. Find d/dt when x=10 miles and x=3 miles.92E Angular Rate of Change In a free-fall experiment, an abject is dropped from a height of 256 feet. A camera on the ground 500 feet from the point of impact records the fall of the object (see figure). (a) Find the position function that yields the height of the object at time t, assuming the object is released at time t=0. At what lime will the object reach ground level? (b) Find the rates of change of the angle of elevation of the camera when t=1 and t=2.Angular Rate of Change A television camera at ground level is filming the lift-off of a rocket at a point 800 meters from the launch pad. Let be the angle of elevation of the rocket and let s be the distance between the camera and the rocket (see figure). Write as a function of s for the period of time when the rocket is moving vertically. Differentiate the result to find d/dt in terms of s and ds/dt.Maximizing an Angle A billboard 85 feet wide is perpendicular to a straight road and is 40 feet from the road (see figure). Find the point on the road at which the angle subtended by the billboard is a maximum.Angular Speed A patrol car is parked 50 feet from a long ware house (see figure). The revolving light on top of the car turns at a rate of 30 revolutions per minute. Write as a function of x. How fast is the light beam moving along the wall when the beam makes an angle of =45 with the line perpendicular from the light to the wall?97E 98E 99E 100E Maximizing an Angle In the figure, find the value of c in the interval [0,4] on the x-axis that maximizes angle .102E 103E 104E 1E 2E 3E 4E Finding an Indefinite Integral In Exercises 3-22, find the indefinite integral. 11(x+1)2dx Finding an Indefinite Integral In Exercises 120, find the indefinite integral. 14+(x3)2dx 7E 8E 9E 10E Finding an Indefinite Integral In Exercises 3-22, find the indefinite integral. e2x4+e4xdx 12E 13E 14E 15E 16E 17E 18E 19E Finding an Indefinite Integral In Exercises 3-22, find the indefinite integral. x2(x+1)2+4dx 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E Evaluating a Definite Integral In Exercises 23-34, evaluate the definite integral. 012arcsinx1x2dx Evaluating a Definite Integral In Exercises 23-34, evaluate the definite integral. 012arccosx1x2dx 33E 34E 35E 36E 37E 38E Completing the Square In Exercises 35-42, find or evaluate the integral by completing the square. 232x34xx2dx 40E 41E 42E 43E 44E 45E 46E Comparing Integration Problems In Exercises 47-50, find the indefinite integrals, if possible, using the formulas and techniques you have studied so far in the text. (a) 11x2dx (b) x1x2dx (c) 1x1x2dx Comparing Integration Problems In Exercises 47-50, find the indefinite integrals, if possible, using the formulas and techniques you have studied so far in the text. (a) ex2dx (b) xex2dx (c) 1x2e1/xdx 49E 50E 51E 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E Area In Exercises 63-66, find the area of the given region. Use a graphing utility to verify your result. y=3cosx1+sin2x 66E 67E 68E 69E 70E 71E 72E 73E True or False? In Exercises 7174, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. One way to find 2e2x9e2xdx is to use the Arcsine Rule.75E 76E 77E 78E 79E 80E 1E 2E 3E 4E 5E 6E 7E Verifying an Identity In Exercises 11-18, verify the identity. coth2xcsch2x=1 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E Finding a Limit In Exercises 1722, find the limit. limxcschx 21E Finding a Limit In Exercises 21-24, find the limit. limx0cothx 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E Finding Relative Extrema In Exercises 39-42, find the relative extrema of the function. Use a graphing utility to confirm your result. f(x)=sinxsinhxcosxcoshx,4x4 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 53E 51E Finding an Indefinite Integral In Exercises 4354, find the indefinite integral. coshx9sinh2xdx 52E 55E 56E 57E 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E Finding an Indefinite Integral In Exercises 75-82, find the indefinite integral using the formulas from Theorem 5.22. 11+e2xdx Finding an Indefinite Integral In Exercises 75-82, find the indefinite integral using the formulas from Theorem 5.22. x9x4dx 79E 80E 81E 82E 83E 84E 85E 86E Differential Equation In Exercises 8790, solve the differential equation. dydx=180+8x16x2 Differential Equation In Exercises 8790, solve the differential equation. dydx=1(x1)4x2+8x1 Differential Equation In Exercises 87 and 88, find the general solution of the differential equation. dydx=x321x5+4xx2 Differential Equation In Exercises 87 and 88, find the general solution of the differential equation. dydx=12x4xx2 Area In Exercises 89-92, find the area of the given region y=sechx2 Area In Exercises 8992, find the area of the given region. y=tanh2x 93E 94E 95E 96E Tractrix Consider the equation of a tractrix y=asech1(xa)a2x2,a0 (a) Find dy/dx. (b) Let L be the tangent line to the tractrix at the point P. When L intersects the y-axis at the point Q, show that the distance between P and Q is a.98E 99E 100E 101E 102E 103E 104E 105E 106E Verifying a Differentiation Formula In Exercises 102-104, verify the differentiation formula. ddx[sinh1x]=1x2+1 108E 109E PUTNAM EXAM CHALLENGE Prove or disprove: there is at least one straight line normal to the graph of y=coshx at a point (a,cosha) and also normal to the graph of y=sinhx at a point (c,sinhc). [At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, coshx=(ex+ex)/2 and sinhx=(ex+ex)/2.]Sketching a Graph In Exercises 1 and 2, sketch the graph of the function and state its domain. f(x)=lnx3 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE Finding an Indefinite Integral In Exercises 21-26, find the indefinite integral. lnxxdx 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE Evaluating the Derivative of an Inverse Function In Exercises 39-42, verify that f has an inverse function. Then use the function f and the given real number a to find (f1) (a). (Hint: Use Theorem 5.9.) f(x)=x3+2,a=1 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50RE 51RE 52RE 53RE

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