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All Textbook Solutions for Single Variable Calculus: Concepts and Contexts, Enhanced Edition
44E45E46E47EA semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A() is the area of the semicircle and B() is the area of the triangle, find. lim0+A()B()The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle . Find lim0+sd50EWrite the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y=1+4x3Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y = (2x3 + 5)43EWrite the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y = sin( cot x)5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49EAt what point on the curve y=1+2x is the tangent line perpendicular to the line 6x + 2y = 1?51E52EA table of values for f, g, f, and g is given. (a) If h(x) = f(g(x)), find h(1). (b) If H(x) = g(f(x)), find H(1).Let f and g be the functions in Exercise 63. (a) If F(x) = f(f(x)), find F(2). (b) If G(x) = g(g(x)), find G(3).55E56E57E58E59E60E61E62E63E64E65E66E67EFind the 1000th derivative of f(x) = xex.The displacement of a particle on a vibrating string is given by the equation s(t)=10+14sin(10t) where sis measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.If the equation of motion of a particle is given by s = A cos(t + ), the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?71E72EThe motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is s(t) = 2e1.5t sin 2t where s is measured in centimeters and t in seconds. Find the velocity after t seconds and graph both the position and velocity functions for 0 1 2.74E75E76E77EThe table gives the US population from 1790 to 1860. (a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines. (c) Use the exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b). (d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,558,000. Can you explain the discrepancy?79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18ERegard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. 23. x4y2 x3y + 2xy3 = 0Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. 24. y sec x = x tan y21EUse implicit differentiation to find an equation of the tangent line to the curve at the given point. 26. sin(x + y) = 2x 2y, (, )23E24EUse implicit differentiation to find an equation of the tangent line to the curve at the given point. 29. x2 + y2 = (2x2 + 2y2 x)2, (0,12), (cardioid)26EUse implicit differentiation to find an equation of the tangent line to the curve at the given point. 31. 2(x2 + y2)2 = 25(x2 y2), (3, 1), (lemniscate)28E(a) The curve with equation y2 = 5x4 x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)30E31E32E33E34E35EIf x2 + xy + y3 = 1, find the value of y at the point where x = 1.37E38E39E40E41E42E43ETwo curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. 68. y = ax3, x2 + 3y2 = bShow that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2/A2 y2/ B2 = 1 are orthogonal trajectories if A2 a2 and a2 b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).46E47E48E49E(a) Where does the normal line to the ellipse x2 xy + y2 = 3 at the point (1, 1) intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line.51E52E53E54EThe Bessel function of order 0, y = J(x), satisfies the differential equation xy + y + xy = 0 for all values of x and its value at 0 is J(0) = 1. (a) Find J(0). (b) Use implicit differentiation to find J(0).The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 5. If the point (5. 0) is on the edge of the shadow, how far above the x-axis is the lamp located?1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44EExplain why the natural logarithmic function y = ln x is used much more frequently in calculus than the other logarithmic functions y = logb x.Differentiate the function. f(x) = x ln x xDifferentiate the function. f(x ) = sin(ln x)Differentiate the function. f(x) = ln(sin2x)5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E40E41E42E43E44E45E46E47E48EA particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t) = t3 8t2 + 24tA particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t)=9tt2+9A particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t) = sin(t/2)4E5E6E7E8E9E10E11E12E13E14E15E(a) The volume of a growing spherical cell is V=43r3, where the radius r is measured in micrometers (1 m = 106 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8m (ii) 5 to 6m (iii) 5 to 5.1m (b) Find the instantaneous rate of change of V with respect to r when r = 5m. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c).17E18EThe quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Q(t) = t3 2t2 + 6t + 2. Find the current when (a) t = 0.5 sand (b) t = 1 s. [See Example 3. The unit of current is an ampere (1 A = 1 C/s).) At what time is the current lowest? EXAMPLE 3 FIGURE620E21E22E23E24E25EThe table shows how the average age of first marriage of Japanese women has varied since 1950. (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for A(t). (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for A and A.Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2, and viscosity = 0.027. (a) Find the velocity of the blood along the center-line r = 0, at radius r = 0.005 cm, and at the wall r = R = 0.01 cm. (b) Find the velocity gradient at r = 0, r = 0.005, and r = 0.01. (c) Where is the velocity the greatest? Where is the velocity changing most? EXAMPLE 728E29EThe cost function for a certain commodity is C(q) = 84 + 0.16q 0.0006q2 + 0.000003q3 (a) Find and interpret C(100). (b) Compare C(100) with the cost of producing the 101st item.31E32EPatients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c I, is given by the equation t=ln(3c+9c28c2) Calculate the derivative of t with respect to c and interpret it.Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f(r)=2Dr, where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate r and explain its meaning.35E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E