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All Textbook Solutions for Calculus (MindTap Course List)
16E17E18E19EFinding an Indefinite Integral In Exercises 5-28, find the indefinite integral. x36x20x+5dx21E22E23EFinding an Indefinite Integral In Exercises 5-28, find the indefinite integral. dxx(lnx2)325E26EFinding an Indefinite Integral In Exercises 528, find the indefinite integral. 6x(x5)2dxFinding an Indefinite Integral In Exercises 528, find the indefinite integral. x(x2)(x1)3dx29E30E31E32E33E36EFinding an Indefinite Integral of a Trigonometric Function In Exercises 33-42, find the indefinite integral. tan22d35E38E37E39E40E41E42E43E44E45E46EFinding a Particular Solution In Exercises 47 and 48, find the particular solution of the differential equation that satisfies the initial conditions. f(x)=2x2,f(1)=1,f(1)=1,x048E49E50EEvaluating a Definite Integral In Exercises 5158, evaluate the definite integral. Use a graphing utility to verify your result. 0453x+1dx52EEvaluating a Definite Integral In Exercises 51-58, evaluate the definite integral. Use a graphing utility to verify your result. 1e(1+lnx)2xdx54E55E56E57E58E59E60E61E62E63E64E65EArea In Exercises 6568, find the area of the given region. Use a graphing utility to verify your result. y=1+lnx3x67E68E69E70E71E72EFinding the Average Value of a Function In Exercises 73-76, find the average value of the function over the given interval. f(x)=8x2,[2,4]Finding the Average Value of a Function In Exercises 73-76, find the average value of the function over the given interval. f(x)=4(x+1)x2,[2,4]75E76E77E78E79E80E81E82E83E84E85EProof Prove that cscudu=lncscu+cotu+C.87EUsing Properties of Logarithms and Trigonometric Identities In Exercises 8790, show that the two formulas are equivalent. cotxdx=lnsinx+Ccotxdx=ln| cscx |+C89E90E91ESales The rate of change in sales S is inversely proportional to time t(t1), measured in weeks. Find S as a function of t when the sales after 2 and 4 weeks are 200 units and 300 units, respectively.93E94E95E96E97E98E99E100E101E1E2E3E4EMatching In Exercises 5-8, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).]6E7EMatching In Exercises 5-8, match the graph of the function with the graph of its inverse function. [The graphs of the inverse functions are labeled (a), (b), (c), and (d).]Verifying Inverse Functions In Exercises 9-16, show that f and g are inverse functions (a) analytically and (b) graphically. f(x)=5x+1,g(x)=x1510E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26EDetermining Whether a Function Has an Inverse Function In Exercises 2530, use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. f(x)=8x3+x2128EDetermining Whether a Function Has an Inverse Function In Exercises 25-30, use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. f(x)=ln(x3)30E31E32E33E34EFinding an Inverse Function In Exercises 35-46, (a) find the inverse function of f, (b) graph f and f1 on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of f and f1. f(x)=2x336E37E38E39E40EFinding an Inverse Function In Exercises 35-46, (a) find the inverse function of f, (b) graph f and f1 on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of f and f1. f(x)=4x2,0x242E43E44EFinding an Inverse Function In Exercises 35-46, (a) find the inverse function of f , (b) graph/ and f1 on the same set of coordinate axes,(c) describe the relationship between the graphs, and (d) state the domains and ranges of f and f1. f(x)=xx2+746E47E48E49E50E51E52E53ETesting Whether a Function Is One-to-One In Exercises 51-54, determine whether the function is one-to-one. If it is, find its inverse function. f(x)=ax+b,a055E56E57E58EThink About It In Exercises 59-62, decide whether the function has an inverse function. If so, describe what the inverse function represents. g (t) is the volume of water that has passed through a water line t minutes after a control valve is opened.60E61E62E63EEvaluating the Derivative of an Inverse Function In Exercises 63-70, verify that f has an inverse function. Then use the function f and the given real number a to find (f1)(a) (Hint: See Example 5.) f(x)=x3+3x1,a=5Evaluating the Derivative of an Inverse Function In Exercises 63-70, verify that f has an inverse function. Then use the function f and the given real number a to find (f1)(a) (Hint: See Example 5.) f(x)=127(x5+2x3),a=1166E67EEvaluating the Derivative of an Inverse Function In Exercises 63-70, verify that f has an inverse function. Then use the function f and the given real number a to find (f1)(a) (Hint: See Example 5.) f(x)=cos2x,0x2,a=169E70E71E72E73E74E75E76E77E78E79E80E81EUsing Composite and Inverse Functions In Exercises 79-82, use the functions f(x)=x+4 and g(x)=2x5 to find the given function. (gf)183E84E85E86E87E88E89E90E91E92E93E94E95E96EDerivative of an Inverse Function Show that f(x)=2x1+t2dt is one-to-one and find (f1)(0).Derivative of an Inverse FunctionShow that f(x)=2xdt1+t4 is one-to-one and find (f-1)'(0).99E100E101ENatural Exponential Function of Describe the graph of f(x)=ex2ESolving an Exponential or Logarithmic Equation In Exercises 3-18, solve for xaccurate to three decimal places. elnx=44E5E6E7E8E9E10E11E12E13E14E15E16ESolving an Exponential or Logarithmic Equation In Exercises 3-18, solve for x accurate to three decimal places. lnx+2=118E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45EFinding a Derivative In Exercises 33-54, find the derivative of the function. y=ln(1+ex1ex)47EFinding a Derivative In Exercises 33-54, find the derivative of the function. y=exex249E50E51E52E53E54E55E56E57E58EFinding an Equation of a Tangent Line In Exercises 55-62, find an equation of the tangent line to the graph of the function at the given point f(x)=exlnx,(1,0)60E61E62E63E64E65E66E67E68E69E70ERelative Extrema and Points of Inflection In Exercises 71-78, find the relative extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. f(x)=ex+ex272ERelative Extrema and Points of Inflection In Exercises 71-78, find the relative extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. g(x)=12e(x2)2/2Relative Extrema and Points of Inflection In Exercises 71-78, find the relative extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. g(x)=12e(x3)2/275E76E77E78EArea Find the area of the largest rectangle that can be inscribed under the curve y=ex2 in the first and second quadrants.Area Perform the following steps to find the maximum area of the rectangle shown in the figure. (a) Solve for c in the equation f(c)=f(c+x) (b) Use the result in part (a) to write the area A as a function of x [Hint:A=xf(c)] (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions of the rectangle of maximum area. Determine the maximum area. (d) Use a graphing utility to graph the expression for c found in part (a). Use the graph to approximate limx0+c and limxc Use this result to describe the changes in dimensions and position of the rectangle for 0x.81E82E83EHarmonic Motion The displacement from equilibrium of a mass oscillating on the end of a spring suspended from a ceiling is y=1.5e0.22tcos4.9t, where y is the displacement(in feet) and f is the time (in seconds). Use a graphing utility to graph the displacement function on the interval [0, 10]. Find a value of t past which the displacement is less than 3 inches from equilibrium.Atmospheric Pressure A meteorologist measures the atmospheric pressure P (in millibars) at altitude h (in kilometers). The data are shown below. h 0 5 10 15 20 P 1013.2 547.5 233.0 121.6 50.7 (a) Use a graphing utility to plot the points (h, In P). Use the regression capabilities of the graphing utility to find a linear model for the revised data points. (b) The line in part (a) has the form In P=ah+b. Write the equation in exponential form. (c) Use a graphing utility to plot the original data and graph the exponential model in part (b). (d) Find the rates of change of the pressure when h=5 and h=18.Modeling Data The table lists the approximate values V of a mid-sized sedan for the years 2010 through 2016.The variable t represents the time (in years), with t=10 corresponding to 2010. t 10 11 12 13 V $23,046 $20,596 $18,851 $17,001 t 14 15 16 V $15,226 $14,101 $12,841 (a) Use the regression capabilities of a graphing utility to fit linear and quadratic models to the data. Plot the data and graph the models. (b) What does the slope represent in the linear model in part (a)? (c) Use the regression capabilities of a graphing utility to fit an exponential model to the data. (d) Determine the horizontal asymptote of the exponential model found in part (c). Interpret its meaning in the context of the problem. (e) Use the exponential model to find the rates of decrease in the value of the sedan when t=12 and t=15.87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104E105E106E107E108E109E110E111E112E113E