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All Textbook Solutions for Calculus
14ESolving an Exponential or Logarithmic Equation In Exercises 3-18, solve for x accurate to three decimal places. lnx+2=116E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45EFinding 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=18.86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104E105E106E107E108E109E110E111E112E113E114E115E116E117E118E119E120E121E122E123EDifferential Equation In Exercises 121 and 122, find the general solution of the differential equation. f(x)=sinx+e2x,f(0)=14,f(0)=12125E126E127E128E129E130E131E132EModeling Data A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given in the table. t 0 1 2 3 4 R 425 240 118 71 36 (a) Use the regression capabilities of a graphing utility to find a linear model for the points (t, In R). Write the resulting equation of the form lnR=at+b in exponential form. (b) Use a graphing utility to plot the data and graph the exponential model. (c) Use a definite integral to approximate the number of liters of chemical released during the 4 hours.133E135E136E137E138E139E141E142E140E143E1E2E3E4EExponential and Logarithmic Forms of Equations In Exercises 11-14, write the exponential equation as a logarithmic equation or vice versa. (a) 23=8 (b) 31=136EExponential and Logarithmic Forms of Equations In Exercises 11-14, write the exponential equation as a logarithmic equation or vice versa. (a) log100.01=2 (b) log0.58=38E9E10E11E12E13E14E15E16E17E18E19ESolving an Equation In Exercises 21-26, solve for x. (a) log3181=x (b) log636=x21E22E23E24E25E26E27E28E29E30E31E32E33E34EInverse Functions In Exercises 37 and 38, illustrate that the functions are inverse functions of each other by sketching their graphs on the same set of coordinate axes. f(x)=4xg(x)=log4x36E37E38E39E40E41E42E43E44E45E46E47E48EFinding a Derivative In Exercises 39-60, find the derivative of the function. h(t)=log5(4t)250EFinding a Derivative In Exercises 39-60, find the derivative of the function. y=log5x2152E53E54E55E56E57E58E59E60E61E62ELogarithmic Differentiation In Exercises 65-68, use logarithmic differentiation to find dy/dx. y=x2/x64E65E66E67E68E69E70E71E72E73E74EFinding an Indefinite Integral In Exercises 69-76, find the indefinite integral. x(5x2)dx76EFinding an Indefinite Integral In Exercises 69-76, find the indefinite integral. 32x1+32xdxFinding an Indefinite Integral In Exercises 69-76, find the indefinite integral. 2sinxcosxdx79E80E81E82E83E84E85E86E87EDepreciation After t years, the value of a car purchased for $25.000 is V(t)=25,000(34)t. (a) Use a graphing utility to graph the function and determine the value of the car 2 years after it was purchased. (b) Find the rates of change of V with respect to t when t=1 and t=4. (c) Use a graphing utility to graph V'(t) and determine the horizontal asymptote of V'(t). Interpret its meaning in the context of the problem.89E90E91E92E93E94E95E96E97E98ETimber Yield The yield V (in millions of cubic feet per acre) for a stand of timber at age t is V=6.7e48.1/t, where t is measured in years. (a) Find the limiting volume of wood per acre as t approaches infinity. (b) Find the rates at which the yield is changing when t=20 and t=60.100E101EModeling Data The breaking strengths B (in tons) of steel cables of various diameters d (in inches) are shown in the table. d 0.50 0.75 1.00 1.25 1.50 1.75 B 9.85 21.8 38.3 59.2 84.4 114.0 (a) Use the regression capabilities of a graphing utility to fit an exponential model to the data. (b) Use a graphing utility to plot the data and graph the model. (c) Find the rates of growth of the model when d=0.8 and d=1.5.103E104E105E106E107E108E109E110E111E112E113E114E115ETrue or False? In Exercises 111116, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = g(x)ex, then the only zeros of f are the zeros of g.117E118E119E120E121E122E123ERestricted Domain What is a restricted domain? Why are restricted domains necessary to define inverse trigonometric functions?1E2E3EEvaluating Inverse Trigonometric Functions In Exercises 7-14, evaluate the expression without using a calculator. arcsin05E6E7EEvaluating Inverse Trigonometric Functions In Exercises 7-14, evaluate the expression without using a calculator. arccot(3)9E10E11EApproximating Inverse Trigonometric Functions In Exercises 15-18, use a calculator to approximate the value. Round your answer to two decimal places. arcsin(0.39)Approximating Inverse Trigonometric Functions In Exercises 15-18, use a calculator to approximate the value. Round your answer to two decimal places. arcsec1.26914EUsing a Right Triangle In Exercises 19-24, use the figure to write the expression in algebraic form given y=arccosx, where 0y/2. cosy16E17E18E19E20E21E22E23E24E25E26E27E28E29E30ESimplifying an Expression Using a Right Triangle In Exercises 29-36, write the expression in algebraic form.( Hint:Sketch a right triangle, as demonstrated in Example 3.) csc(arctanx2)32E33E34E35ESolving an Equation In Exercises 37-40, solve the equation for x. arccosx=arcsecx37E38E39E40E41E42E43E44E45EFinding a Derivative In Exercises 41-56, find the derivative of the function. h(x)=x2arctan5x