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All Textbook Solutions for Calculus
34E35E36EFinding a Limit In Exercises 39-42, find the limit. limx3+ln(x3)38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54EFinding a Derivative In Exercises 43-66, find the derivative of the function. y=ln(lnx2)56E57E58E59E60E61E62E63E64E65E66EFinding an Equation of a Tangent Line In Exercises 67-74, (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=3x2lnx,(1,3)68E69E70E71EFinding an Equation of a Tangent Line In Exercises 67-74, (a) find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the tangent feature of a graphing utility to confirm your results. f(x)=12xlnx2,(1,0)89ELogarithmic Differentiation In Exercises 75-80, use logarithmic differentiation to find dy / dx. y=x2(x+1)(x+2),x091ELogarithmic Differentiation In Exercises 75-80, use logarithmic differentiation to find dy / dx. y=x21x2+1,x1Logarithmic Differentiation In Exercises 75-80, use logarithmic differentiation to find dy / dx. y=x(x1)3/2x+1,x194EImplicit Differentiation In Exercises 81-84, use implicit differentiation to find dy/dx. x23lny+y2=10Implicit Differentiation In Exercises 81-84, use implicit differentiation to find dy/dx. lnxy+5x=3075E76E77E78E79ERelative Extrema and Points of Inflection In Exercises 87-92, locate any relative extrema and points of inflection. Use a graphing utility to confirm your results. y=2xln2x81ERelative Extrema and Points of Inflection In Exercises 87-92, locate any relative extrema and points of inflection. Use a graphing utility to confirm your results. y=lnxx83ERelative Extrema and Points of Inflection In Exercises 87-92, locate any relative extrema and points of inflection. Use a graphing utility to confirm your results. y=x2lnx487E88E85E86E95E96E97EHOW DO YOU SEE IT? The graph shows the temperature T (in degrees Celsius) of an object h hours after it is removed from a furnace. (a) Find limhT. What does this limit represent? (b) When is the temperature changing most rapidly?99E100E101E102EHome Mortgage The term t (in years) of a $200,000 home mortgage at 7.5% interest can be approximated by t=13.375ln(xx1250),x1250 where x is the monthly payment in dollars. (a) Use a graphing utility to graph the model. (b) Use the model to approximate the term of a home mortgage for which the monthly payment is $1398.43. What is the total amount paid? (c) Use the model to approximate the term of a home mortgage for which the monthly payment is $1611.19. What is the total amount paid? (d) Find the instantaneous rates of change of t with respect to x when x=1398.43 and x=1611.19. (e) Write a short paragraph describing the benefit of the higher monthly payment.104EModeling Data The table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch). (Source: Standard Handbook of Mechanical Engineers) p 5 10 14.696(1 atm) 20 T 162.24 193.21 212.00 227.96 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81 A model that approximates the data is T=87.97+34.96lnp+7.91p (a) Use a graphing utility to plot the data and graph the model. (b) Find the rates of change of T with respect to p when p=10 and p=70. (c) Use a graphing utility to graph T'. Find limpT(p) and interpret the result in the context of the problem.Modeling Data The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is one atmosphere (1.033227 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers). h 0 5 10 15 20 25 P 1 0.55 0.25 0.12 0.06 0.02 (a) Use a graphing utility to find a model of the form p=a+blnh for the data. Explain why the result is an error message. (b) Use a graphing utility to find the logarithmic model h=a+blnp for the data. (c) Use a graphing utility to plot the data and graph the model from part (b). (d) Use the model to estimate the altitude when p=0.75. (e) Use the model to estimate the pressure when h=13. (f) Use the model to find the rates of change of pressure when h=5 and h=20. Interpret the results.Tractrix A person walking along a dock drags a boat by a 10-meter rope. The boat travels along a path known as a tractrix (see figure). The equation of this path is y=10ln(10+100x2x)100x2 (a) Use a graphing utility to graph the function. (b) What are the slopes of this path when x=5 and x=9? (c) What does the slope of the path approach as x approaches 10 from the left?108E109E1EFinding an Indefinite Integral In Exercises 126, find the indefinite integral. 10xdx3E4E5EFinding an Indefinite Integral In Exercises 5-28, find the indefinite integral. 954xdx7E8E9E10E11E12E13E14E15E16E17EFinding an Indefinite Integral In Exercises 5-28, find the indefinite integral. x36x20x+5dx19E20E21E22E23E24E25EFinding an Indefinite Integral In Exercises 528, find the indefinite integral. x(x2)(x1)3dx27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44EFinding 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,x046E47E48EEvaluating a Definite Integral In Exercises 5158, evaluate the definite integral. Use a graphing utility to verify your result. 0453x+1dx50EEvaluating a Definite Integral In Exercises 51-58, evaluate the definite integral. Use a graphing utility to verify your result. 1e(1+lnx)2xdx52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74EFinding 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]95E96E75E76E77E78E83E84E108E79E80E81E82E85E86E87EProof Prove that cscudu=lncscu+cotu+C.89EUsing Properties of Logarithms and Trigonometric Identities In Exercises 8790, show that the two formulas are equivalent. cotxdx=lnsinx+Ccotxdx=ln| cscx |+C91E92E97ESales 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.99E100E101E102E103E104E105E106E107EMatching 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).]10E11EMatching 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)=x152E3E4E5E6E7E8E13E14E15E16E17E18E19E20E21E22E23E24E25E26EDetermining 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)28E29E30E31E32E33E34EFinding 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)=2x3Finding an Inverse Function In Exercises 3546, (a) find the inverse function of f, (b) graph f and f-l on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domain and range of f and f-l. f(x)=74x37E38E39E40EFinding 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+746E47E48ECost You need 50 pounds of two commodities costing $1.25 and $1.60 per pound. (a) Verify that the total cost is y = 1.25x + 1.60(50 x), where x is the number of pounds of the less expensive commodity. (b) Find the inverse function of the cost function. What does each variable represent in the inverse function? (c) What is the domain of the inverse function? Validate or explain your answer using the context of the problem. (d) Determine the number of pounds of the less expensive commodity purchased when the total cost is $73.50E51E52E53ETesting 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,a055E58E57E56EThink 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.60E61E62E63E64EEvaluating 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)183E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98EDerivative 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).101E102E103ESolving an Exponential or Logarithmic Equation In Exercises 3-18, solve for xaccurate to three decimal places. elnx=42E3E4E5E6E7E8E9E10E11E12E13E