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All Textbook Solutions for Calculus (MindTap Course List)
80E81E82E83E84ESales The sales S (in thousands of units) of a seasonal product are given by the model S=74.50+43.75sint6 where t is the time in months, with t = 1 corresponding to January. Find the average sales for each time period. (a) The first quarter (0t3) (b) The second quarter (3t6) (c) The entire year (0t12)86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104ESketching a Graph In Exercises 1 and 2, sketch the graph of the function and state its domain. f(x)=lnx32RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23REFinding an Indefinite Integral In Exercises 21-26, find the indefinite integral. lnxxdx25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38REEvaluating 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=140RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77RE78RE79RE80RE81RE82RE83RE84RE85RE86RE87RE88RE89RE90RE91RE92RE93RE94RE95RE96RE97RE98RE99RE100REFinding a Derivative In Exercises 99-104, find the derivative of the function. y=xarcsecx102RE103RE104RE105RE106RE107RE108RE109RE110RE111RE112RE113RE114RE115RE116RE117RE118RE119RE120RE121RE122RE123RE124RE125RE126RE127RE128RE129RE130RE131RE132RE133RE134RE1PS2PS3PS4PSFinding Limits Use a graphing utility to estimate each limit. Then calculate each limit using LHpitals Rule. What can you conclude about the form 0? (a) limx0+(cotx+1x) (b) limx0+(cotx1x) (c) limx0+[ (cotx+1x)(cotx1x) ]6PS7PS8PS9PS10PS11PSGudermannian Function The Gudcrmannian function of x is gd(x)=arctan(sinhx). (a) Graph g d using a graphing utility. (b) Show that g d is an odd function. (c) Show that g d is monotonic and therefore has an inverse. (d) Find the point of inflection of g d. (e) Verify that gd(x)=arcsin(tanhx).Decreasing Function Show that f(x)=lnxnx is a decreasing function for xe and n0.14PSArea Use integration by substitution to find the area under the curve y=1x+x between x = 1 and x = 4.16PS17PSNatural Logarithmic Function Explain why ln x is positive for x1 and negative for 0x1.2E3E4EEvaluating a Logarithm Using Technology In Exercises 5-8, use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral 1x(1/t)dt. ln456EEvaluating a Logarithm Using Technology In Exercises 5-8, use a graphing utility to evaluate the logarithm by (a) using the natural logarithm key and (b) using the integration capabilities to evaluate the integral 1x(1/t)dt. ln0.88EMatching In Exercises 912, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x)=lnx+1Matching In Exercises 912, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x)=lnx11E12E13ESketching the Graph In Exercises 13-18, sketch the graph of the function and state its domain. f(x)=2lnx15ESketching the Graph In Exercises 13-18, sketch the graph of the function and state its domain. f(x)=lnx17E18EUsing Properties of Logarithms In Exercises 19 and 20, use the properties of logarithms to approximate the indicated logarithms, given that In 20.6931 and In 31.0986. (a) ln6 (b) ln23 (c) ln81 (d) ln3Using Properties of Logarithms In Exercises 19 and 20, use the properties of logarithms to approximate the indicated logarithms, given that ln 2 = 0.6931 and In 3 = 1.0968. (a). ln 0.25 (b). ln 24 (c). ln123 (d). ln172Expanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx4Expanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx523EExpanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. ln(xyz)25E26EExpanding a Logarithmic Expression In Exercises 21-30, use the properties of logarithms to expand the logarithmic expression. lnx1xExpanding a Logarithmic Expression In Exercises 2130, use the properties of logarithms to expand the logarithmic expression. ln(3e2)29E30E31E32E33E34ECondensing a Logarithmic Expression In Exercises 31-36, write the expression as a logarithm of a single quantity. 4ln212ln(x3+6x)36E37E38EFinding a Limit In Exercises 39-42, find the limit. limx3+ln(x3)40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56EFinding a Derivative In Exercises 43-66, find the derivative of the function. y=ln(lnx2)58E59E60E61E62E63E64E65E66E67EFinding 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. y=lnx2/3,(1,0)Finding 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)70E71E72E73EFinding 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)75ELogarithmic Differentiation In Exercises 75-80, use logarithmic differentiation to find dy / dx. y=x2(x+1)(x+2),x077ELogarithmic 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,x180EImplicit 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=3083E84E85E86E87ERelative 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=2xln2x89ERelative 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=lnxx91ERelative 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=x2lnx493E94E95E96E97EHOW 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?108E109ECONCEPT CHECK Log Rule for Integration Can you use the Log Rule to find the integral below? Explain. x(x24)3dx2E3E4E5E6E7EFinding an Indefinite Integral In Exercises 5-28, find the indefinite integral. 954xdx9E10E11E12E13E14E15E