Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
2E3E4E5E6E7ESolve each equation for x. a ln(lnx)=1b eex=109E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27-32 Find the limit. limxe3xe3xe3x+e3x27-32 Find the limit. limxex229E27-32 Find the limit. limx2e3/(2x)27-32 Find the limit. limx(e2xcosx)32E33-52 Differentiate the function. f(x)=e534E35E36E33-52 Differentiate the function. y=eax338E33-52 Differentiate the function. y=etan40E33-52 Differentiate the function. f(x)=x2exx2+ex42E33-52 Differentiate the function. y=x2e3x44E33-52 Differentiate the function. f(t)=eatsinbt33-52 Differentiate the function. f(z)=ez/(z1)33-52 Differentiate the function. F(t)=etsin2t48E33-52 Differentiate the function. g(u)=esecu250E33-52 Differentiate the function. y=cos(1e2x1+e2x)52E53E54E55E56E57E58E59E60E61E62E63E64E65EAn object is attached to the end of a vibrating spring and its displacement from its equilibrium position is y=8et/2sin4t, where t is measured in seconds and y is measured in centimeters. a Graph the displacement function together with the functions y=8et/2 and y=8et/2. How are these graphs related? Can you explain why? b Use the graph to estimate the maximum value of the displacement. Does it occur when the graph touches the graph of y=8et/2 c What is the velocity of the object when it first returns to its equilibrium position? d Use the graph to estimate the time after which the displacement is no more than 2 cm from equilibrium.67E68E69E70E71E72E73E74E75E76EA drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t)=Atpekt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A =0.01, p =4, k =0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.78EAfter the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream blood alcohol concentration, or BAC surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function C(t)=1.35te2.802t models the average BAC, measured in mg/mL, of a group of eight male subjects t hours after rapid consumption of 15 mL of ethanol corresponding to one alcoholic drink. What is the maximum average BAC during the first 3 hours? When does it occur?80E81E82E83-94 Evaluate the integral. 01(xe+ex)dx84E83-94 Evaluate the integral. 02dxex86E83-94 Evaluate the integral. ex1+exdx88E89E90E91E92E93E94E95E96E97E98E99EShow that the function y=ex2erf(x) satisfies the differential equation y=2xy+2/An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t)=100e0.01t liters per minute. How much oil leaks out during the first hour?102E103E104E105E106E107E108E109E110E111E112EExplain why the natural logarithmic function y=lnx is used much more frequently in calculus than the other logarithmic functions y=logbx.2E2-26 Differentiate the function. f(x)=sin(lnx)2-26 Differentiate the function. f(x)=ln(sin2x)2-26 Differentiate the function. f(x)=ln1x2-26 Differentiate the function. y=1lnx2-26 Differentiate the function. f(x)=log10(1+cosx)2-26 Differentiate the function. f(x)=log10x2-26 Differentiate the function. g(x)=ln(xe2x)2-26 Differentiate the function. g(t)=1+lnt2-26 Differentiate the function. F(t)=(lnt2)sint12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43-54 Use logarithmic differentiation to find the derivative of the function. y=(x2+2)2(x4+4)443-54 Use logarithmic differentiation to find the derivative of the function. y=excos2xx2+x+143-54 Use logarithmic differentiation to find the derivative of the function. y=x1x4+143-54 Use logarithmic differentiation to find the derivative of the function. y=xex2x(x+1)2/343-54 Use logarithmic differentiation to find the derivative of the function. y=xx43-54 Use logarithmic differentiation to find the derivative of the function. y=xcosx43-54 Use logarithmic differentiation to find the derivative of the function. y=xsinx43-54 Use logarithmic differentiation to find the derivative of the function. y=(x)x43-54 Use logarithmic differentiation to find the derivative of the function. y=(cosx)x52E43-54 Use logarithmic differentiation to find the derivative of the function. y=(tanx)1/x54E55E56E57E58E59E59-60 Use a graph to estimate the roots of the equation correct to one decimal place. Then use these estimates as the initial approximations in Newtons method to find the roots correct to six decimal places. ln(4x2)=x61E62E63E64E65E66E67E68E69EThe table gives the US population from 1790 to 1860. Year Population 1790 3, 929, 000 1800 5, 308, 000 1810 7, 240, 000 1820 9, 639, 000 1830 12, 861, 000 1840 17, 063, 000 1850 23, 192, 000 1860 31, 443, 000 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?71-82 Evaluate the integral. 243xdx72E71-82 Evaluate the integral. 12dt83t74E71-82 Evaluate the integral. 1ex2+x+1xdx76E71-82 Evaluate the integral. (lnx)22dx78E71-82 Evaluate the integral. sin2x1+cos2xdx80E71-82 Evaluate the integral. 042sds82E83E84E85E86E87E88E89E90E91E92E93E94Ea Write an equation that defines bx when b is a positive number and x is a real number. b What is the domain of the function ff(x)=bx? c If b1, what is the range of this function? d Sketch the general shape of the graph of the exponential function for each of the following cases. i b1 ii b=1 iii 0b12E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E25-42 Differentiate the function. y=xlog4sinx25-42 Differentiate the function. y=log2(xlog5x)35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60EAfter the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream blood alcohol concentration, or BAC surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function C(t)=1.35te2.802t models the average BAC, measured in mg/mL, of a group of eight male subjects t hours after rapid consumption of 15 mL of ethanol corresponding to one alcoholic drink.In this section we modeled the world population from 1900 to 2010 with the exponential function P(t)=(1436.53)(1.0135)t where t=0 corresponds to the year 1900 and P(t) is measured in millions. According to this model, what was the rate of increase of world population in 1920? In 1950? In 2000?63EA researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. Timehours 0 4 8 12 16 20 24 Bacteria count CFU/mL 37 47 63 78 105 130 173 a Make a scatter plot of the data. b Use a graphing calculator to find an exponential curve f(t)=abt that models the bacteria population t hours later. Graph the model from part b together with the scatter plot in part a. Use the TRACE feature to determine how long it takes for the bacteria count to double.65E66E67E68E69E70EA population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. a Find the relative growth rate. b Find an expression for the number of cells after t hours. c Find the number of cells after 6 hours. d Find the rate of growth after 6 hours. e When will the population reach a million cells?A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. a Find an expression for the number of bacteria after t hours. b Find the number of bacteria after 3 hours. c Find the rate of growth after 3 hours. d When will the population reach 10, 000?A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25, 600 after 6 hours. a What is the relative growth rate? Express your answer as a percentage. b What was the initial size of the culture? c Find an expression for the number of bacteria after t hours. d Find the number of cells after 4.5 hours. e Find the rate of growth after 4.5 hours. f When will the population reach 50, 000?The table gives estimates of the world population, in millions from 1750 to 2000. Year Population 1750 1800 1850 1900 1950 2000 790 980 1260 1650 2560 6080 a Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. b Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. c Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.The table gives the population of Indonesia, in millions, for the second half of the 20th century. Year Population 1950 1960 1970 1980 1990 2000 83 100 122 150 182 214 a Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and 1960 to predict the population in 1980. Compare with the actual figure. b Use the census figures for 1960 and 1980 to predict the population in 2000. Compare with the actual population. c Use the census figures for 1980 and 2000 to predict the population in 2010 and compare with the actual population of 243 million. d Use the model in part c to predict the population in 2020. Do you think the prediction will be too high or too low? Why?7EStrontium-90 has a half-life of 28 days. a A sample has a mass of 50 mg initially. Find a formula for the mass remaining after t days. b Find the mass remaining after 40 days. c How long does it take the sample to decay to a mass of 2 mg? d Sketch the graph of the mass function.The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. a Find the mass that remains after t years. b How much of the sample remains after 100 years? c After how long will only 1 mg remain?10E11EDinosaur fossils are too old to be reliably dated using carbon-14. See Exercise 11. Suppose we had a 68-million- year-old dinosaur fossil. What fraction of the living dinosaurs 14C would be remaining today? Suppose the minimum detectable amount is 0.1. What is the maximum age of a fossil that we could date using 14C?13E14EA roast turkey is taken from an oven when its temperature has reached 185F and is placed on a table in a room where the temperature is 75F. a If the temperature of the turkey is 150F after half an hour, what is the temperature after 45 minutes? b When will the turkey have cooled to 100F?16EWhen a cold drink is taken from a refrigerator, its temperature is 5C. After 25 minutes in a 20C room its temperature has increased to 10C. a What is the temperature of the drink after 50 minutes? b When will its temperature be 15C?18EThe rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 15C the pressure is 101.3 kPa at sea level and 87.14 kPa at h = 1000 m. a What is the pressure at an altitude of 3000 m? b What is the pressure at the top of Mount McKinley, at an altitude of 6187 m?a If 1000 is borrowed at 8 interest, find the amounts due at the end of 3 years if the interest is compounded i annually, ii quarterly, iii monthly, iv weekly, v daily, vi hourly, and vii continuously. b Suppose 1000 is borrowed and the interest is compounded continuously. If A(t) is the amount due after t years, where 0t3, graph A(t) for each of the interest rates 6, 8, and 10 on a common screen.21Ea How long will it take an investment to double in value if the interest rate is 6 compounded continuously? b What is the equivalent annual interest rate?Find the exact value of each expression. a sin1(0.5) b cos1(1)2EFind the exact value of each expression. a csc12 b cos1(3/2)