Essential Calculus: Early Transcendentals
Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Chapter 3.7, Problem 46E

If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is

v = m g c ( 1 − e − c t / m )

where g is the acceleration due to gravity and c is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; c is the proportionality constant.)

(a) Calculate lim t → ∞ v . What is the meaning of this limit?

(b) For fixed t, use l’Hospital’s Rule to calculate lim c → 0 + v . What can you conclude about the velocity of a falling object in a vacuum?

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Chapter 3 Solutions

Essential Calculus: Early Transcendentals

Ch. 3.1 - (a) Write an equation that defines the exponential...Ch. 3.1 - (a) How is the number e defined? (b) What is an...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - 7-12 Make a rough sketch of the graph of the...Ch. 3.1 - 7-12. Make a rough sketch of the graph of the...Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Make a rough sketch of the graph of the function....
Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Starting with the graph of y = ex, write the...Ch. 3.1 - Starting with the graph of y = ex, find the...Ch. 3.1 - Find the domain of each function. 19. (a)...Ch. 3.1 - Find the domain of each function. (a) g(t) =...Ch. 3.1 - 21–22 Find the exponential function f(x) = Cb2...Ch. 3.1 - Find the exponential function f(x) = Cax whose...Ch. 3.1 - Prob. 19ECh. 3.1 - Compare the rates of growth of the functions f(x)...Ch. 3.1 - Compare the functions f(x) = x10 and g(x) = ex by...Ch. 3.1 - Use a graph to estimate the values of x such that...Ch. 3.1 - Find the limit. limx(1.001)xCh. 3.1 - Prob. 24ECh. 3.1 - Find the limit. limxe3xe3xe3x+e3xCh. 3.1 - Find the limit. limx2+10x310xCh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - If you graph the function f(x)=1e1/x1+e1/x you' ll...Ch. 3.1 - Graph several members of the family of functions...Ch. 3.2 - (a) What is a one-to-one function? (b) How can you...Ch. 3.2 - (a) Suppose f is a one-to-one function with domain...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - 3-14 A function is given by a table of values, a...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - Assume that f is a one-to-one function. (a) If...Ch. 3.2 - 16. If f(x) = x5 + x3 + x, find f‒1(3) and f(f...Ch. 3.2 - 17. If g(x) = 3 + x + ex, find g−1(4). Ch. 3.2 - 18. The graph of f is given. (a) Why is f...Ch. 3.2 - The formula C=59(F32), where F 459.67, expresses...Ch. 3.2 - 20. In the theory of relativity, the mass of a...Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - 21- 26 Find a formula for the inverse of the...Ch. 3.2 - 21- 26 Find a formula for the inverse of the...Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find an explicit formula for f1 and use it to...Ch. 3.2 - Find an explicit formula for f1 and use it to...Ch. 3.2 - Use the given graph of f to sketch the graph of...Ch. 3.2 - Use the given graph of f to sketch the graph of...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 35-38 Find (f1)(a). 35. f(x) = 2x3 + 3x2 + 7x + 4,...Ch. 3.2 - 35-38 Find(f1)(a). 36. f(x) = x3 + 3 sin x + 2 cos...Ch. 3.2 - 35-38 Find(f1)(a). 37. f(x)=3+x2+tan(x/2),1x1,a=3Ch. 3.2 - 35-38 Find(f1)(a). 38. f(x)=x3+x2+x+1,a=2Ch. 3.2 - Suppose f1 is the inverse function of a...Ch. 3.2 - Suppose f−1 is the inverse function of a...Ch. 3.2 - (a) How is the logarithmic function y = logax...Ch. 3.2 - (a) What is the natural logarithm? (b) What is the...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Use Formula 14 to evaluate each logarithm correct...Ch. 3.2 - Use Formula 14 to graph the given functions on a...Ch. 3.2 - Use Formula 14 to graph the given functions on a...Ch. 3.2 - 45. Suppose that the graph of y = log2 x is drawn...Ch. 3.2 - Compare the functions f(x)=x0.1 and g(x) = ln x by...Ch. 3.2 - Make a rough sketch of the graph of each function....Ch. 3.2 - Make a rough sketch of the graph of each function....Ch. 3.2 - (a) What are the domain and range of f? (b) What...Ch. 3.2 - (a) What are the domain and range of f? (b) What...Ch. 3.2 - Solve each equation for x. 51. (a) e74x=6 (b)...Ch. 3.2 - Solve each equation for x. 52. (a) ln(x2 1) = 3...Ch. 3.2 - Solve each equation for x. 53. (a) 2x5 = 3 (b) ln...Ch. 3.2 - Solve each equation for x. 54. (a) ln(ln x) = 1...Ch. 3.2 - Solve each inequality for x. 55. (a) ln x 0 (b)...Ch. 3.2 - Solve each inequality for x. 56. (a) 1 e3x1 2...Ch. 3.2 - (a) Find the domain of f(x) = ln(ex 3). (b) Find...Ch. 3.2 - (a) What are the values of eln 300 and ln(e300)?...Ch. 3.2 - 71-76 Find the limit. 71. limx3+ln(x29)Ch. 3.2 - 71-76 Find the limit. 72. limx2log5(8xx4)Ch. 3.2 - Prob. 73ECh. 3.2 - 7176 Find the limit. 74. limx0+ln(sinx)Ch. 3.2 - Find the limit. limx[ln(1+x2)ln(1+x)]Ch. 3.2 - Find the limit. limx[ln(2+x)ln(1+x)]Ch. 3.2 - When a camera flash goes off, the batteries...Ch. 3.2 - Let a 1. Prove, using precise definitions, that...Ch. 3.2 - (a) If we shift a curve to the left, what happens...Ch. 3.3 - Differentiate the function. f(x) = log10 (x3 + 1)Ch. 3.3 - Differentiate the function. f(x) = x ln x xCh. 3.3 - Differentiate the function. f(x ) = sin(ln x)Ch. 3.3 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.3 - Differentiate the function. f(x)=ln1xCh. 3.3 - Differentiate the function. y=1lnxCh. 3.3 - Differentiate the function. f(x) = sin x ln(5x)Ch. 3.3 - Differentiate the function. 8. f(x) = log5 (xex)Ch. 3.3 - Differentiate the function. Ch. 3.3 - Differentiate the function. 10. f(u)=u1+lnuCh. 3.3 - Differentiate the function. g(x)=ln(xx21)Ch. 3.3 - Differentiate the function. 12. h(x)=ln(x+x21)Ch. 3.3 - Differentiate the function. G(y)=ln(2y+1)5y2+1Ch. 3.3 - Differentiate the function. 14. g(r) = r2 ln(2r +...Ch. 3.3 - Differentiate the function. F(s) = ln ln sCh. 3.3 - Differentiate the function. 16. y=ln|cos(lnx)|Ch. 3.3 - Differentiate the function. 20. g(x)=xexCh. 3.3 - Differentiate the function. y=xexCh. 3.3 - Differentiate the function. f(x) = (x3 + 2x)exCh. 3.3 - Differentiate the function. H(z)=a2z2a2+z2Ch. 3.3 - Differentiate the function. y = tan[ln(ax + b)]Ch. 3.3 - Differentiate the function. 22. y=ex1exCh. 3.3 - Differentiate the function. y=1+2e3xCh. 3.3 - Differentiate the function. 24. y=e2tcos4tCh. 3.3 - Differentiate the function. 25. y = 5 1/xCh. 3.3 - Differentiate the function. 26. y=101x2Ch. 3.3 - Differentiate the function. 27. F(t) = et sin 2tCh. 3.3 - Differentiate the function. 28. y=eueueu+euCh. 3.3 - Differentiate the function. 29. y=ln|2x5x2|Ch. 3.3 - Differentiate the function. 30. y=1+xe2xCh. 3.3 - Differentiate the function. 31. f(t) = tan (et) +...Ch. 3.3 - Differentiate the function. 32. y=ektanxCh. 3.3 - Differentiate the function. 33. y=ln(ex+xex)Ch. 3.3 - Differentiate the function. 34. y=[ln(1+ex)]2Ch. 3.3 - Differentiate the function. 35. y=2xlog10xCh. 3.3 - Differentiate the function. 36. y = x2 e1/xCh. 3.3 - Differentiate the function. 37. f(t)=sin2(esin2t)Ch. 3.3 - Differentiate the function. 38. y=log2(excosx)Ch. 3.3 - Differentiate the function. 39. g(x) = (2rarx +...Ch. 3.3 - Differentiate the function. 40. y=23x2Ch. 3.3 - Find y and y. 41. y = eax sin xCh. 3.3 - Find y and y. 42. y=lnxx2Ch. 3.3 - Find y and y. 43. y = x ln xCh. 3.3 - Find y and y. 44. y = ln (sec x + tan x)Ch. 3.3 - Differentiate f and find the domain of f....Ch. 3.3 - Differentiate f and find the domain of f. f(x) ln...Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Let f(x) = cx + ln(cos x). For what value of c is...Ch. 3.3 - Let f(x) = loga(3x2 2). For what value of a is...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - 46. Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Find y if 2x2y=x+y.Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Find y if y = ln(x2 + y2).Ch. 3.3 - Find y if xy = yx.Ch. 3.3 - The motion of a spring that is subject to a...Ch. 3.3 - Under certain circumstances a rumor spreads...Ch. 3.3 - Show that the function y = Aex + Bxex satisfies...Ch. 3.3 - For what values of r does the function y = erx...Ch. 3.3 - If f(x) = e2x, find a formula for f(n)(x). Ch. 3.3 - Find the thousandth derivative of f(x) = xe–x. Ch. 3.3 - Find a formula for f(n)(x) if f(x) = ln(x 1).Ch. 3.3 - Find d9dx9(x8lnx).Ch. 3.3 - If f(x) = 3 + x + ex, find (f1)(4).Ch. 3.3 - Evaluate . Ch. 3.4 - A population of protozoa develops with a constant...Ch. 3.4 - A common inhabitant of human intestines is the...Ch. 3.4 - A bacteria culture initially contains 100 cells...Ch. 3.4 - A bacteria culture grows with constant relative...Ch. 3.4 - The table gives estimates of the world population,...Ch. 3.4 - The table gives the population of India, in...Ch. 3.4 - Experiments show that if the chemical reaction...Ch. 3.4 - Strontium-90 has a half-life of 28 days. (a) A...Ch. 3.4 - The half-life of cesium-137 is 30 years. Suppose...Ch. 3.4 - A sample oflritium-3 decayed to 94.5% of its...Ch. 3.4 - 11. Scientists can determine the age of ancient...Ch. 3.4 - A curve passes through the point (0, 5) and has...Ch. 3.4 - 15. A roast turkey is taken from an oven when its...Ch. 3.4 - In a murder investigation, the temperature of the...Ch. 3.4 - When a cold drink is taken from a refrigerator,...Ch. 3.4 - 18. A freshly brewed cup of coffee has temperature...Ch. 3.4 - The rate of change of atmospheric pressure P with...Ch. 3.4 - (a) If 1000 is borrowed at 8% interest, find the...Ch. 3.4 - If 3000 is invested at 5% interest, find the value...Ch. 3.4 - (a) How long will it take an investment to double...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Prove that cos(sin1x)=1x2.Ch. 3.5 - Simplify the expression. tan(sin1x)Ch. 3.5 - Simplify the expression. sin(tan1x)Ch. 3.5 - Simplify the expression. cos(2 tan1x)Ch. 3.5 - Prove Formula 6 for the derivative of cos1 by the...Ch. 3.5 - (a) Prove that sin1x + cos1x = /2. (b) Use part...Ch. 3.5 - Prove that ddx(cot1x)=11+x2.Ch. 3.5 - Prove that ddx(sec1x)=1xx21.Ch. 3.5 - Prove that ddx(csc1x)=1xx21.Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - 1629 Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - 3031 Find the derivative of the function. Find the...Ch. 3.5 - Find the derivative of the function. Find the...Ch. 3.5 - Find y if tan1(xy) = 1 + x2y.Ch. 3.5 - If g(x)=xsin1(x/4)+16x2, find g(2).Ch. 3.5 - Find an equation of the tangent line to the curve...Ch. 3.5 - Prob. 35ECh. 3.5 - Find the limit. limxarccos(1+x21+2x2)Ch. 3.5 - Find the limit. limxarctan(ex)Ch. 3.5 - Prob. 38ECh. 3.5 - A ladder 10 ft long leans against a vertical wall....Ch. 3.5 - A lighthouse is located on a small island, 3 km...Ch. 3.5 - Some authors define y = sec1x sec y = x and y ...Ch. 3.5 - (a) Sketch the graph of the function f(x) =...Ch. 3.6 - Find the numerical value of each expression. 1....Ch. 3.6 - Find the numerical value of each expression. (a)...Ch. 3.6 - 1-6 Find the numerical value of each...Ch. 3.6 - Find the numerical value of each expression. 4....Ch. 3.6 - Find the numerical value of each expression. 5....Ch. 3.6 - Find the numerical value of each expression. 6....Ch. 3.6 - Prove the identity. 7. sinh(x) = sinh x (This...Ch. 3.6 - Prove the identity. 8. cosh(x) = cosh x (This...Ch. 3.6 - Prove the identity. 9. cosh x + sinh x = exCh. 3.6 - Prove the identity. 10. cosh x sinh r = exCh. 3.6 - Prove the identity. 11. sinh(x + y) = sinh x cosh...Ch. 3.6 - Prove the identity. 12. cosh(x + y) = cosh x cosh...Ch. 3.6 - Prove the identity. 15. sinh 2x = 2 sinh x cosh xCh. 3.6 - Prove the identity. 18. 1+tanhx1tanhx=e2xCh. 3.6 - Prove the identity. 19. (cosh x + sinh x)n = cosh...Ch. 3.6 - If x=1213 find the values of the other hyperbolic...Ch. 3.6 - If cosh=53 and x 0. find the values of the other...Ch. 3.6 - (a) Use the graphs of sinh, cosh, and tanh in...Ch. 3.6 - Use the definitions of the hyperbolic functions to...Ch. 3.6 - Prove the formulas given in Table 1 for the...Ch. 3.6 - Give an alternative solution 10 Example 3 by...Ch. 3.6 - Prove Equation 4.Ch. 3.6 - Prove Formula 5 using (a) the method of Example 3...Ch. 3.6 - For each of I he following functions (i) give a...Ch. 3.6 - Prove the formulas given in Table 6 for the...Ch. 3.6 - Find the derivative. Simplify where possible. f(x)...Ch. 3.6 - Find the derivative. Simplify where possible. f(x)...Ch. 3.6 - Find the derivative. Simplify where possible. g(x)...Ch. 3.6 - Find the derivative. Simplify where possible. h(x)...Ch. 3.6 - Find the derivative. Simplify where possible. f(t)...Ch. 3.6 - Find the derivative. Simplify where possible. f(t)...Ch. 3.6 - Find the derivative. Simplify where possible. y =...Ch. 3.6 - Find the derivative. Simplify where possible. 37....Ch. 3.6 - Find the derivative. Simplify where possible....Ch. 3.6 - 26-41 Find the derivative. Simplify where...Ch. 3.6 - Find the derivative. Simplify where possible. 40....Ch. 3.6 - 30-45 Find the derivative. Simplify where...Ch. 3.6 - Find the derivative. Simplify where possible. 42....Ch. 3.6 - Find the derivative. Simplify where possible. 43....Ch. 3.6 - Find the derivative. Simplify where possible. 44....Ch. 3.6 - Find the derivative. Simplify where possible. 45....Ch. 3.6 - Show that ddx1+tanhx1tanhx4=12ex/2.Ch. 3.6 - Show that ddx arctan(tanh x) = sech 2x.Ch. 3.6 - The Gateway Arch in St. Louis was designed by Eero...Ch. 3.6 - If a water wave with length L. moves with velocity...Ch. 3.6 - A flexible cable always hangs in the shape of a...Ch. 3.6 - Prob. 47ECh. 3.6 - Using principles from physics it can be shown that...Ch. 3.6 - A cable with linear density = 2 kg/m is strung...Ch. 3.6 - Evaluate limxsinhxex.Ch. 3.6 - (a) Show that any function of the form y = A sinh...Ch. 3.6 - If x = ln( sec + tan ), show that sec = cosh x.Ch. 3.6 - 57. At what point of the curve y = cosh x does the...Ch. 3.6 - Show that if a 0 and b 0, then there exist...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 5ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 7ECh. 3.7 - Prob. 8ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 12ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - 33. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - 34. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - Prob. 17ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - 49. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 37ECh. 3.7 - 1-38 Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 39ECh. 3.7 - Prob. 40ECh. 3.7 - Prob. 41ECh. 3.7 - Prob. 42ECh. 3.7 - Prob. 43ECh. 3.7 - Prob. 44ECh. 3.7 - If an object with mass m is dropped from rest, one...Ch. 3.7 - If an initial amount A0 of money is invested at an...Ch. 3.7 - If an electrostatic field E acts on a liquid or a...Ch. 3.7 - 82. A metal cable has radius r and is covered by...Ch. 3.7 - Prob. 49ECh. 3.7 - The figure shows a sector of a circle with central...Ch. 3.7 - Evaluate limx[xx2ln(1+xx)].Ch. 3.7 - 86. Suppose f is a positive function. If and ,...Ch. 3.7 - If f is continuous, f(2) = 0, and f(2) = 7,...Ch. 3.7 - For what values of a and b is the following...Ch. 3.7 - If f is continuous, use lHospitals Rule to show...Ch. 3.7 - If f is continuous, show that...Ch. 3.7 - Let f(x)={e1/x2ifx00ifx=0 (a) Use the definition...Ch. 3.7 - Let f(x)={xxifx01ifx=0 (a) Show that f is...Ch. 3 - Prob. 1RCCCh. 3 - (a) How is the inverse sine function f(x) = sin1 x...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - The graph of g is given. (a) Why is g one-to-one?...Ch. 3 - 1112 Find the exact value of each expression. 11....Ch. 3 - 1112 Find the exact value of each expression. 12....Ch. 3 - 1316 Solve the equation for x. 13. (a) ex = 5 (b)...Ch. 3 - 1316 Solve the equation for x. 14. (a) eex=2 (b)...Ch. 3 - 1316 Solve the equation for x. 15. (a) ln(x + 1) +...Ch. 3 - 1316 Solve the equation for x. 16. (a) ln(1 + ex)...Ch. 3 - (a) Express e as a limit. (b) What is the value of...Ch. 3 - (a) What are the domain and range of the natural...Ch. 3 - (a) Write a differential equation that expresses...Ch. 3 - State the derivative of each function. (a) y = ex...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - 1-50 Calculate y'. 8. xey = y sin x Ch. 3 - Calculate y'. 9. y = ln(x ln x)Ch. 3 - Calculate y'. 10. y = emx' cos nxCh. 3 - Calculate y'. 12. y = (arcsin 2x)2Ch. 3 - Calculate y'. 13. y=e1/xx2Ch. 3 - Calculate y'. 14. y = ln sec xCh. 3 - Write the definitions of the hyperbolic functions...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - The graph of f is shown. Is f one-to-one? Explain.Ch. 3 - Suppose f is one-to-one, f(7) = 3, and f(7) = 8....Ch. 3 - Find the inverse function of f(x)=x+12x+1.Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - Let a 1. For large values of x, which of the...Ch. 3 - 1743 Differentiate. 22. y = x cos1xCh. 3 - 1743 Differentiate. 23. f(t) = t2 ln tCh. 3 - 1743 Differentiate. 24. g(t)=et1+etCh. 3 - 1743 Differentiate. 29. h() = etan 2Ch. 3 - 1743 Differentiate. 36. y = sin1(ex)Ch. 3 - Show that ddx(12tan1x+14ln(x+1)2x2+1)=1(1+x)(1+x2)Ch. 3 - 4548 Find f in terms of g. 45. f(x)=eg(x)Ch. 3 - 4548 Find f in terms of g. 46. f(x) = g(ex)Ch. 3 - 4548 Find f in terms of g. 47. f(x) = ln |g(x)|Ch. 3 - 4548 Find f in terms of g. 48. f(x) = g(ln x)Ch. 3 - 4950 Find f(n)(x). 49. f(x) = 2xCh. 3 - 4950 Find f(n)(x). 50. f(x) = ln(2x)Ch. 3 - Find an equation of the tangent to the curve y = x...Ch. 3 - A bacteria culture contains 200 cells initially...Ch. 3 - Cobalt-60 has a half-life of 5.24 years. (a) Find...Ch. 3 - Let C(t) be the concentration of a drug in the...Ch. 3 - A cup of hot chocolate has temperature 80C in a...Ch. 3 - 6176 Evaluate the limit. 61. limx0+tan1(1/x)Ch. 3 - 6176 Evaluate the limit. 62. limxexx2Ch. 3 - 6176 Evaluate the limit. 63. limx3e2/(x3)Ch. 3 - 6176 Evaluate the limit. 64. limxarctan(x3x)Ch. 3 - 6176 Evaluate the limit. 65. limx0+ln(sinhx)Ch. 3 - Prob. 66RECh. 3 - 6176 Evaluate the limit. 67. limx1+2x12xCh. 3 - 6176 Evaluate the limit. 68. limx(1+4x)xCh. 3 - 6176 Evaluate the limit. 69. limx0ex1tanxCh. 3 - 6176 Evaluate the limit. 70. limx0tan4xx+sin2xCh. 3 - Prob. 71RECh. 3 - Prob. 72RECh. 3 - 6176 Evaluate the limit. 73. limx(x2x3)e2xCh. 3 - 6176 Evaluate the limit. 74. limx(x)cscxCh. 3 - Prob. 75RECh. 3 - 6176 Evaluate the limit. 76. limx(/2)(tanx)cosxCh. 3 - If f(x) = ln x + tan1x, find (f1)(/4).Ch. 3 - Show that cosarctan[sin(arccotx)]=x2+1x2+2Ch. 3 - Calculate y'. 17. y=arctanCh. 3 - Calculate y'. 21. y = 3x ln xCh. 3 - Calculate y'. 27. y = log5(1 + 2x)Ch. 3 - Calculate y'. 28. y = (cos x)xCh. 3 - Calculate y'. 29. y=lnsinx12sin2xCh. 3 - Calculate y'. 30. y=(x2+1)4(2x+1)3(3x1)5Ch. 3 - Calculate y'. 31. y = x tan1(4x)Ch. 3 - Calculate y'. 32. y = ecos x + cos(ex)Ch. 3 - Calculate y'. 34. y = 10tanCh. 3 - Calculate y'. 38. y=arctan(arcsinx)Ch. 3 - Calculate y'. 41. y=x+1(2x)5(x+3)7Ch. 3 - Calculate y'. 43. y = x sinh(x2)Ch. 3 - Calculate y'. 45. y = ln( cosh 3x)Ch. 3 - 1-50 Calculate y'. 47. y = cosh–1(sinh x) Ch. 3 - Calculate y'. 48. y=xtanh1xCh. 3 - Calculate y'. 49. y=cos(etan3x)Ch. 3 - Use mathematical induction (page 72) to show that...Ch. 3 - If f(x) = xesin x find f(x). Graph f and f on the...Ch. 3 - At what point on the curve y = [ln(x + 4)]2 is the...Ch. 3 - (a) Find an equation of the tangent to the curve y...Ch. 3 - The function C(t) = K(eat ebt), where a, b, and K...Ch. 3 - (a) What does lHospitals Rule say? (b) How can you...
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