BuyFind

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
BuyFind

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

Solutions

Chapter
Section
Chapter 6, Problem 77RE
Textbook Problem

Evaluate the limit.

77. lim x 1 + ( x x 1 1 ln x )

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Chapter 6 Solutions

Single Variable Calculus
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Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - Assume that f is a one-to-one function. (a) If...Ch. 6.1 - If f(x) = x5 + x3 +x, find f 1 (3) and f(f 1(2)).Ch. 6.1 - If h(x)=x+x, find h1 (6).Ch. 6.1 - The graph of f is given. (a) Why is f one-to-one?...Ch. 6.1 - The formula , where F = 459.67, expresses the...Ch. 6.1 - In the theory of relativity, the mass of a...Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find a formula for the inverse of the function....Ch. 6.1 - Find an explicit formula for f 1 and use it to...Ch. 6.1 - Find an explicit formula for f 1 and use it to...Ch. 6.1 - Use the given graph of f to sketch the graph of f...Ch. 6.1 - Use the given graph of f to sketch the graph of f...Ch. 6.1 - Let f(x)=1x2, 0 x 1. (a) Find f 1. How is it...Ch. 6.1 - Let g(x)=1x33. (a) Find g1. How is it related to...Ch. 6.1 - (a) Show that f is one-to-one. (b) Use Theorem 7...Ch. 6.1 - (a) Show that f is one-to-one. (b) Use Theorem 7...Ch. 6.1 - (a) Show that f is one-to-one. (b) Use Theorem 7...Ch. 6.1 - (a) Show that f is one-to-one. (b) Use Theorem 7...Ch. 6.1 - Find (f 1)(a). 39.f(x) = 3x3 + 4x2 +6x +5, a = 5Ch. 6.1 - Find (f 1)(a). 40. f(x) = x3 +3 sin x + 2 cos x, a...Ch. 6.1 - Find (f 1)(a). 41.f(x) = 3 + x2 + tan(x/2), 1 x ...Ch. 6.1 - Find (f 1)(a). 42. f(x)=x3+4x+4, a = 3Ch. 6.1 - Suppose f 1 is the inverse function of a...Ch. 6.1 - If g is an increasing function such that g(2) = 8...Ch. 6.1 - If f(x)=3x1+t3dt, find (f 1)(0).Ch. 6.1 - Suppose f1 is the inverse function of a...Ch. 6.1 - Show that h(x) = sin x, x, is not one-to-one, but...Ch. 6.1 - (a) If we shift a curve to the left, what happens...Ch. 6.1 - (a) If f is a one-to-one, twice differentiable...Ch. 6.2 - (a) Write an equation that defines the exponential...Ch. 6.2 - (a) How is the number e defined? (b) What is an...Ch. 6.2 - Graph the given functions on a common screen. How...Ch. 6.2 - Graph the given functions on a common screen. How...Ch. 6.2 - Graph the given functions on a common screen. How...Ch. 6.2 - Graph the given functions on a common screen. How...Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Make a rough sketch of the graph of the function....Ch. 6.2 - Starting with the graph of y = ex, write the...Ch. 6.2 - Starting with the graph of y = ex, find the...Ch. 6.2 - Find the domain of each function. 15. (a)...Ch. 6.2 - Find the domain of each function. 16.(a)...Ch. 6.2 - Find the exponential function f(x) = Cbx whose...Ch. 6.2 - Find the exponential function f(x) = Cbx whose...Ch. 6.2 - Suppose the graphs of f(x) = x2 and g(x) = 2x are...Ch. 6.2 - Compare the functions f(x) = x5 and g(x) = 5x by...Ch. 6.2 - Compare the functions f(x) = x10 and g(x) = ex by...Ch. 6.2 - Use a graph to estimate the values of x such that...Ch. 6.2 - Find the limit. 23. limx(1.001)xCh. 6.2 - Find the limit. 24. limx(1.001)xCh. 6.2 - Find the limit. 25. limxe3xe3xe3xe3xCh. 6.2 - Find the limit. 26. limxex2Ch. 6.2 - Find the limit. 27. limx2+e3/(2x)Ch. 6.2 - Find the limit. 28. limx2e3/(2x)Ch. 6.2 - Find the limit. 29. limx(e2xcosx)Ch. 6.2 - Find the limit. 30. limx(/2)+etanxCh. 6.2 - Differentiate the function. 31. f(x)=e5Ch. 6.2 - Differentiate the function. 32. k(r)=er+rcCh. 6.2 - Differentiate the function. 33. f(x)=(3x25x)exCh. 6.2 - Differentiate the function. 34. y=ex1exCh. 6.2 - Differentiate the function. 35. y=eax3Ch. 6.2 - Differentiate the function. 36. g(x)=ex2xCh. 6.2 - Differentiate the function. 37. y=etanCh. 6.2 - Differentiate the function. 38. V(t)=4+ttetCh. 6.2 - Differentiate the function. 39. f(x)=x2exx2+exCh. 6.2 - Differentiate the function. 40. y=x2e1/xCh. 6.2 - Differentiate the function. 41. y=x2e3xCh. 6.2 - Differentiate the function. 42. f(t)=tan(1+e2t)Ch. 6.2 - Differentiate the function. 43. f(t)=eatsinbtCh. 6.2 - Differentiate the function. 44. f(z)=ez/(z1)Ch. 6.2 - Differentiate the function. 45. F(t)=etsin2tCh. 6.2 - Differentiate the function. 46. y=esin2x+sin(e2x)Ch. 6.2 - Differentiate the function. 47. g(u)=esecu2Ch. 6.2 - Differentiate the function. 48. y=1+xe2xCh. 6.2 - Differentiate the function. 49. y=cos(1e2x1+e2x)Ch. 6.2 - Differentiate the function. 50. f(t)=sin2(esin2t)Ch. 6.2 - Find an equation of the tangent line to the curve...Ch. 6.2 - Find an equation of the tangent line to the curve...Ch. 6.2 - Find y if ex/y=xy.Ch. 6.2 - Find an equation of the tangent line to the curve...Ch. 6.2 - Show that the function y=ex+ex/2 satisfies the...Ch. 6.2 - Show that the function y=Aex+Bxex satisfies the...Ch. 6.2 - For what values of r does the function y = erx...Ch. 6.2 - Find the values of for which y = ex satisfies the...Ch. 6.2 - If f(x) = e2x, find a formula for f(n) (x).Ch. 6.2 - Find the thousandth derivative of f(x)=xex.Ch. 6.2 - (a) Use the Intermediate Value Theorem to show...Ch. 6.2 - Use a graph to find an initial approximation (to...Ch. 6.2 - Use the graph of V in Figure 11 to estimate the...Ch. 6.2 - Under certain circumstances a rumor spreads...Ch. 6.2 - An object is attached to the end of a vibrating...Ch. 6.2 - Find the absolute maximum value of the function...Ch. 6.2 - Find the absolute minimum value of the function...Ch. 6.2 - Find the absolute maximum and absolute minimum...Ch. 6.2 - Find the absolute maximum and absolute minimum...Ch. 6.2 - Find (a) the intervals of increase or decrease,...Ch. 6.2 - Find (a) the intervals of increase or decrease,...Ch. 6.2 - Discuss the curve using the guidelines of Section...Ch. 6.2 - Discuss the curve using the guidelines of Section...Ch. 6.2 - Discuss the curve using the guidelines of Section...Ch. 6.2 - Letg(x)=ecx+f(x) and h(x)=ekxf(x), where...Ch. 6.2 - A drug response curve describes the level of...Ch. 6.2 - After an antibiotic tablet is taken, the...Ch. 6.2 - After the consumption of an alcoholic beverage,...Ch. 6.2 - Draw a graph of f that shows all the important...Ch. 6.2 - Draw a graph of f that shows all the important...Ch. 6.2 - The family of bell-shaped curves y=12e(x)2/(22)...Ch. 6.2 - Evaluate the integral. 83. 01(xe+ex)dxCh. 6.2 - Evaluate the integral. 84. 55edxCh. 6.2 - Evaluate the integral. 85. 02dxexCh. 6.2 - Evaluate the integral. 86. x2ex3dxCh. 6.2 - Evaluate the integral. 87. ex1+exdxCh. 6.2 - Evaluate the integral. 88. (1+ex)2exdxCh. 6.2 - Evaluate the integral. 89. (ex+ex)2dxCh. 6.2 - Evaluate the integral. 90.ex(4+ex)5dxCh. 6.2 - Evaluate the integral. 91. eu(1eu)2duCh. 6.2 - Evaluate the integral. 92. esincosdCh. 6.2 - Evaluate the integral. 93. 12e1/xx2dxCh. 6.2 - Evaluate the integral. 94. 011+exexdxCh. 6.2 - Find, correct to three decimal places, the area of...Ch. 6.2 - Find f(x) if f(x)=3ex+5sinx, f(0) = 1, and f(0)=2.Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - The error function erf(x)=20xet2dt is used in...Ch. 6.2 - Show that the function y=ex2erf(x) satisfies the...Ch. 6.2 - An oil storage tank ruptures at time t = 0 and oil...Ch. 6.2 - A bacteria population starts with 400 bacteria and...Ch. 6.2 - Dialysis treatment removes urea and other waste...Ch. 6.2 - The rate of growth of a fish population was...Ch. 6.2 - If f(x) = 3 + x + ex, find (f1)(4).Ch. 6.2 - Evaluate limxesinx1x.Ch. 6.2 - If you graph the function f(x)=1e1x1+e1x youll see...Ch. 6.2 - Graph several members of the family of functions...Ch. 6.2 - (a) Show that ex 1 + x if x 0. [Hint: Show that...Ch. 6.2 - (a) Use the inequality of Exercise 109(a) to show...Ch. 6.2 - (a) Use mathematical induction to prove that for x...Ch. 6.2 - Use the Laws of Logarithms to expand the quantity....Ch. 6.2 - Use the Laws of Logarithms to expand the quantity....Ch. 6.2 - Use the Laws of Logarithms to expand the quantity....Ch. 6.2 - Use the Laws of Logarithms to expand the quantity....Ch. 6.2 - Express the quantity as a single logarithm. 5.2 ln...Ch. 6.2 - Express the quantity as a single logarithm. 6....Ch. 6.2 - Express the quantity as a single logarithm. 7.ln...Ch. 6.2 - Express the quantity as a single logarithm. 8....Ch. 6.2 - 13ln(x+2)3+12[lnxln(x2+3x+2)2]Ch. 6.2 - ln b + 2 ln c 3 ln dCh. 6.2 - Make a rough sketch of the graph of each function....Ch. 6.2 - Make a rough sketch of the graph of each function....Ch. 6.2 - Make a rough sketch of the graph of each function....Ch. 6.2 - Make a rough sketch of the graph of each function....Ch. 6.2 - Find the limit. 15. limx3+ln(x29)Ch. 6.2 - Find the limit. 16. limx[ln(2+x)ln(1+x)]Ch. 6.2 - Differentiate the function. 17.f(x) = x3 ln xCh. 6.2 - Differentiate the function. 18.f(x) = x ln x xCh. 6.2 - Differentiate the function. 19.f(x) = sin(ln x)Ch. 6.2 - Differentiate the function. 20.f(x) = ln(sin2x)Ch. 6.2 - Differentiate the function. 21. f(x)=ln1xCh. 6.2 - Differentiate the function. 22. y=1lnxCh. 6.2 - Differentiate the function. 23.f(x) = sin x ln(5x)Ch. 6.2 - Differentiate the function. 24. h(x)=ln(x+x21)Ch. 6.2 - Differentiate the function. 25. g(x)=lnaxa+xCh. 6.2 - Differentiate the function. 26. g(t)=1+lntCh. 6.2 - Differentiate the function. 27. G(y)=ln(2y+1)5y2+1Ch. 6.2 - Differentiate the function. 28. H(z)=lna2z2a2+z2Ch. 6.2 - Differentiate the function. 29. F(t)=(lnt)2sintCh. 6.2 - Differentiate the function. 30. P(v)=lnv1vCh. 6.2 - Differentiate the function. 31. f(u)=lnu1+ln(2u)Ch. 6.2 - Differentiate the function. 32. y=(lntanx)2Ch. 6.2 - Differentiate the function. 33. y=ln|2x5x2|Ch. 6.2 - Differentiate the function. 34. y=lntan2xCh. 6.2 - Differentiate the function. 35. y=tan[ln(ax+b)]Ch. 6.2 - Differentiate the function. 36. y=ln(cscxcotx)Ch. 6.2 - Find y and y. 37. y=xlnxCh. 6.2 - Find y and y. 38. y=ln(1+lnx)Ch. 6.2 - Differentiate f and find the domain of f. 39....Ch. 6.2 - Differentiate f and find the domain of f. 40....Ch. 6.2 - Differentiate f and find the domain of f. 41....Ch. 6.2 - Differentiate f and find the domain of f. 42....Ch. 6.2 - If f(x)=ln(x+lnx), find f (1).Ch. 6.2 - If f(x)=lnxx, find f (e).Ch. 6.2 - Find f(x). Check that your answer is reasonable by...Ch. 6.2 - Find f(x). Check that your answer is reasonable by...Ch. 6.2 - Find an equation of the tangent line to the curve...Ch. 6.2 - Find an equation of the tangent line to the curve...Ch. 6.2 - Find y if y=ln(x2+y2).Ch. 6.2 - Find y if ln xy = y sin x.Ch. 6.2 - Find the formula for f (n)(x) if f(x)=ln(x1)Ch. 6.2 - Find d9dx9(x8lnx).Ch. 6.2 - Use a graph to estimate the roots of the equation...Ch. 6.2 - Use a graph to estimate the roots of the equation...Ch. 6.2 - Discuss the curve under the guidelines of Section...Ch. 6.2 - Discuss the curve under the guidelines of Section...Ch. 6.2 - Discuss the curve under the guidelines of Section...Ch. 6.2 - Discuss the curve under the guidelines of Section...Ch. 6.2 - Investigate the family of curves f(x)=ln(x2+c)....Ch. 6.2 - Use logarithmic differentiation to find the...Ch. 6.2 - Use logarithmic differentiation to find the...Ch. 6.2 - Use logarithmic differentiation to find the...Ch. 6.2 - Use logarithmic differentiation to find the...Ch. 6.2 - Evaluate the integral. 65. 243xdxCh. 6.2 - Evaluate the integral. 66. 03dx5x+1Ch. 6.2 - Evaluate the integral. 67. 12dt83tCh. 6.2 - Evaluate the integral. 68. 49(x+1x)2dxCh. 6.2 - Evaluate the integral. 69. 1ex2+x+1xdxCh. 6.2 - Evaluate the integral. 70. e6dxxlnxCh. 6.2 - Evaluate the integral. 71. (lnx)2xdxCh. 6.2 - Evaluate the integral. 72. cosx2+sinxdxCh. 6.2 - Evaluate the integral. 73. sin2x1+cos2xdxCh. 6.2 - Evaluate the integral. 74. cos(lnt)tdtCh. 6.2 - Show that cotxdx=ln|sinx|+C by (a) differentiating...Ch. 6.2 - Sketch the region enclosed by the curves...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - The work done by a gas when it expands from volume...Ch. 6.2 - Find f if f(x)=x2, x 0, f(1) = 0, and f(2) = 0.Ch. 6.2 - If g is the inverse function of f(x)=2x+lnx, find...Ch. 6.2 - (a) Find the linear approximation to f(x) = ln x...Ch. 6.2 - (a) By comparing areas, show that 13ln1.5512 (b)...Ch. 6.2 - Refer to Example 1. (a) Find an equation of the...Ch. 6.2 - By comparing areas, show that...Ch. 6.2 - Prove the third law of logarithms. [Hint: Start by...Ch. 6.2 - For what values of m do the line y = mx and the...Ch. 6.2 - (a) Compare the rates of growth of f(x) = x0.1 and...Ch. 6.2 - Use the definition of derivative to prove that...Ch. 6.3 - (a) How is the logarithmic function y = logb x...Ch. 6.3 - (a) What is the natural logarithm? (b) What is the...Ch. 6.3 - Find the exact value of each expression. 3....Ch. 6.3 - Find the exact value of each expression. 4....Ch. 6.3 - Find the exact value of each expression. 5....Ch. 6.3 - Find the exact value of each expression. 6....Ch. 6.3 - Find the exact value of each expression. 7....Ch. 6.3 - Find the exact value of each expression. 8....Ch. 6.3 - Use the properties of logarithms to expand the...Ch. 6.3 - Use the properties of logarithms to expand the...Ch. 6.3 - Use the properties of logarithms to expand the...Ch. 6.3 - Use the properties of logarithms to expand the...Ch. 6.3 - Express the quantity as a single logarithm. 13....Ch. 6.3 - Express the quantity as a single logarithm. 14....Ch. 6.3 - Express the quantity as a single logarithm. 15....Ch. 6.3 - Express the quantity as a single logarithm. 16....Ch. 6.3 - Express the quantity as a single logarithm. 17....Ch. 6.3 - Express the quantity as a single logarithm. 18....Ch. 6.3 - Use Formula 7 to evaluate each logarithm correct...Ch. 6.3 - Use Formula 7 to graph the given functions on a...Ch. 6.3 - Use Formula 7 to graph the given functions on a...Ch. 6.3 - Use Formula 7 to graph the given functions on a...Ch. 6.3 - Make a rough sketch of the graph of each function....Ch. 6.3 - Make a rough sketch of the graph of each function....Ch. 6.3 - (a) What are the domain and range of f? (b) What...Ch. 6.3 - (a) What are the domain and range of f? (b) What...Ch. 6.3 - Solve each equation for x. 27. (a)e74x = 6 (b)...Ch. 6.3 - Solve each equation for x. 28.(a) ln(x2 1) = 3(b)...Ch. 6.3 - Solve each equation for x. 29. (a) 2x5 = 3(b) ln x...Ch. 6.3 - Solve each equation for x. 30....Ch. 6.3 - Solve each equation for x. 31. e e2x = 1Ch. 6.3 - Solve each equation for x. 32.10(1 + ex)1 = 3Ch. 6.3 - Solve each equation for x. 33.ln(ln x) = 1Ch. 6.3 - Solve each equation for x. 34. eex=10Ch. 6.3 - Solve each equation for x. 35.e2x ex 6 = 0Ch. 6.3 - Solve each equation for x. 36. ln(2x + 1) = 2 ln...Ch. 6.3 - Find the solution of the equation correct to four...Ch. 6.3 - Find the solution of the equation correct to four...Ch. 6.3 - Solve each inequality for x. 39. (a) ln x 0(b) ex...Ch. 6.3 - Solve each inequality for x. 40.(a) 1 e3x1 2(b)...Ch. 6.3 - Suppose that the graph of y = log2x is drawn on a...Ch. 6.3 - The velocity of a particle that moves in a...Ch. 6.3 - The geologist C. F. Richter defined the magnitude...Ch. 6.3 - A sound so faint that it can just be heard has...Ch. 6.3 - If a bacteria population starts with 100 bacteria...Ch. 6.3 - When a camera flash goes off, the batteries...Ch. 6.3 - Find the limit. 47. limx3+ln(x29)Ch. 6.3 - Find the limit. 48. limx2log5(8xx4)Ch. 6.3 - Find the limit. 49. limx0ln(cosx)Ch. 6.3 - Find the limit. 50. limx0+ln(sinx)Ch. 6.3 - Find the limit. 51. limx[ln(1+x2)ln(1+x)]Ch. 6.3 - Find the limit. 52. limx[ln(2+x)ln(1+x)]Ch. 6.3 - Find the domain of the function. 53. f(x) = ln(4 ...Ch. 6.3 - Find the domain of the function. 54....Ch. 6.3 - Find (a) the domain of f and (b) f1 and its...Ch. 6.3 - Find (a) the domain of f and (b) f1 and its...Ch. 6.3 - Find (a) the domain of f and (b) f1 and its...Ch. 6.3 - (a) What are the values of eln 300 and ln(e300)?...Ch. 6.3 - Find the inverse function. 59. y = 2 ln(x 1)Ch. 6.3 - Find the inverse function. 60. g(x)=log4(x3+2)Ch. 6.3 - Find the inverse function. 61. f(x)=ex3Ch. 6.3 - Find the inverse function. 62. y = (ln x)2, x 1Ch. 6.3 - Find the inverse function. 63. y = 32x4Ch. 6.3 - Find the inverse function. 64. y=1ex1+exCh. 6.3 - On what interval is the function f(x) = e3x ex...Ch. 6.3 - On what interval is the curve y = 2ex e3x concave...Ch. 6.3 - (a) Show that the function f(x)=ln(x+x2+1) is an...Ch. 6.3 - Find an equation of the tangent to the curve y =...Ch. 6.3 - Show that the equation x1/ln x = 2 has no...Ch. 6.3 - Any function of the form f(x) = [g(x)]h(x), where...Ch. 6.3 - Let b 1. Prove, using Definitions 3.4.6 and...Ch. 6.3 - (a) Compare the rates of growth of f(x) = x0.1 and...Ch. 6.3 - Solve the inequality ln(x2 2x 2) 0.Ch. 6.3 - A prime number is a positive integer that has no...Ch. 6.3 - Sketch, by hand, the graph of the function f(x) =...Ch. 6.3 - Simplify each expression. 2. (a)eln15 (b) ln(1/e2)Ch. 6.3 - Simplify each expression. 3. (a)eln2 (b) eln(lne3)Ch. 6.3 - Simplify each expression. 4. (a)lnesinx (b) ex+lnxCh. 6.3 - Solve each equation for x. 5. (a)e74x=6 (b)...Ch. 6.3 - Solve each equation for x. 6. (a)ln(x21)=3 (b)...Ch. 6.3 - Solve each equation for x. 7. (a)e3x+1=k (b)...Ch. 6.3 - Solve each equation for x. 8. (a)ln(lnx)=1 (b)...Ch. 6.3 - Solve each equation for x. 9. ee2x=1Ch. 6.3 - Solve each equation for x. 10. 10(1+ex)1=3Ch. 6.3 - Solve each equation for x. 11. e2xex6=0Ch. 6.3 - Solve each equation for x. 12. ln(2x+1)=2lnxCh. 6.3 - Find the solution of the equation correct to four...Ch. 6.3 - Find the solution of the equation correct to four...Ch. 6.3 - Solve each inequality for x. 15. (a)lnx0 (b) ex5Ch. 6.3 - Solve each inequality for x. 16. (a)1e3x12 (b)...Ch. 6.3 - Make a rough sketch of the graph of the function....Ch. 6.3 - Make a rough sketch of the graph of the function....Ch. 6.3 - Make a rough sketch of the graph of the function....Ch. 6.3 - Make a rough sketch of the graph of the function....Ch. 6.3 - Find (a) the domain of f and (b) f 1 and its...Ch. 6.3 - Find (a) the domain of f and (b) f 1 and its...Ch. 6.3 - Find the inverse function. 23.y = 2 ln(x 1)Ch. 6.3 - Find the inverse function. 24.y = (ln x)2, x 1Ch. 6.3 - Find the inverse function. 25. f(x)=ex3Ch. 6.3 - Find the inverse function. 26. y=1ex1+exCh. 6.3 - Find the limit. 27. limxe3xe3xe3x+e3xCh. 6.3 - Find the limit. 28. limxex2Ch. 6.3 - Find the limit. 29. limx2+e3/(2x)Ch. 6.3 - Find the limit. 30. limx2e3/(2x)Ch. 6.3 - Find the limit. 31. limx(e2xcosx)Ch. 6.3 - Find the limit. 32. limx(/2)+etanxCh. 6.3 - Differentiate the function. 33.f(x) = e5Ch. 6.3 - Differentiate the function. 34.k(r) = er + reCh. 6.3 - Differentiate the function. 35. f(x) = (3x2 5x)exCh. 6.3 - Differentiate the function. 36. y=ex1exCh. 6.3 - Differentiate the function. 37. y=eax3Ch. 6.3 - Differentiate the function. 38. g(x)=ex2xCh. 6.3 - Differentiate the function. 39.y = etanCh. 6.3 - Differentiate the function. 40. V(t)=4+ttetCh. 6.3 - Differentiate the function. 41. f(x)=x2exx2+exCh. 6.3 - Differentiate the function. 42. y = x2 e1/xCh. 6.3 - Differentiate the function. 43. y = x2 e3xCh. 6.3 - Differentiate the function. 44.f(t) = tan(1 + e2t)Ch. 6.3 - Differentiate the function. 45.f(t) = eat sin btCh. 6.3 - Differentiate the function. 46. f(z) = ez/(z 1)Ch. 6.3 - Differentiate the function. 47. F(t) = et sin 2tCh. 6.3 - Differentiate the function. 48.y = esin 2x +...Ch. 6.3 - Differentiate the function. 49. g(u)=esecu2Ch. 6.3 - Differentiate the function. 50. y=1+xe2xCh. 6.3 - Differentiate the function. 51. y=cos(1e2x1+e2x)Ch. 6.3 - Differentiate the function. 52. f(t)=sin2(esin2t)Ch. 6.3 - Find an equation of the tangent line to the curve...Ch. 6.3 - Find an equation of the tangent line to the curve...Ch. 6.3 - Find y if ex/y = x y.Ch. 6.3 - Find an equation of the tangent line to the curve...Ch. 6.3 - Show that the function y = ex + ex/2 satisfies the...Ch. 6.3 - Show that the function y = Aex + Bxex satisfies...Ch. 6.3 - For what values of r does the function y = erx...Ch. 6.3 - Find the values of for which y = ex satisfies the...Ch. 6.3 - If f(x) = e2x find a formula for f(n)(x).Ch. 6.3 - Find the thousandth derivative of f(x) = xex.Ch. 6.3 - (a) Use the Intermediate Value Theorem to show...Ch. 6.3 - Use a graph to find an initial approximation (to...Ch. 6.3 - Under certain circumstances a rumor spreads...Ch. 6.3 - An object is attached to the end of a vibrating...Ch. 6.3 - Find the absolute maximum value of the function...Ch. 6.3 - Find the absolute minimum value of the function...Ch. 6.3 - Find the absolute maximum and absolute minimum...Ch. 6.3 - Find the absolute maximum and absolute minimum...Ch. 6.3 - Find (a) the intervals of increase or decrease,...Ch. 6.3 - Find (a) the intervals of increase or decrease,...Ch. 6.3 - Discuss the curve using the guidelines of Section...Ch. 6.3 - Discuss the curve using the guidelines of Section...Ch. 6.3 - Discuss the curve using the guidelines of Section...Ch. 6.3 - Let g(x) = ecx + f(x) and h(x) = ekxf(x), where...Ch. 6.3 - A drug response curve describes the level of...Ch. 6.3 - After an antibiotic tablet is taken, the...Ch. 6.3 - After the consumption of an alcoholic beverage,...Ch. 6.3 - Draw a graph of f that shows all the important...Ch. 6.3 - Draw a graph of f that shows all the important...Ch. 6.3 - The family of bell-shaped curves y=12e(x)2/(22)...Ch. 6.3 - Evaluate the integral. 83. 01(xe+ex)dxCh. 6.3 - Evaluate the integral. 84. 55edxCh. 6.3 - Evaluate the integral. 85. 02dxexCh. 6.3 - Evaluate the integral. 86. x2ex3dxCh. 6.3 - Evaluate the integral. 87. ex1+exdxCh. 6.3 - Evaluate the integral. 88. (1+ex)2exdxCh. 6.3 - Evaluate the integral. 89. (ex+ex)2dxCh. 6.3 - Evaluate the integral. 90. ex(4+ex)5dxCh. 6.3 - Evaluate the integral. 91. eu(1eu)2duCh. 6.3 - Evaluate the integral. 92. esincosdCh. 6.3 - Evaluate the integral. 93. 12e1/xx2dxCh. 6.3 - Evaluate the integral. 94. 011+exexdxCh. 6.3 - Find, correct to three decimal places, the area of...Ch. 6.3 - Find f(x) if f(x) = 3ex + 5 sin x, f(0) = 1, and...Ch. 6.3 - Find the volume of the solid obtained by rotating...Ch. 6.3 - Find the volume of the solid obtained by rotating...Ch. 6.3 - The error function erf(x)=20xet2dt is used in...Ch. 6.3 - Show that the function y=ex2erf(x) satisfies the...Ch. 6.3 - An oil storage tank ruptures at time t = 0 and oil...Ch. 6.3 - A bacteria population starts with 400 bacteria and...Ch. 6.3 - Dialysis treatment removes urea and other waste...Ch. 6.3 - The rate of growth of a fish population was...Ch. 6.3 - If you graph the function f(x)=1e1/x1+e1/x youll...Ch. 6.3 - Graph several members of the family of functions...Ch. 6.3 - Prove the second law of exponents [see (7)] Laws...Ch. 6.3 - Prove the third law of exponents [see (7)]. Laws...Ch. 6.3 - (a) Show that ex 1 + x if x 0. [Hint: Show that...Ch. 6.3 - (a) Use the inequality of Exercise 109(a) to show...Ch. 6.3 - (a) Use mathematical induction to prove that for x...Ch. 6.4 - Explain why the natural logarithmic function y =...Ch. 6.4 - Differentiate the function. 2.f(x) = x ln x xCh. 6.4 - Differentiate the function. 3.f(x) = sin (ln x)Ch. 6.4 - Differentiate the function. 4.f(x) = ln(sin2x)Ch. 6.4 - Differentiate the function. 5. f(x)=ln1xCh. 6.4 - Differentiate the function. 6. y=1lnxCh. 6.4 - Differentiate the function. 7. f(x)=log10(1+cosx)Ch. 6.4 - Differentiate the function. 8. f(x)=log10xCh. 6.4 - Differentiate the function. 9.g(x) = ln (xe2x)Ch. 6.4 - Differentiate the function. 10. g(t)=1+lntCh. 6.4 - Differentiate the function. 11. F(t)=(lnt)2sintCh. 6.4 - Differentiate the function. 12. h(x)=ln(x+x21)Ch. 6.4 - Differentiate the function. 13. G(y)=ln(2y+1)5y2+1Ch. 6.4 - Differentiate the function. 14. P(v)=lnv1vCh. 6.4 - Differentiate the function. 15. f(u)=lnu1+ln(2u)Ch. 6.4 - Differentiate the function. 16. y=ln|1+tt3|Ch. 6.4 - Differentiate the function. 17.f(x) = x5 + 5xCh. 6.4 - Differentiate the function. 18.g(x) = x sin(2x)Ch. 6.4 - Differentiate the function. 19. T(z)=2zlog2zCh. 6.4 - Differentiate the function. 20.y =ln(csc x cot x)Ch. 6.4 - Differentiate the function. 21.y = ln(ex + xex)Ch. 6.4 - Differentiate the function. 22. H(z)=lna2z2a2+z2Ch. 6.4 - Differentiate the function. 23.y = tan[ln(ax + b)Ch. 6.4 - Differentiate the function. 24. y=log2(xlog5x)Ch. 6.4 - Differentiate the function. 25. G(x)=4C/xCh. 6.4 - Differentiate the function. 26.F(t) = 3cos2tCh. 6.4 - Find y and y. 27. y=xlnxCh. 6.4 - Find y and y. 28. y=lnx1+lnxCh. 6.4 - Find y and y. 29. y = ln |sec x|Ch. 6.4 - Find y and y. 30.y = ln(1 + ln x)Ch. 6.4 - Differentiate f and find the domain of f. 31....Ch. 6.4 - Differentiate f and find the domain of f. 32....Ch. 6.4 - Differentiate f and find the domain of f. 33. f(x)...Ch. 6.4 - Differentiate f and find the domain of f. 34.f(x)...Ch. 6.4 - If f(x) = ln(x + ln x), find f(1).Ch. 6.4 - If f(x) = cos(ln x2), find f(1).Ch. 6.4 - Find an equation of the tangent line to the curve...Ch. 6.4 - Find an equation of the tangent line to the curve...Ch. 6.4 - If f(x)=sinx+lnx, find f(x). Check that your...Ch. 6.4 - Find equations of the tangents lines to the curve...Ch. 6.4 - Let f(x)=cx+ln(cosx). For what value of c is...Ch. 6.4 - Let f(x) = logb(3x2 2). For what value of b is...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Find y if y=ln(x2+y2).Ch. 6.4 - Find y if xy=yx.Ch. 6.4 - Find a formula for f(n)(x) if f(x)=ln(x1).Ch. 6.4 - Find d9dx9(x8lnx).Ch. 6.4 - Use a graph to estimate the roots of the equation...Ch. 6.4 - Use a graph to estimate the roots of the equation...Ch. 6.4 - Find the intervals of concavity and the inflection...Ch. 6.4 - Find the absolute minimum value of the function...Ch. 6.4 - Discuss the curve under the guidelines of Section...Ch. 6.4 - Discuss the curve under the guidelines of Section...Ch. 6.4 - Discuss the curve under the guidelines of Section...Ch. 6.4 - Discuss the curve under the guidelines of Section...Ch. 6.4 - Investigate the family of curves f(x) = ln(x2 +...Ch. 6.4 - Evaluate the integral. 71. 243xdxCh. 6.4 - Evaluate the integral. 72. 03dx5x+1Ch. 6.4 - Evaluate the integral. 73. 12dt83tCh. 6.4 - Evaluate the integral. 74. 49(x+1x)2dxCh. 6.4 - Evaluate the integral. 75. 1ex2+x+1xdxCh. 6.4 - Evaluate the integral. 76. cos(lnt)tdtCh. 6.4 - Evaluate the integral. 77. (lnx)2xdxCh. 6.4 - Evaluate the integral. 78. cosx2+sinxdxCh. 6.4 - Evaluate the integral. 79. sin2x1+cos2xdxCh. 6.4 - Evaluate the integral. 80. exex+1dxCh. 6.4 - Evaluate the integral. 81. 042sdsCh. 6.4 - Evaluate the integral. 82. x2x2dxCh. 6.4 - Show that cotxdx=ln|sinx|+C by (a) differentiating...Ch. 6.4 - Sketch the region enclosed by the curves y=lnxx...Ch. 6.4 - Find the volume of the solid obtained by rotating...Ch. 6.4 - Find the volume of the solid obtained by rotating...Ch. 6.4 - The work done by a gas when it expands from volume...Ch. 6.4 - Find f if f(x)=x2,x0,f(1)=0, and f(2) = 0.Ch. 6.4 - If g is the inverse function of f(x)=2x+lnx, find...Ch. 6.4 - If f(x)=ex+lnx and h(x)=f1(x), find h(e)Ch. 6.4 - For what values of m do the line y = mx and the...Ch. 6.4 - (a) Find the linear approximation to f(x) = In x...Ch. 6.4 - Use the definition of derivative to prove that...Ch. 6.4 - Show that limn(1+xn)n=ex for any x 0.Ch. 6.4 - (a) Write an equation that defines bx when b is a...Ch. 6.4 - (a) If b is a positive number and b 1, how is...Ch. 6.4 - Write the expression as a power of e. 3.4Ch. 6.4 - Write the expression as a power of e. 4. x5Ch. 6.4 - Write the expression as a power of e. 5. 10x2Ch. 6.4 - Write the expression as a power of e. 6.(tan x)sec...Ch. 6.4 - Evaluate the expression. 7.(a) log232 (b) log82Ch. 6.4 - Evaluate the expression. 8.(a) log1010 (b)...Ch. 6.4 - Evaluate the expression. 9. (a)log10 40 + log10...Ch. 6.4 - Evaluate the expression. 10. (a) loga1a (b)...Ch. 6.4 - Graph the given functions on a common screen. How...Ch. 6.4 - Graph the given functions on a common screen. How...Ch. 6.4 - Use Formula 6 to evaluate each logarithm correct...Ch. 6.4 - Use Formula 6 to graph the given functions on a...Ch. 6.4 - Use Formula 6 to graph the given functions on a...Ch. 6.4 - Use Formula 6 to graph the given functions on a...Ch. 6.4 - Find the exponential function f(x) = Cbx whose...Ch. 6.4 - Find the exponential function f(x) = Cbx whose...Ch. 6.4 - (a) Suppose the graphs of f(x) = x2 and g(x) = 2x...Ch. 6.4 - Compare the rates of growth of the functions f(x)...Ch. 6.4 - Find the limit. 21. limx(1.001)xCh. 6.4 - Find the limit. 22. limx(1.001)xCh. 6.4 - Find the limit. 23.limt2t2Ch. 6.4 - Find the limit. 24. limx3+log10(x25x+6)Ch. 6.4 - Differentiate the function. 25.f(x) = x5 + 5xCh. 6.4 - Differentiate the function. 26.g(x) = x sin(2x)Ch. 6.4 - Differentiate the function. 27. G(x)=4C/xCh. 6.4 - Differentiate the function. 28. F(t) = 3cos 2tCh. 6.4 - Differentiate the function. 29. L(v)=tan(4v2)Ch. 6.4 - Differentiate the function. 30. G(u) = (1 + 10ln...Ch. 6.4 - Differentiate the function. 31.f(x) = log2(1 3x)Ch. 6.4 - Differentiate the function. 32. f(x)=log10xCh. 6.4 - Differentiate the function. 33.y = x log4 sin xCh. 6.4 - Differentiate the function. 34.y = log2(x log5x)Ch. 6.4 - Differentiate the function. 35.y = xxCh. 6.4 - Differentiate the function. 36. y=xcosxCh. 6.4 - Differentiate the function. 37. y=xsinxCh. 6.4 - Differentiate the function. 38. y=(x)xCh. 6.4 - Differentiate the function. 39. y=(cosx)xCh. 6.4 - Differentiate the function. 40. y=(sinx)lnxCh. 6.4 - Differentiate the function. 41. y=(tanx)1/xCh. 6.4 - Differentiate the function. 42. y=(lnx)cosxCh. 6.4 - Find an equation of the tangent line to the curve...Ch. 6.4 - If f(x)=xcosx, find f(x). Check that your answer...Ch. 6.4 - Evaluate the integral. 45. 042sdsCh. 6.4 - Evaluate the integral. 46. (x5+5x)dxCh. 6.4 - Evaluate the integral. 47. log10xxdxCh. 6.4 - Evaluate the integral. 48. x2x2dxCh. 6.4 - Evaluate the integral. 49. 3sincosdCh. 6.4 - Evaluate the integral. 50. 2x2x+1dxCh. 6.4 - Find the area of the region bounded by the curves...Ch. 6.4 - The region under the curve y = 10x from x = 0 to x...Ch. 6.4 - Use a graph to find the root of the equation 2x= 1...Ch. 6.4 - Find y if xy = yx.Ch. 6.4 - Find the inverse function of g(x)=log4(x3+2).Ch. 6.4 - Calculate limx0+xlnxCh. 6.4 - The geologist C. F. Richter defined the magnitude...Ch. 6.4 - A sound so faint that it can just be heard has...Ch. 6.4 - Referring to Exercise 58, find the rate of change...Ch. 6.4 - According to the Beer-Lambert Law, the light...Ch. 6.4 - After the consumption of an alcoholic beverage,...Ch. 6.4 - In this section we modeled the world population...Ch. 6.4 - Use the graph of V in Figure 9 to estimate the...Ch. 6.4 - Prove the second law of exponents [see (3)].Ch. 6.4 - Prove the fourth law of exponents [see (3)].Ch. 6.4 - Deduce the following laws of logarithms from (3):...Ch. 6.4 - Show that limn(1+xn)n=ex for any x 0.Ch. 6.5 - A population of protozoa develops with a constant...Ch. 6.5 - A common inhabitant of human intestines is the...Ch. 6.5 - A bacteria culture initially contains 100 cells...Ch. 6.5 - A bacteria culture grows with constant relative...Ch. 6.5 - The table gives estimates of the world population,...Ch. 6.5 - The table gives the population of Indonesia, in...Ch. 6.5 - Experiments show that if the chemical reaction...Ch. 6.5 - Strontium-90 has a halt-life of 28 days. (a) A...Ch. 6.5 - The half-life of cesium-1 37 is 30 years. Suppose...Ch. 6.5 - A sample of tritium-3 decayed to 94.5% of its...Ch. 6.5 - Scientists can determine the age of ancient...Ch. 6.5 - Dinosaur Fossils are too old to be reliably dated...Ch. 6.5 - Dinosaur fossils are often dated by using an...Ch. 6.5 - A curve passes through the point (0, 5) and has...Ch. 6.5 - A roast turkey is taken from an oven when its...Ch. 6.5 - In a murder investigation, the temperature of the...Ch. 6.5 - When a cold drink is taken from a refrigerator,...Ch. 6.5 - A freshly brewed cup of coffee has temperature 95C...Ch. 6.5 - The rate of change of atmospheric pressure P with...Ch. 6.5 - (a) It 1000 is borrowed at 8% interest, find the...Ch. 6.5 - (a) If 3000 is invested at 5% interest, find the...Ch. 6.5 - (a) How long will it take an investment to double...Ch. 6.6 - Find the exact value of each expression. 1....Ch. 6.6 - Find the exact value of each expression. 2....Ch. 6.6 - Find the exact value of each expression....Ch. 6.6 - Find the exact value of each expression....Ch. 6.6 - Find the exact value of each expression....Ch. 6.6 - Find the exact value of each expression....Ch. 6.6 - Find the exact value of each expression. 7....Ch. 6.6 - Find the exact value of each expression. 8....Ch. 6.6 - Find the exact value of each expression. 9....Ch. 6.6 - Find the exact value of each expression. 10....Ch. 6.6 - Prove that cos(sin1x)=1x2.Ch. 6.6 - Simplify the expression. 12. tan(sin1x)Ch. 6.6 - Simplify the expression. 13. sin(tan1x)Ch. 6.6 - Simplify the expression. 14. sin(2arccosx)Ch. 6.6 - Graph the given functions on the same screen. How...Ch. 6.6 - Graph the given functions on the same screen. How...Ch. 6.6 - Prove Formula 6 for the derivative of cos1 by the...Ch. 6.6 - (a) Prove that sin1x+cos1x=/2. (b) Use part (a) to...Ch. 6.6 - Prove that ddx(cot1x)=11+x2.Ch. 6.6 - Prove that ddx(sec1x)=1xx21.Ch. 6.6 - Prove that ddx(csc1x)=1xx21.Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - Find the derivative of the function. Simplify...Ch. 6.6 - y=arccos(b+acosxa+bcosx),0x,ab0Ch. 6.6 - Find the derivative of the function. Find the...Ch. 6.6 - Find the derivative of the function. Find the...Ch. 6.6 - Find yif tan1(x2y)=x+xy2.Ch. 6.6 - If g(x)=xsin1(x/4)+16x2, find g(2).Ch. 6.6 - Find an equation of the tangent line to the curve...Ch. 6.6 - Find f(x). Check that your answer is reasonable by...Ch. 6.6 - Find f(x). Check that your answer is reasonable by...Ch. 6.6 - Find the limit. 43. limx1+sin1xCh. 6.6 - Find the limit. 44. limxarccos(1+x21+2x2)Ch. 6.6 - Find the limit. 45. limxarctan(ex)Ch. 6.6 - Find the limit. 46. limx0+tan1(lnx)Ch. 6.6 - Where should the point P be chosen on the line...Ch. 6.6 - A painting in an art gallery has height h and is...Ch. 6.6 - A ladder 10 ft long leans against a vertical wall....Ch. 6.6 - A lighthouse is located on a small island, 3 km...Ch. 6.6 - Sketch the curve using the guidelines of Section...Ch. 6.6 - Sketch the curve using the guidelines of Section...Ch. 6.6 - Sketch the curve using the guidelines of Section...Ch. 6.6 - Sketch the curve using the guidelines of Section...Ch. 6.6 - Find the most general antiderivative of the...Ch. 6.6 - Find g(t) if g(t)=2/1t2 and g(1)=5.Ch. 6.6 - Evaluate the integral. 59. 1/3381+x2dxCh. 6.6 - Evaluate the integral. 60. 1/21/261p2dpCh. 6.6 - Evaluate the integral. 61. 01/2sin1x1x2dxCh. 6.6 - Evaluate the integral. 62. 03/4dx1+16x2Ch. 6.6 - Evaluate the integral. 63. 1+x1x2dxCh. 6.6 - Evaluate the integral. 64. 0/2sinx1+cos2xdxCh. 6.6 - Evaluate the integral. 65. dx1x2sin1xCh. 6.6 - Evaluate the integral. 66. 1xx24dxCh. 6.6 - Evaluate the integral. 67. t21t6dtCh. 6.6 - Evaluate the integral. 68. e2x1e4xdxCh. 6.6 - Evaluate the integral. 69. dxx(1+x)Ch. 6.6 - Evaluate the integral. 70. x1+x4dxCh. 6.6 - Use the method of Example 8 to show that, if a 0,...Ch. 6.6 - The region under the curve y=1/x2+4 from x = 0, x...Ch. 6.6 - Evaluate 01sin1xdx by interpreting it as an area...Ch. 6.6 - Prove that, for xy1, arctanx+arctany=arctanx+y1xy...Ch. 6.6 - Use the result of Exercise 74 to prove the...Ch. 6.6 - (a) Sketch the graph of the function...Ch. 6.6 - Use the method of Example 6 to prove the identity...Ch. 6.6 - Prove the identity arcsinx1x+1=2arctanx2Ch. 6.6 - Some authors define y=sec1xsecy=x and...Ch. 6.6 - Let f(x)=xarctan(1/x) if x0 and f(0) = 0. (a) Is f...Ch. 6.7 - Find the numerical value of each expression. 1....Ch. 6.7 - Find the numerical value of each expression. 2....Ch. 6.7 - Find the numerical value of each expression. 3....Ch. 6.7 - Find the numerical value of each expression. 4....Ch. 6.7 - Find the numerical value of each expression. 5....Ch. 6.7 - Find the numerical value of each expression. 6....Ch. 6.7 - Prove the identity. 7. sinh(x)=sinhx (This shows...Ch. 6.7 - Prove the identity. 8. cosh(x)=coshx (This shows...Ch. 6.7 - Prove the identity. 9. coshx+sinhx=exCh. 6.7 - Prove the identity. 10. coshxsinhx=exCh. 6.7 - Prove the identity. 11....Ch. 6.7 - Prove the identity. 12....Ch. 6.7 - Prove the identity. 13. coth2x1=csch2xCh. 6.7 - Prove the identity. 14....Ch. 6.7 - Prove the identity. 15. sinh 2x = 2 sinh x cosh xCh. 6.7 - Prove the identity. 16. cosh 2x = cosh2x + sinh2xCh. 6.7 - Prove the identity. 17. tanh(lnx)=x21x2+1Ch. 6.7 - Prove the identity. 18. 1+tanhx1tanhx=e2xCh. 6.7 - Prove the identity. 19. (cosh x + sinh x)n = cosh...Ch. 6.7 - If x=1213, find the values of the other hyperbolic...Ch. 6.7 - If x=53 and x 0, find the values of the other...Ch. 6.7 - (a) Use the graphs of sinh, cosh, and tanh in...Ch. 6.7 - Use the definitions of the hyperbolic functions to...Ch. 6.7 - Prove the formulas given in Table 1 for the...Ch. 6.7 - Give an alternative solution to Example 3 by...Ch. 6.7 - Prove Equation 4.Ch. 6.7 - Prove Equation 5 using (a) the method of Example 3...Ch. 6.7 - For each of the following functions (i) give a...Ch. 6.7 - Prove the formulas given in Table 6 for the...Ch. 6.7 - Find the derivative. Simplify where possible. 30....Ch. 6.7 - Find the derivative. Simplify where possible. 31....Ch. 6.7 - Find the derivative. Simplify where possible. 32....Ch. 6.7 - Find the derivative. Simplify where possible. 33....Ch. 6.7 - Find the derivative. Simplify where possible. 34....Ch. 6.7 - Find the derivative. Simplify where possible. 35....Ch. 6.7 - Find the derivative. Simplify where possible. 36....Ch. 6.7 - Find the derivative. Simplify where possible. 37....Ch. 6.7 - Find the derivative. Simplify where possible. 38....Ch. 6.7 - Find the derivative. Simplify where possible. 39....Ch. 6.7 - Find the derivative. Simplify where possible. 40....Ch. 6.7 - Find the derivative. Simplify where possible. 41....Ch. 6.7 - Find the derivative. Simplify where possible. 42....Ch. 6.7 - Find the derivative. Simplify where possible. 43....Ch. 6.7 - Find the derivative. Simplify where possible. 44....Ch. 6.7 - Find the derivative. Simplify where possible. 45....Ch. 6.7 - Show thatddx1+tanx1tanx4=12ex/2.Ch. 6.7 - Show that ddxarctan(tanhx)=sech2x.Ch. 6.7 - The Gateway Arch in St. Louis was designed by Eero...Ch. 6.7 - If a water wave with length L moves with velocity...Ch. 6.7 - A flexible cable always hangs in the shape of a...Ch. 6.7 - A telephone line hangs between two poles 14 m...Ch. 6.7 - Using principles from physics it can be shown that...Ch. 6.7 - A cable with linear density = 2 kg/m is strung...Ch. 6.7 - A model for the velocity of a falling object after...Ch. 6.7 - (a) Show that any function of the form y = A sinh...Ch. 6.7 - If x=ln(sec+tan), show that sec = cosh x.Ch. 6.7 - At what point of the curve y = cosh x does the...Ch. 6.7 - Investigate the family of functions fN(x) = tanh(n...Ch. 6.7 - Evaluate the integral. 59. sinhxcosh2xdxCh. 6.7 - Evaluate the integral. 60. sinh(1+4x)dxCh. 6.7 - Evaluate the integral. 61. sinhxxdxCh. 6.7 - Evaluate the integral. 62. tanhxdxCh. 6.7 - Evaluate the integral. 63. coshxcosh2x1dxCh. 6.7 - Evaluate the integral. 64. sech2x2+tanhxdxCh. 6.7 - Evaluate the integral. 65. 461t29dtCh. 6.7 - Evaluate the integral. 66. 01116t2+1dtCh. 6.7 - Evaluate the integral. 67. ex1e2xdxCh. 6.7 - Estimate the value of the number c such that the...Ch. 6.7 - (a) Use Newtons method or a graphing device to...Ch. 6.7 - Show that the area of the shaded hyperbolic sector...Ch. 6.7 - Show that if a 0 and b 0, then there exist...Ch. 6.8 - Given that limxaf(x)=0limxag(x)=0limxah(x)=1...Ch. 6.8 - Given that limxaf(x)=0limxag(x)=0limxah(x)=1...Ch. 6.8 - Given that limxaf(x)=0limxag(x)=0limxah(x)=1...Ch. 6.8 - Given that limxaf(x)=0limxag(x)=0limxah(x)=1...Ch. 6.8 - Use the graphs of f and g and their tangent lines...Ch. 6.8 - Use the graphs of f and g and their tangent lines...Ch. 6.8 - The graph of a function f and its tangent line at...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Use a graph to estimate the value of the limit....Ch. 6.8 - Use a graph to estimate the value of the limit....Ch. 6.8 - Illustrate lHospitals Rule by graphing both...Ch. 6.8 - Illustrate lHospitals Rule by graphing both...Ch. 6.8 - Prove that limxexxn= for any positive integer n....Ch. 6.8 - Prove that limxlnxxp=0 for any number p 0. This...Ch. 6.8 - What happens if you try to use lHospitals Rule to...Ch. 6.8 - What happens if you try to use lHospitals Rule to...Ch. 6.8 - Use 1Hospitals Rule to help sketch the curve. Use...Ch. 6.8 - Use 1Hospitals Rule to help sketch the curve. Use...Ch. 6.8 - Use 1Hospitals Rule to help sketch the curve. Use...Ch. 6.8 - Use lHospitals Rule to help sketch the curve. Use...Ch. 6.8 - Use 1Hospitals Rule to help sketch the curve. Use...Ch. 6.8 - Use 1Hospitals Rule to help sketch the curve. Use...Ch. 6.8 - Investigate the family of curves given by...Ch. 6.8 - Investigate the family of curvesf(x)=excx. In...Ch. 6.8 - If an object with mass m is dropped from rest, one...Ch. 6.8 - If an initial amount A0 of money is invested at an...Ch. 6.8 - Light enters the eye through the pupil and strikes...Ch. 6.8 - Some populations initially grow exponentially but...Ch. 6.8 - A metal cable has radius r and is covered by...Ch. 6.8 - In Section 4.3 we investigated the Fresnel...Ch. 6.8 - Suppose that the temperature in a long thin rod...Ch. 6.8 - The first appearance in prim of lHospitals Rule...Ch. 6.8 - The figure shows a sector of a circle with central...Ch. 6.8 - Evaluate limx[xx2ln(1+xx)]Ch. 6.8 - Suppose f is a positive function. If limxaf(x)=0...Ch. 6.8 - If f is continuous, f(2) = 0, and f'(2) = 7,...Ch. 6.8 - For what values of a and b is the following...Ch. 6.8 - If f is continuous, use 1Hospitals Rule to show...Ch. 6.8 - If fis continuous, show that...Ch. 6.8 - Let f(x)={e1/x2ifx00ifx=0 (a) Use the definition...Ch. 6.8 - Let f(x)={|x|xifx01ifx=0 (a) Show that f is...Ch. 6 - (a) What is a one-to-one function? How can you...Ch. 6 - (a) What are the domain and range of the natural...Ch. 6 - (a) How is the inverse sine function f(x) = sin1 x...Ch. 6 - Write the definitions of the hyperbolic functions...Ch. 6 - State the derivative of each function. (a) y = ex...Ch. 6 - (a) How is the number e defined? (b) Express e as...Ch. 6 - (a) Write a differential equation that expresses...Ch. 6 - (a) What does lHospitals Rule say? (b) How can you...Ch. 6 - State whether each of the following limit forms is...Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - Determine whether the statement is true or false....Ch. 6 - 1. The graph of f is shown. Is f one-to-one?...Ch. 6 - The graph of g is given. (a) Why is g one-to-one?...Ch. 6 - Suppose f is one-to-one, f(7) = 3, and f'(7) = 8....Ch. 6 - Find the inverse function of f(x)=x+12x+1.Ch. 6 - Sketch a rough graph of the function without using...Ch. 6 - Sketch a rough graph of the function without using...Ch. 6 - Sketch a rough graph of the function without using...Ch. 6 - Sketch a rough graph of the function without using...Ch. 6 - Sketch a rough graph of the function without using...Ch. 6 - Let b 1. For large values of x, which of the...Ch. 6 - Find the exact value of each expression. 11. (a)...Ch. 6 - Find the exact value of each expression. 12.(a)...Ch. 6 - Solution the equation for x. 13. lnx=13Ch. 6 - Solve the equation for x. 14. ex=13Ch. 6 - Solve the equation for x. 15. eex=17Ch. 6 - Solve the equation for x. 16. ln(1+ex)=3Ch. 6 - Solve the equation for x. 17. ln(x+1)+ln(x1)=1Ch. 6 - Solve the equation for x. 18. log5(cx)=dCh. 6 - Solve the equation for x. 19. tan1x=1Ch. 6 - Solve the equation for x. 20.sin x = 0.3Ch. 6 - Differentiate. 21. f(t)=t2lntCh. 6 - Differentiate. 22. g(t)=et1+etCh. 6 - Differentiate. 23. h()=etan2Ch. 6 - Differentiate. 24. h(u)=10uCh. 6 - Differentiate. 25. y=ln|sec5x+tan5x|Ch. 6 - Differentiate. 26. y=xcos1xCh. 6 - Differentiate. 27. y=xtan1(4x)Ch. 6 - Differentiate. 28. y=emxcosnxCh. 6 - Differentiate. 29. y=ln(sec2x)Ch. 6 - Differentiate. 30. y=tln(t4)Ch. 6 - Differentiate. 31. y=e1/xx2Ch. 6 - Differentiate. 32.y = (arcsin 2x)2Ch. 6 - Differentiate. 33. y=3xlnxCh. 6 - Differentiate. 34. y=ecosx+cos(ex)Ch. 6 - Differentiate. 35. H(v)=vtan1vCh. 6 - Differentiate. 36. F(z)=log10(1+z2)Ch. 6 - Differentiate. 37. y=xsinh(x2)Ch. 6 - Differentiate. 38. y=(cosx)xCh. 6 - Differentiate. 39. y=lnsinx12sin2xCh. 6 - Differentiate. 40. y=arctan(arcsinx)Ch. 6 - Differentiate. 41. y=ln(1x)+1lnxCh. 6 - Differentiate. 42. xey=y1Ch. 6 - Differentiate. 43. y=ln(cosh3x)Ch. 6 - Differentiate. 44. y=(x2+1)2(2x+1)3(3x1)5Ch. 6 - Differentiate. 45. y=cosh1(sinhx)Ch. 6 - Differentiate. 46. y=xtanh1xCh. 6 - Differentiate. 47. y=cos(etan3x)Ch. 6 - Show that ddx(12tan1x+14ln(x+1)2x2+1)=1(1+x)(1+x2)Ch. 6 - Find f in terms of g. 49. f(x)=eg(x)Ch. 6 - Find f in terms of g. 50. f(x)=g(ex)Ch. 6 - Find f in terms of g. 51. f(x)=ln|g(x)|Ch. 6 - Find f in terms of g. 52. f(x)=g(lnx)Ch. 6 - Find f(n)(x). 53. f(x)=2xCh. 6 - Find f(n)(x). 54. f(x)=ln(2x)Ch. 6 - Use mathematical induction to show that if...Ch. 6 - Find y if y = x + arctan y.Ch. 6 - Find an equation of the tangent to the curve at...Ch. 6 - Find an equation of the tangent to the curve at...Ch. 6 - At what point on the curve y = [ln(x + 4)]2 is the...Ch. 6 - If f(x)=xesinx, find f(x). Graph f and f on the...Ch. 6 - (a) Find an equation of the tangent to the curve y...Ch. 6 - The function C(t)=K(eatebt), where a, b, and K are...Ch. 6 - Evaluate the limit. 63. limxe3xCh. 6 - Evaluate the limit. 64. limx10ln(100x2)Ch. 6 - Evaluate the limit. 65. limx3e2/(x3)Ch. 6 - Evaluate the limit. 66. limxarctan(x3x)Ch. 6 - Evaluate the limit. 67. limx0+ln(sinhx)Ch. 6 - Evaluate the limit. 68. limxexsinxCh. 6 - Evaluate the limit. 69. limx1+2x12xCh. 6 - Evaluate the limit. 70. limx(1+4x)xCh. 6 - Evaluate the limit. 71. limxex1tanxCh. 6 - Evaluate the limit. 72. limx1cosxx2+xCh. 6 - Evaluate the limit. 73. limxe2xe2xln(x+1)Ch. 6 - Evaluate the limit. 74. limxe2xe2xln(x+1)Ch. 6 - Evaluate the limit. 75. limx(x2x3)e2xCh. 6 - Evaluate the limit. 76. limx0+x2lnxCh. 6 - Evaluate the limit. 77. limx1+(xx11lnx)Ch. 6 - Evaluate the limit. 78. limx(/2)(tanx)cosxCh. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Sketch the curve using the guidelines of Section...Ch. 6 - Investigate the family of curves given by...Ch. 6 - Investigate the family of functions f(x)=cxecx2...Ch. 6 - An equation of motion of the form s=Aectcos(t+)...Ch. 6 - (a) Show that there is exactly one root of the...Ch. 6 - A bacteria culture contains 200 cells initially...Ch. 6 - Cobalt-60 has a half-life of 5.24 years. (a) Find...Ch. 6 - The biologist G. F. Gause conducted an experiment...Ch. 6 - Evaluate the integral. 92. 04116+t2dtCh. 6 - Evaluate the integral. 93. 01ye2y2dyCh. 6 - Evaluate the integral. 94. 25dr1+2rCh. 6 - Evaluate the integral. 95. 01ex1+e2xdxCh. 6 - Evaluate the integral. 96. 0/2cosx1+sin2xdxCh. 6 - Evaluate the integral. 97. exxdxCh. 6 - Evaluate the integral. 98. sin(lnx)xdxCh. 6 - Evaluate the integral. 99. x+1x2+2xdxCh. 6 - Evaluate the integral. 100. csc2x1+cotxdxCh. 6 - Evaluate the integral. 101. tanxln(cosx)dxCh. 6 - Evaluate the integral. 102. x1x4dxCh. 6 - Evaluate the integral. 103. 2tansec2dCh. 6 - Evaluate the integral. 104. sinhauduCh. 6 - Evaluate the integral. 105. (1xx)2dxCh. 6 - Use Properties of integrals to prove the...Ch. 6 - Use Properties of integrals to prove the...Ch. 6 - Use Properties of integrals to prove the...Ch. 6 - Find f(x). 109. f(x)=1xessdsCh. 6 - Find f(x). 110.f(x)=lnx2xet2dtCh. 6 - Find the average value of the function f(x) = 1/x...Ch. 6 - Find the area of the region bounded by the curves...Ch. 6 - Find the volume of the solid obtained by rotating...Ch. 6 - If f(x) = x + x2 + ex, find (f1)(1).Ch. 6 - If f(x) = ln x + tan1 x, find (f1)(/4).Ch. 6 - What is the area of the largest rectangle in the...Ch. 6 - What is the area of the largest triangle in the...Ch. 6 - Evaluate 01exdx without using the Fundamental...Ch. 6 - If F(x)=abtxdt, where a, b 0, then, by the...Ch. 6 - Show that cos{arctan[sin(arccotx)]}=x2+1x2+2Ch. 6 - If f is a continuous function such that...Ch. 6 - The figure shows two regions in the first...Ch. 6 - 1. If a rectangle has its base on the x-axis and...Ch. 6 - Prove that log2 5 is an irrational number.Ch. 6 - Does the function f(x)=e10|x2|x2 an absolute...Ch. 6 - If 04e(x2)4dx=k, find the value of 04xe(x2)4dx.Ch. 6 - Show that dndxn(eaxsinbx)=rneaxsin(bx+n) where a...Ch. 6 - Show that sin1(tanh x) = tan1(sinh x).Ch. 6 - Show that for x 0, x1+x2tan1xxCh. 6 - Suppose f is continuous, f(0) = 0, f(1) = 1, f(x) ...Ch. 6 - Show that f(x)=1x1+t3dt is one-to-one and find...Ch. 6 - If y=xa212a21arctansinxa+a21+cosx show that...Ch. 6 - For what value of a is the following equation...Ch. 6 - Evaluate limx(x+2)1/xx1/x(x+3)1/xx1/xCh. 6 - Evaluate limx01x0x(1tan2t)1/tdt. [Assume that the...Ch. 6 - Sketch the set of all points (x, y) such that...Ch. 6 - Prove that cosh(sinh x) sinh(cosh x) for all x.Ch. 6 - Show that, for all positive value of x and y,...Ch. 6 - For what value of k does the equation e2x=kx have...Ch. 6 - For which positive numbers a is it true that ax1+x...Ch. 6 - For which positive numbers a does the curve y = ax...Ch. 6 - For what values of c does the curve y = cx3 + ex...

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