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 6.4, Problem 49E
Textbook Problem

Use logarithmic differentiation to find the derivative of the function.

49. y = x sin x

Expert Solution

Want to see this answer and more?

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

See solution

Chapter 6 Solutions

Single Variable Calculus
Show all chapter solutions
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...

Additional Math Textbook Solutions

Find more solutions based on key concepts
Show solutions
Solve the equations in Exercises 126. 14x2=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 39-50, sketch the graph of the function with the given rule. Find the domain and range of the func...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 7-34, perform the indicated operations and simplify. 22.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate each expression: 7+6(3+2)75(4+2)

Elementary Technical Mathematics

True or False: f(x) = 10x − x2 is increasing on (4,8).

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The normal plane to at t = 1 has equation:

Study Guide for Stewart's Multivariable Calculus, 8th

A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

21. A U.S. Senate Judiciary Committee report showed the number of homicides in each state. In Indiana. Ohio, an...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Quartic Regression In Exercise S-8 through S-14, use regression to find a quartic model for the given data set....

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)