Single Variable Calculus
Single Variable Calculus
8th Edition
ISBN: 9781305266636
Author: James Stewart
Publisher: Cengage Learning
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Chapter 6.2, Problem 65E

Under certain circumstances a rumor spreads according to the equation

p ( t )   =   1 1 + a e − k t

where p(t) is the proportion of the population that has heard the rumor at time t and a and k are positive constants. [In Section 9.4 we will see that this is a reasonable model for p(t).]

(a) Find lim t → ∞ p ( t ) .

(b) Find the rate of spread of the rumor.

(c) Graph p for the case a = 10, k = 0.5 with t measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.

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Chapter 6 Solutions

Single Variable Calculus

Ch. 6.1 - (a) What is a one-to-one function? (b) How can you...Ch. 6.1 - (a) Suppose f is a one-to-one function with domain...Ch. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - Prob. 9ECh. 6.1 - Prob. 10E
Ch. 6.1 - A function is given by a table of values, a graph,...Ch. 6.1 - Prob. 12ECh. 6.1 - Prob. 13ECh. 6.1 - Prob. 14ECh. 6.1 - Prob. 15ECh. 6.1 - Prob. 16ECh. 6.1 - Assume that f is a one-to-one function. (a) If...Ch. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Prob. 21ECh. 6.1 - Prob. 22ECh. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - Prob. 28ECh. 6.1 - Prob. 29ECh. 6.1 - Prob. 30ECh. 6.1 - Prob. 31ECh. 6.1 - Prob. 32ECh. 6.1 - Prob. 33ECh. 6.1 - Prob. 34ECh. 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 - Prob. 40ECh. 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 - Prob. 48ECh. 6.1 - Prob. 49ECh. 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 - Prob. 2ECh. 6.2 - Prob. 3ECh. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Prob. 7ECh. 6.2 - Prob. 8ECh. 6.2 - Prob. 9ECh. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Prob. 16ECh. 6.2 - Find the exponential function f(x) = Cbx whose...Ch. 6.2 - Prob. 18ECh. 6.2 - Prob. 19ECh. 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 - Prob. 22ECh. 6.2 - Prob. 23ECh. 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 - Prob. 30ECh. 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 - Prob. 50ECh. 6.2 - Prob. 51ECh. 6.2 - Prob. 52ECh. 6.2 - Find y if ex/y=xy.Ch. 6.2 - Prob. 54ECh. 6.2 - Prob. 55ECh. 6.2 - Prob. 56ECh. 6.2 - For what values of r does the function y = erx...Ch. 6.2 - Prob. 58ECh. 6.2 - If f(x) = e2x, find a formula for f(n) (x).Ch. 6.2 - Prob. 60ECh. 6.2 - Prob. 61ECh. 6.2 - Prob. 62ECh. 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 - Prob. 66ECh. 6.2 - Find the absolute maximum value of the function...Ch. 6.2 - Prob. 68ECh. 6.2 - Prob. 69ECh. 6.2 - Prob. 70ECh. 6.2 - Prob. 71ECh. 6.2 - Find (a) the intervals of increase or decrease,...Ch. 6.2 - Prob. 73ECh. 6.2 - Prob. 74ECh. 6.2 - Prob. 75ECh. 6.2 - Prob. 76ECh. 6.2 - A drug response curve describes the level of...Ch. 6.2 - Prob. 78ECh. 6.2 - After the consumption of an alcoholic beverage,...Ch. 6.2 - Prob. 80ECh. 6.2 - Prob. 81ECh. 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 - Prob. 90ECh. 6.2 - Prob. 91ECh. 6.2 - Prob. 92ECh. 6.2 - Prob. 93ECh. 6.2 - Prob. 94ECh. 6.2 - Find, correct to three decimal places, the area of...Ch. 6.2 - Prob. 96ECh. 6.2 - Prob. 97ECh. 6.2 - Find the volume of the solid obtained by rotating...Ch. 6.2 - Prob. 99ECh. 6.2 - Prob. 100ECh. 6.2 - Prob. 101ECh. 6.2 - Prob. 102ECh. 6.2 - Prob. 103ECh. 6.2 - Prob. 104ECh. 6.2 - Prob. 105ECh. 6.2 - Prob. 106ECh. 6.2 - Prob. 107ECh. 6.2 - Prob. 108ECh. 6.2 - Prob. 109ECh. 6.2 - Prob. 110ECh. 6.2 - Prob. 111ECh. 6.2 - Prob. 1AECh. 6.2 - Prob. 2AECh. 6.2 - Prob. 3AECh. 6.2 - Prob. 4AECh. 6.2 - Express the quantity as a single logarithm. 5.2 ln...Ch. 6.2 - Prob. 6AECh. 6.2 - Prob. 7AECh. 6.2 - Express the quantity as a single logarithm. 8....Ch. 6.2 - Prob. 9AECh. 6.2 - Prob. 10AECh. 6.2 - Prob. 11AECh. 6.2 - Prob. 12AECh. 6.2 - Prob. 13AECh. 6.2 - Prob. 14AECh. 6.2 - Prob. 15AECh. 6.2 - Prob. 16AECh. 6.2 - Prob. 17AECh. 6.2 - Prob. 18AECh. 6.2 - Prob. 19AECh. 6.2 - Prob. 20AECh. 6.2 - Prob. 21AECh. 6.2 - Prob. 22AECh. 6.2 - Prob. 23AECh. 6.2 - Prob. 24AECh. 6.2 - Prob. 25AECh. 6.2 - Prob. 26AECh. 6.2 - Prob. 27AECh. 6.2 - Prob. 28AECh. 6.2 - Prob. 29AECh. 6.2 - Prob. 30AECh. 6.2 - Prob. 31AECh. 6.2 - Prob. 32AECh. 6.2 - Prob. 33AECh. 6.2 - Prob. 34AECh. 6.2 - Prob. 35AECh. 6.2 - Prob. 36AECh. 6.2 - Prob. 37AECh. 6.2 - Prob. 38AECh. 6.2 - Prob. 39AECh. 6.2 - Prob. 40AECh. 6.2 - Prob. 41AECh. 6.2 - Prob. 42AECh. 6.2 - Prob. 43AECh. 6.2 - Prob. 44AECh. 6.2 - Prob. 45AECh. 6.2 - Prob. 46AECh. 6.2 - Prob. 47AECh. 6.2 - Prob. 48AECh. 6.2 - Prob. 49AECh. 6.2 - Prob. 50AECh. 6.2 - Prob. 51AECh. 6.2 - Prob. 52AECh. 6.2 - Prob. 53AECh. 6.2 - Prob. 54AECh. 6.2 - Prob. 55AECh. 6.2 - Prob. 56AECh. 6.2 - Prob. 57AECh. 6.2 - Prob. 58AECh. 6.2 - Prob. 60AECh. 6.2 - Prob. 61AECh. 6.2 - Prob. 62AECh. 6.2 - Prob. 63AECh. 6.2 - Prob. 64AECh. 6.2 - Prob. 65AECh. 6.2 - Prob. 66AECh. 6.2 - Prob. 67AECh. 6.2 - Prob. 68AECh. 6.2 - Prob. 69AECh. 6.2 - Evaluate the integral. 70. e6dxxlnxCh. 6.2 - Prob. 71AECh. 6.2 - Prob. 72AECh. 6.2 - Prob. 73AECh. 6.2 - Prob. 74AECh. 6.2 - Show that cotxdx=ln|sinx|+C by (a) differentiating...Ch. 6.2 - Sketch the region enclosed by the curves...Ch. 6.2 - Prob. 77AECh. 6.2 - Prob. 78AECh. 6.2 - Prob. 79AECh. 6.2 - Prob. 80AECh. 6.2 - Prob. 81AECh. 6.2 - Prob. 82AECh. 6.2 - (a) By comparing areas, show that 13ln1.5512 (b)...Ch. 6.2 - Prob. 84AECh. 6.2 - Prob. 85AECh. 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 - Prob. 88AECh. 6.2 - Prob. 89AECh. 6.3 - (a) How is the logarithmic function y = logb x...Ch. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - Prob. 8ECh. 6.3 - Prob. 9ECh. 6.3 - Prob. 10ECh. 6.3 - Prob. 11ECh. 6.3 - Prob. 12ECh. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Express the quantity as a single logarithm. 16....Ch. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Use Formula 7 to graph the given functions on a...Ch. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Prob. 33ECh. 6.3 - Solve each equation for x. 34. eex=10Ch. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Prob. 38ECh. 6.3 - Prob. 39ECh. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - The velocity of a particle that moves in a...Ch. 6.3 - Prob. 43ECh. 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 - Prob. 47ECh. 6.3 - Find the limit. 48. limx2log5(8xx4)Ch. 6.3 - Prob. 49ECh. 6.3 - Prob. 50ECh. 6.3 - Prob. 51ECh. 6.3 - Prob. 52ECh. 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 - Prob. 55ECh. 6.3 - Prob. 56ECh. 6.3 - Prob. 57ECh. 6.3 - Prob. 58ECh. 6.3 - Prob. 59ECh. 6.3 - Prob. 60ECh. 6.3 - Prob. 61ECh. 6.3 - Prob. 62ECh. 6.3 - Prob. 63ECh. 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 - Prob. 66ECh. 6.3 - Prob. 67ECh. 6.3 - Find an equation of the tangent to the curve y =...Ch. 6.3 - Prob. 69ECh. 6.3 - Prob. 70ECh. 6.3 - Prob. 71ECh. 6.3 - Prob. 72ECh. 6.3 - Prob. 73ECh. 6.3 - Prob. 74ECh. 6.3 - Sketch, by hand, the graph of the function f(x) =...Ch. 6.3 - Prob. 2AECh. 6.3 - Prob. 3AECh. 6.3 - Prob. 4AECh. 6.3 - Prob. 5AECh. 6.3 - Prob. 6AECh. 6.3 - Prob. 7AECh. 6.3 - Prob. 8AECh. 6.3 - Prob. 9AECh. 6.3 - Prob. 10AECh. 6.3 - Prob. 11AECh. 6.3 - Prob. 12AECh. 6.3 - Prob. 13AECh. 6.3 - Prob. 14AECh. 6.3 - Prob. 15AECh. 6.3 - Prob. 16AECh. 6.3 - Prob. 17AECh. 6.3 - Prob. 18AECh. 6.3 - Prob. 19AECh. 6.3 - Prob. 20AECh. 6.3 - Prob. 21AECh. 6.3 - Prob. 22AECh. 6.3 - Prob. 23AECh. 6.3 - Prob. 24AECh. 6.3 - Prob. 25AECh. 6.3 - Find the inverse function. 26. y=1ex1+exCh. 6.3 - Prob. 27AECh. 6.3 - Prob. 28AECh. 6.3 - Prob. 29AECh. 6.3 - Prob. 30AECh. 6.3 - Prob. 31AECh. 6.3 - Prob. 32AECh. 6.3 - Prob. 33AECh. 6.3 - Prob. 34AECh. 6.3 - Prob. 35AECh. 6.3 - Prob. 36AECh. 6.3 - Prob. 37AECh. 6.3 - Prob. 38AECh. 6.3 - Prob. 39AECh. 6.3 - Prob. 40AECh. 6.3 - Prob. 41AECh. 6.3 - Prob. 42AECh. 6.3 - Prob. 43AECh. 6.3 - Prob. 44AECh. 6.3 - Prob. 45AECh. 6.3 - Prob. 46AECh. 6.3 - Prob. 47AECh. 6.3 - Prob. 48AECh. 6.3 - Prob. 49AECh. 6.3 - Prob. 50AECh. 6.3 - Prob. 51AECh. 6.3 - Prob. 52AECh. 6.3 - Prob. 53AECh. 6.3 - Prob. 54AECh. 6.3 - Prob. 55AECh. 6.3 - Find an equation of the tangent line to the curve...Ch. 6.3 - Prob. 57AECh. 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 - Prob. 61AECh. 6.3 - Prob. 62AECh. 6.3 - Prob. 63AECh. 6.3 - Prob. 64AECh. 6.3 - Under certain circumstances a rumor spreads...Ch. 6.3 - Prob. 66AECh. 6.3 - Prob. 67AECh. 6.3 - Find the absolute minimum value of the function...Ch. 6.3 - Prob. 69AECh. 6.3 - Prob. 70AECh. 6.3 - Prob. 71AECh. 6.3 - Prob. 72AECh. 6.3 - Discuss the curve using the guidelines of Section...Ch. 6.3 - Prob. 74AECh. 6.3 - Discuss the curve using the guidelines of Section...Ch. 6.3 - Prob. 76AECh. 6.3 - Prob. 77AECh. 6.3 - After an antibiotic tablet is taken, the...Ch. 6.3 - After the consumption of an alcoholic beverage,...Ch. 6.3 - Prob. 80AECh. 6.3 - Prob. 81AECh. 6.3 - Prob. 82AECh. 6.3 - Prob. 83AECh. 6.3 - Prob. 84AECh. 6.3 - Prob. 85AECh. 6.3 - Prob. 86AECh. 6.3 - Prob. 87AECh. 6.3 - Prob. 88AECh. 6.3 - Prob. 89AECh. 6.3 - Prob. 90AECh. 6.3 - Prob. 91AECh. 6.3 - Prob. 92AECh. 6.3 - Prob. 93AECh. 6.3 - Prob. 94AECh. 6.3 - Prob. 95AECh. 6.3 - Prob. 96AECh. 6.3 - Prob. 97AECh. 6.3 - Prob. 98AECh. 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 - Prob. 101AECh. 6.3 - Prob. 102AECh. 6.3 - Prob. 103AECh. 6.3 - The rate of growth of a fish population was...Ch. 6.3 - Prob. 105AECh. 6.3 - Prob. 106AECh. 6.3 - Prob. 107AECh. 6.3 - Prob. 108AECh. 6.3 - Prob. 109AECh. 6.3 - Prob. 110AECh. 6.3 - (a) Use mathematical induction to prove that for x...Ch. 6.4 - Explain why the natural logarithmic function y =...Ch. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Prob. 23ECh. 6.4 - Differentiate the function. 24. y=log2(xlog5x)Ch. 6.4 - Differentiate the function. 25. G(x)=4C/xCh. 6.4 - Prob. 26ECh. 6.4 - Prob. 27ECh. 6.4 - Prob. 28ECh. 6.4 - Prob. 29ECh. 6.4 - Prob. 30ECh. 6.4 - Prob. 31ECh. 6.4 - Prob. 32ECh. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Prob. 35ECh. 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 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - Prob. 40ECh. 6.4 - Prob. 41ECh. 6.4 - Let f(x) = logb(3x2 2). For what value of b is...Ch. 6.4 - Prob. 43ECh. 6.4 - Prob. 44ECh. 6.4 - Prob. 45ECh. 6.4 - Prob. 46ECh. 6.4 - Prob. 47ECh. 6.4 - Prob. 48ECh. 6.4 - Use logarithmic differentiation to find the...Ch. 6.4 - Prob. 50ECh. 6.4 - Prob. 51ECh. 6.4 - Prob. 52ECh. 6.4 - Prob. 53ECh. 6.4 - Prob. 54ECh. 6.4 - Prob. 55ECh. 6.4 - Prob. 56ECh. 6.4 - Prob. 57ECh. 6.4 - Prob. 58ECh. 6.4 - Prob. 59ECh. 6.4 - Prob. 60ECh. 6.4 - Prob. 61ECh. 6.4 - Find the absolute minimum value of the function...Ch. 6.4 - Prob. 63ECh. 6.4 - Discuss the curve under the guidelines of Section...Ch. 6.4 - Prob. 65ECh. 6.4 - Prob. 66ECh. 6.4 - Prob. 68ECh. 6.4 - Prob. 71ECh. 6.4 - Prob. 72ECh. 6.4 - Prob. 73ECh. 6.4 - Prob. 74ECh. 6.4 - Prob. 75ECh. 6.4 - Prob. 76ECh. 6.4 - Prob. 77ECh. 6.4 - Prob. 78ECh. 6.4 - Prob. 79ECh. 6.4 - Evaluate the integral. 80. exex+1dxCh. 6.4 - Evaluate the integral. 81. 042sdsCh. 6.4 - Prob. 82ECh. 6.4 - Prob. 83ECh. 6.4 - Prob. 84ECh. 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 - Prob. 87ECh. 6.4 - Prob. 88ECh. 6.4 - Prob. 89ECh. 6.4 - If f(x)=ex+lnx and h(x)=f1(x), find h(e)Ch. 6.4 - Prob. 91ECh. 6.4 - Prob. 92ECh. 6.4 - Prob. 93ECh. 6.4 - Prob. 94ECh. 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 - Prob. 4AECh. 6.4 - Prob. 5AECh. 6.4 - Prob. 6AECh. 6.4 - Prob. 7AECh. 6.4 - Prob. 8AECh. 6.4 - Evaluate the expression. 9. (a)log10 40 + log10...Ch. 6.4 - Prob. 10AECh. 6.4 - Prob. 11AECh. 6.4 - Prob. 12AECh. 6.4 - Prob. 13AECh. 6.4 - Prob. 14AECh. 6.4 - Prob. 15AECh. 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 - Prob. 18AECh. 6.4 - (a) Suppose the graphs of f(x) = x2 and g(x) = 2x...Ch. 6.4 - Prob. 20AECh. 6.4 - Prob. 21AECh. 6.4 - Prob. 22AECh. 6.4 - Prob. 23AECh. 6.4 - Prob. 24AECh. 6.4 - Prob. 25AECh. 6.4 - Prob. 26AECh. 6.4 - Prob. 27AECh. 6.4 - Prob. 28AECh. 6.4 - Prob. 29AECh. 6.4 - Prob. 30AECh. 6.4 - Prob. 31AECh. 6.4 - Prob. 32AECh. 6.4 - Prob. 33AECh. 6.4 - Prob. 34AECh. 6.4 - Prob. 35AECh. 6.4 - Prob. 36AECh. 6.4 - Prob. 37AECh. 6.4 - Prob. 38AECh. 6.4 - Differentiate the function. 39. y=(cosx)xCh. 6.4 - Prob. 40AECh. 6.4 - Prob. 41AECh. 6.4 - Prob. 42AECh. 6.4 - Find an equation of the tangent line to the curve...Ch. 6.4 - Prob. 44AECh. 6.4 - Prob. 45AECh. 6.4 - Prob. 46AECh. 6.4 - Prob. 47AECh. 6.4 - Prob. 48AECh. 6.4 - Prob. 49AECh. 6.4 - Prob. 50AECh. 6.4 - Prob. 51AECh. 6.4 - The region under the curve y = 10x from x = 0 to x...Ch. 6.4 - Prob. 53AECh. 6.4 - Prob. 54AECh. 6.4 - Prob. 55AECh. 6.4 - Prob. 56AECh. 6.4 - Prob. 57AECh. 6.4 - Prob. 58AECh. 6.4 - Prob. 59AECh. 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 - Prob. 67AECh. 6.4 - Prob. 68AECh. 6.4 - Prob. 69AECh. 6.4 - Prob. 70AECh. 6.5 - A population of protozoa develops with a constant...Ch. 6.5 - A common inhabitant of human intestines is the...Ch. 6.5 - Prob. 3ECh. 6.5 - A bacteria culture grows with constant relative...Ch. 6.5 - The table gives estimates of the world population,...Ch. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Strontium-90 has a halt-life of 28 days. (a) A...Ch. 6.5 - Prob. 9ECh. 6.5 - Prob. 10ECh. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Dinosaur fossils are often dated by using an...Ch. 6.5 - Prob. 14ECh. 6.5 - Prob. 15ECh. 6.5 - In a murder investigation, the temperature of the...Ch. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - The rate of change of atmospheric pressure P with...Ch. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.6 - Find the exact value of each expression. 1....Ch. 6.6 - Prob. 2ECh. 6.6 - Prob. 3ECh. 6.6 - Prob. 4ECh. 6.6 - Prob. 5ECh. 6.6 - Prob. 6ECh. 6.6 - Prob. 7ECh. 6.6 - Find the exact value of each expression. 8....Ch. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Prob. 11ECh. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Prob. 14ECh. 6.6 - Prob. 15ECh. 6.6 - Prob. 16ECh. 6.6 - Prob. 17ECh. 6.6 - (a) Prove that sin1x+cos1x=/2. (b) Use part (a) to...Ch. 6.6 - Prob. 19ECh. 6.6 - Prob. 20ECh. 6.6 - Prob. 21ECh. 6.6 - Prob. 22ECh. 6.6 - Prob. 23ECh. 6.6 - Prob. 24ECh. 6.6 - Prob. 25ECh. 6.6 - Prob. 26ECh. 6.6 - Prob. 27ECh. 6.6 - Prob. 28ECh. 6.6 - Prob. 29ECh. 6.6 - Prob. 30ECh. 6.6 - Prob. 31ECh. 6.6 - Prob. 32ECh. 6.6 - Prob. 33ECh. 6.6 - Prob. 34ECh. 6.6 - y=arccos(b+acosxa+bcosx),0x,ab0Ch. 6.6 - Prob. 36ECh. 6.6 - Prob. 37ECh. 6.6 - Prob. 38ECh. 6.6 - Prob. 39ECh. 6.6 - Prob. 40ECh. 6.6 - Prob. 41ECh. 6.6 - Prob. 42ECh. 6.6 - Prob. 43ECh. 6.6 - Prob. 44ECh. 6.6 - Prob. 45ECh. 6.6 - Prob. 46ECh. 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 - Prob. 51ECh. 6.6 - Prob. 52ECh. 6.6 - Sketch the curve using the guidelines of Section...Ch. 6.6 - Prob. 54ECh. 6.6 - Prob. 57ECh. 6.6 - Prob. 58ECh. 6.6 - Prob. 59ECh. 6.6 - Evaluate the integral. 60. 1/21/261p2dpCh. 6.6 - Prob. 61ECh. 6.6 - Prob. 62ECh. 6.6 - Prob. 63ECh. 6.6 - Prob. 64ECh. 6.6 - Prob. 65ECh. 6.6 - Prob. 66ECh. 6.6 - Prob. 67ECh. 6.6 - Prob. 68ECh. 6.6 - Prob. 69ECh. 6.6 - Prob. 70ECh. 6.6 - Prob. 71ECh. 6.6 - Prob. 72ECh. 6.6 - Prob. 73ECh. 6.6 - Prob. 74ECh. 6.6 - Prob. 75ECh. 6.6 - Prob. 76ECh. 6.6 - Prob. 77ECh. 6.6 - Prob. 78ECh. 6.6 - Some authors define y=sec1xsecy=x and...Ch. 6.6 - Prob. 80ECh. 6.7 - Prob. 1ECh. 6.7 - Prob. 2ECh. 6.7 - Find the numerical value of each expression. 3....Ch. 6.7 - Find the numerical value of each expression. 4....Ch. 6.7 - Prob. 5ECh. 6.7 - Prob. 6ECh. 6.7 - Prob. 7ECh. 6.7 - Prob. 8ECh. 6.7 - Prob. 9ECh. 6.7 - Prob. 10ECh. 6.7 - Prob. 11ECh. 6.7 - Prob. 12ECh. 6.7 - Prob. 13ECh. 6.7 - Prob. 14ECh. 6.7 - Prob. 15ECh. 6.7 - Prob. 16ECh. 6.7 - Prob. 17ECh. 6.7 - Prob. 18ECh. 6.7 - Prob. 19ECh. 6.7 - Prob. 20ECh. 6.7 - Prob. 21ECh. 6.7 - Prob. 22ECh. 6.7 - Use the definitions of the hyperbolic functions to...Ch. 6.7 - Prob. 24ECh. 6.7 - Give an alternative solution to Example 3 by...Ch. 6.7 - Prob. 26ECh. 6.7 - Prob. 27ECh. 6.7 - Prob. 28ECh. 6.7 - Prob. 29ECh. 6.7 - Prob. 30ECh. 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 - Prob. 46ECh. 6.7 - Prob. 47ECh. 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 - Prob. 50ECh. 6.7 - Prob. 51ECh. 6.7 - Using principles from physics it can be shown that...Ch. 6.7 - Prob. 53ECh. 6.7 - Prob. 54ECh. 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 - Prob. 58ECh. 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 - Prob. 68ECh. 6.7 - Prob. 69ECh. 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 - Prob. 8ECh. 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 - Prob. 15ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 17ECh. 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 - Prob. 24ECh. 6.8 - Prob. 25ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 27ECh. 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 - Prob. 32ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 34ECh. 6.8 - Prob. 35ECh. 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 - Prob. 39ECh. 6.8 - Prob. 40ECh. 6.8 - Prob. 41ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 43ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 46ECh. 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 - Prob. 50ECh. 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 - Prob. 54ECh. 6.8 - Prob. 55ECh. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Find the limit. Use lHospitals Rule where...Ch. 6.8 - Prob. 58ECh. 6.8 - Prob. 59ECh. 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 - Prob. 66ECh. 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 - Prob. 70ECh. 6.8 - Prob. 71ECh. 6.8 - Prob. 72ECh. 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 - Prob. 75ECh. 6.8 - Prob. 76ECh. 6.8 - Prob. 77ECh. 6.8 - Prob. 78ECh. 6.8 - Prob. 79ECh. 6.8 - Prob. 80ECh. 6.8 - Prob. 81ECh. 6.8 - Prob. 82ECh. 6.8 - Prob. 86ECh. 6.8 - Prob. 87ECh. 6.8 - Prob. 88ECh. 6.8 - Prob. 89ECh. 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 - Prob. 94ECh. 6.8 - Prob. 95ECh. 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 - Prob. 99ECh. 6.8 - Prob. 100ECh. 6.8 - Prob. 101ECh. 6.8 - Prob. 102ECh. 6.8 - Prob. 103ECh. 6.8 - Prob. 104ECh. 6 - (a) What is a one-to-one function? How can you...Ch. 6 - Prob. 2RCCCh. 6 - Prob. 3RCCCh. 6 - Prob. 4RCCCh. 6 - Prob. 5RCCCh. 6 - Prob. 6RCCCh. 6 - Prob. 7RCCCh. 6 - (a) What does lHospitals Rule say? (b) How can you...Ch. 6 - Prob. 9RCCCh. 6 - Prob. 1RQCh. 6 - Prob. 2RQCh. 6 - Prob. 3RQCh. 6 - Prob. 4RQCh. 6 - Prob. 5RQCh. 6 - Prob. 6RQCh. 6 - Determine whether the statement is true or false....Ch. 6 - Prob. 8RQCh. 6 - Prob. 9RQCh. 6 - Prob. 10RQCh. 6 - Prob. 11RQCh. 6 - Prob. 12RQCh. 6 - Prob. 13RQCh. 6 - Determine whether the statement is true or false....Ch. 6 - Prob. 15RQCh. 6 - Prob. 16RQCh. 6 - Prob. 17RQCh. 6 - Prob. 18RQCh. 6 - Prob. 19RQCh. 6 - Prob. 1RECh. 6 - Prob. 2RECh. 6 - Suppose f is one-to-one, f(7) = 3, and f'(7) = 8....Ch. 6 - Prob. 4RECh. 6 - Prob. 5RECh. 6 - Prob. 6RECh. 6 - Prob. 7RECh. 6 - Prob. 8RECh. 6 - Prob. 9RECh. 6 - Prob. 10RECh. 6 - Find the exact value of each expression. 11. (a)...Ch. 6 - Prob. 12RECh. 6 - Prob. 13RECh. 6 - Prob. 14RECh. 6 - Prob. 15RECh. 6 - Prob. 16RECh. 6 - Prob. 17RECh. 6 - Prob. 18RECh. 6 - Prob. 19RECh. 6 - Prob. 20RECh. 6 - Prob. 21RECh. 6 - Prob. 22RECh. 6 - Prob. 23RECh. 6 - Prob. 24RECh. 6 - Prob. 25RECh. 6 - Prob. 26RECh. 6 - Prob. 27RECh. 6 - Prob. 28RECh. 6 - Differentiate. 29. y=ln(sec2x)Ch. 6 - Prob. 30RECh. 6 - Prob. 31RECh. 6 - Prob. 32RECh. 6 - Prob. 33RECh. 6 - Prob. 34RECh. 6 - Prob. 35RECh. 6 - Prob. 36RECh. 6 - Prob. 37RECh. 6 - Prob. 38RECh. 6 - Prob. 39RECh. 6 - Prob. 40RECh. 6 - Prob. 41RECh. 6 - Prob. 42RECh. 6 - Prob. 43RECh. 6 - Prob. 44RECh. 6 - Prob. 45RECh. 6 - Prob. 46RECh. 6 - Prob. 47RECh. 6 - Prob. 48RECh. 6 - Prob. 49RECh. 6 - Prob. 50RECh. 6 - Prob. 51RECh. 6 - Prob. 52RECh. 6 - Prob. 53RECh. 6 - Prob. 54RECh. 6 - Prob. 55RECh. 6 - Prob. 56RECh. 6 - Prob. 57RECh. 6 - Prob. 58RECh. 6 - Prob. 59RECh. 6 - Prob. 60RECh. 6 - Prob. 61RECh. 6 - Prob. 62RECh. 6 - Prob. 63RECh. 6 - Prob. 64RECh. 6 - Prob. 65RECh. 6 - Prob. 66RECh. 6 - Prob. 67RECh. 6 - Prob. 68RECh. 6 - Prob. 69RECh. 6 - Prob. 70RECh. 6 - Prob. 71RECh. 6 - Prob. 72RECh. 6 - Prob. 73RECh. 6 - Prob. 74RECh. 6 - Prob. 75RECh. 6 - Prob. 76RECh. 6 - Prob. 77RECh. 6 - Prob. 78RECh. 6 - Prob. 79RECh. 6 - Prob. 80RECh. 6 - Prob. 81RECh. 6 - Prob. 82RECh. 6 - Prob. 83RECh. 6 - Prob. 84RECh. 6 - Prob. 85RECh. 6 - Prob. 86RECh. 6 - Prob. 87RECh. 6 - Prob. 88RECh. 6 - Prob. 89RECh. 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 - Prob. 92RECh. 6 - Prob. 93RECh. 6 - Prob. 94RECh. 6 - Evaluate the integral. 95. 01ex1+e2xdxCh. 6 - Prob. 96RECh. 6 - Evaluate the integral. 97. exxdxCh. 6 - Prob. 98RECh. 6 - Prob. 99RECh. 6 - Prob. 100RECh. 6 - Prob. 101RECh. 6 - Prob. 102RECh. 6 - Prob. 103RECh. 6 - Evaluate the integral. 104. sinhauduCh. 6 - Prob. 105RECh. 6 - Prob. 106RECh. 6 - Prob. 107RECh. 6 - Prob. 108RECh. 6 - Prob. 109RECh. 6 - Prob. 110RECh. 6 - Prob. 111RECh. 6 - Prob. 112RECh. 6 - Prob. 113RECh. 6 - Prob. 114RECh. 6 - Prob. 115RECh. 6 - Prob. 116RECh. 6 - What is the area of the largest triangle in the...Ch. 6 - Prob. 118RECh. 6 - Prob. 119RECh. 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 - Prob. 1PCh. 6 - Prob. 2PCh. 6 - Prob. 3PCh. 6 - If 04e(x2)4dx=k, find the value of 04xe(x2)4dx.Ch. 6 - Prob. 5PCh. 6 - Prob. 6PCh. 6 - Prob. 7PCh. 6 - Prob. 8PCh. 6 - Prob. 9PCh. 6 - Prob. 10PCh. 6 - For what value of a is the following equation...Ch. 6 - Prob. 12PCh. 6 - Prob. 13PCh. 6 - Prob. 14PCh. 6 - Prob. 15PCh. 6 - Show that, for all positive value of x and y,...Ch. 6 - Prob. 17PCh. 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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Derivatives of Trigonometric Functions - Product Rule Quotient & Chain Rule - Calculus Tutorial; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=_niP0JaOgHY;License: Standard YouTube License, CC-BY