Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)
Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)
2nd Edition
ISBN: 9781285101552
Author: James Stewart
Publisher: Cenage
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Textbook Question
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Chapter 3.3, Problem 65E

The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is

s ( t ) = 2 e − 1.5 t sin 2 π t

where s is measured in centimeters and t in seconds. Find the velocity after t seconds and graph both the position and velocity functions for 0 ⩽ t ⩽ 2 .

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Chapter 3 Solutions

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

Ch. 3.1 - (a) Write an equation that defines the exponential...Ch. 3.1 - (a) How is the number e defined? (b) What is an...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - Graph the given functions on a common screen. How...Ch. 3.1 - 7-12 Make a rough sketch of the graph of the...Ch. 3.1 - 7-12. Make a rough sketch of the graph of the...Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Make a rough sketch of the graph of the function....
Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Make a rough sketch of the graph of the function....Ch. 3.1 - Starting with the graph of y = ex, write the...Ch. 3.1 - Starting with the graph of y = ex, find the...Ch. 3.1 - Find the domain of each function. 19. (a)...Ch. 3.1 - Find the domain of each function. (a) g(t) =...Ch. 3.1 - 21–22 Find the exponential function f(x) = Cb2...Ch. 3.1 - Find the exponential function f(x) = Cax whose...Ch. 3.1 - Prob. 19ECh. 3.1 - Compare the rates of growth of the functions f(x)...Ch. 3.1 - Compare the functions f(x) = x10 and g(x) = ex by...Ch. 3.1 - Use a graph to estimate the values of x such that...Ch. 3.1 - Find the limit. limx(1.001)xCh. 3.1 - Prob. 24ECh. 3.1 - Find the limit. limxe3xe3xe3x+e3xCh. 3.1 - Find the limit. limx2+10x310xCh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - If you graph the function f(x)=1e1/x1+e1/x you' ll...Ch. 3.1 - Graph several members of the family of functions...Ch. 3.2 - (a) What is a one-to-one function? (b) How can you...Ch. 3.2 - (a) Suppose f is a one-to-one function with domain...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - 3-14 A function is given by a table of values, a...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - A function is given by a table of values, a graph,...Ch. 3.2 - Assume that f is a one-to-one function. (a) If...Ch. 3.2 - 16. If f(x) = x5 + x3 + x, find f‒1(3) and f(f...Ch. 3.2 - 17. If g(x) = 3 + x + ex, find g−1(4). Ch. 3.2 - 18. The graph of f is given. (a) Why is f...Ch. 3.2 - The formula C=59(F32), where F 459.67, expresses...Ch. 3.2 - 20. In the theory of relativity, the mass of a...Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - 21- 26 Find a formula for the inverse of the...Ch. 3.2 - 21- 26 Find a formula for the inverse of the...Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find a formula for the inverse of the function....Ch. 3.2 - Find an explicit formula for f1 and use it to...Ch. 3.2 - Find an explicit formula for f1 and use it to...Ch. 3.2 - Use the given graph of f to sketch the graph of...Ch. 3.2 - Use the given graph of f to sketch the graph of...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 3134 (a) Show that f is one-to-one. (b) Use...Ch. 3.2 - 35-38 Find (f1)(a). 35. f(x) = 2x3 + 3x2 + 7x + 4,...Ch. 3.2 - 35-38 Find(f1)(a). 36. f(x) = x3 + 3 sin x + 2 cos...Ch. 3.2 - 35-38 Find(f1)(a). 37. f(x)=3+x2+tan(x/2),1x1,a=3Ch. 3.2 - 35-38 Find(f1)(a). 38. f(x)=x3+x2+x+1,a=2Ch. 3.2 - Suppose f1 is the inverse function of a...Ch. 3.2 - Suppose f−1 is the inverse function of a...Ch. 3.2 - (a) How is the logarithmic function y = logax...Ch. 3.2 - (a) What is the natural logarithm? (b) What is the...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Find the exact value of each expression (without a...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Use the properties of logarithms to expand the...Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Express the given quantity as a single logarithm....Ch. 3.2 - Use Formula 14 to evaluate each logarithm correct...Ch. 3.2 - Use Formula 14 to graph the given functions on a...Ch. 3.2 - Use Formula 14 to graph the given functions on a...Ch. 3.2 - 45. Suppose that the graph of y = log2 x is drawn...Ch. 3.2 - Compare the functions f(x)=x0.1 and g(x) = ln x by...Ch. 3.2 - Make a rough sketch of the graph of each function....Ch. 3.2 - Make a rough sketch of the graph of each function....Ch. 3.2 - (a) What are the domain and range of f? (b) What...Ch. 3.2 - (a) What are the domain and range of f? (b) What...Ch. 3.2 - Solve each equation for x. 51. (a) e74x=6 (b)...Ch. 3.2 - Solve each equation for x. 52. (a) ln(x2 1) = 3...Ch. 3.2 - Solve each equation for x. 53. (a) 2x5 = 3 (b) ln...Ch. 3.2 - Solve each equation for x. 54. (a) ln(ln x) = 1...Ch. 3.2 - Solve each inequality for x. 55. (a) ln x 0 (b)...Ch. 3.2 - Solve each inequality for x. 56. (a) 1 e3x1 2...Ch. 3.2 - (a) Find the domain of f(x) = ln(ex 3). (b) Find...Ch. 3.2 - (a) What are the values of eln 300 and ln(e300)?...Ch. 3.2 - 71-76 Find the limit. 71. limx3+ln(x29)Ch. 3.2 - 71-76 Find the limit. 72. limx2log5(8xx4)Ch. 3.2 - Prob. 73ECh. 3.2 - 7176 Find the limit. 74. limx0+ln(sinx)Ch. 3.2 - Find the limit. limx[ln(1+x2)ln(1+x)]Ch. 3.2 - Find the limit. limx[ln(2+x)ln(1+x)]Ch. 3.2 - When a camera flash goes off, the batteries...Ch. 3.2 - Let a 1. Prove, using precise definitions, that...Ch. 3.2 - (a) If we shift a curve to the left, what happens...Ch. 3.3 - Differentiate the function. f(x) = log10 (x3 + 1)Ch. 3.3 - Differentiate the function. f(x) = x ln x xCh. 3.3 - Differentiate the function. f(x ) = sin(ln x)Ch. 3.3 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.3 - Differentiate the function. f(x)=ln1xCh. 3.3 - Differentiate the function. y=1lnxCh. 3.3 - Differentiate the function. f(x) = sin x ln(5x)Ch. 3.3 - Differentiate the function. 8. f(x) = log5 (xex)Ch. 3.3 - Differentiate the function. Ch. 3.3 - Differentiate the function. 10. f(u)=u1+lnuCh. 3.3 - Differentiate the function. g(x)=ln(xx21)Ch. 3.3 - Differentiate the function. 12. h(x)=ln(x+x21)Ch. 3.3 - Differentiate the function. G(y)=ln(2y+1)5y2+1Ch. 3.3 - Differentiate the function. 14. g(r) = r2 ln(2r +...Ch. 3.3 - Differentiate the function. F(s) = ln ln sCh. 3.3 - Differentiate the function. 16. y=ln|cos(lnx)|Ch. 3.3 - Differentiate the function. 20. g(x)=xexCh. 3.3 - Differentiate the function. y=xexCh. 3.3 - Differentiate the function. f(x) = (x3 + 2x)exCh. 3.3 - Differentiate the function. H(z)=a2z2a2+z2Ch. 3.3 - Differentiate the function. y = tan[ln(ax + b)]Ch. 3.3 - Differentiate the function. 22. y=ex1exCh. 3.3 - Differentiate the function. y=1+2e3xCh. 3.3 - Differentiate the function. 24. y=e2tcos4tCh. 3.3 - Differentiate the function. 25. y = 5 1/xCh. 3.3 - Differentiate the function. 26. y=101x2Ch. 3.3 - Differentiate the function. 27. F(t) = et sin 2tCh. 3.3 - Differentiate the function. 28. y=eueueu+euCh. 3.3 - Differentiate the function. 29. y=ln|2x5x2|Ch. 3.3 - Differentiate the function. 30. y=1+xe2xCh. 3.3 - Differentiate the function. 31. f(t) = tan (et) +...Ch. 3.3 - Differentiate the function. 32. y=ektanxCh. 3.3 - Differentiate the function. 33. y=ln(ex+xex)Ch. 3.3 - Differentiate the function. 34. y=[ln(1+ex)]2Ch. 3.3 - Differentiate the function. 35. y=2xlog10xCh. 3.3 - Differentiate the function. 36. y = x2 e1/xCh. 3.3 - Differentiate the function. 37. f(t)=sin2(esin2t)Ch. 3.3 - Differentiate the function. 38. y=log2(excosx)Ch. 3.3 - Differentiate the function. 39. g(x) = (2rarx +...Ch. 3.3 - Differentiate the function. 40. y=23x2Ch. 3.3 - Find y and y. 41. y = eax sin xCh. 3.3 - Find y and y. 42. y=lnxx2Ch. 3.3 - Find y and y. 43. y = x ln xCh. 3.3 - Find y and y. 44. y = ln (sec x + tan x)Ch. 3.3 - Differentiate f and find the domain of f....Ch. 3.3 - Differentiate f and find the domain of f. f(x) ln...Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Let f(x) = cx + ln(cos x). For what value of c is...Ch. 3.3 - Let f(x) = loga(3x2 2). For what value of a is...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - 46. Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Use logarithmic differentiation to find the...Ch. 3.3 - Find y if 2x2y=x+y.Ch. 3.3 - Find an equation of the tangent line to the curve...Ch. 3.3 - Find y if y = ln(x2 + y2).Ch. 3.3 - Find y if xy = yx.Ch. 3.3 - The motion of a spring that is subject to a...Ch. 3.3 - Under certain circumstances a rumor spreads...Ch. 3.3 - Show that the function y = Aex + Bxex satisfies...Ch. 3.3 - For what values of r does the function y = erx...Ch. 3.3 - If f(x) = e2x, find a formula for f(n)(x). Ch. 3.3 - Find the thousandth derivative of f(x) = xe–x. Ch. 3.3 - Find a formula for f(n)(x) if f(x) = ln(x 1).Ch. 3.3 - Find d9dx9(x8lnx).Ch. 3.3 - If f(x) = 3 + x + ex, find (f1)(4).Ch. 3.3 - Evaluate . Ch. 3.4 - A population of protozoa develops with a constant...Ch. 3.4 - A common inhabitant of human intestines is the...Ch. 3.4 - A bacteria culture initially contains 100 cells...Ch. 3.4 - A bacteria culture grows with constant relative...Ch. 3.4 - The table gives estimates of the world population,...Ch. 3.4 - The table gives the population of India, in...Ch. 3.4 - Experiments show that if the chemical reaction...Ch. 3.4 - Strontium-90 has a half-life of 28 days. (a) A...Ch. 3.4 - The half-life of cesium-137 is 30 years. Suppose...Ch. 3.4 - A sample oflritium-3 decayed to 94.5% of its...Ch. 3.4 - 11. Scientists can determine the age of ancient...Ch. 3.4 - A curve passes through the point (0, 5) and has...Ch. 3.4 - 15. A roast turkey is taken from an oven when its...Ch. 3.4 - In a murder investigation, the temperature of the...Ch. 3.4 - When a cold drink is taken from a refrigerator,...Ch. 3.4 - 18. A freshly brewed cup of coffee has temperature...Ch. 3.4 - The rate of change of atmospheric pressure P with...Ch. 3.4 - (a) If 1000 is borrowed at 8% interest, find the...Ch. 3.4 - If 3000 is invested at 5% interest, find the value...Ch. 3.4 - (a) How long will it take an investment to double...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Find the exact value of each expression. (a)...Ch. 3.5 - Prove that cos(sin1x)=1x2.Ch. 3.5 - Simplify the expression. tan(sin1x)Ch. 3.5 - Simplify the expression. sin(tan1x)Ch. 3.5 - Simplify the expression. cos(2 tan1x)Ch. 3.5 - Prove Formula 6 for the derivative of cos1 by the...Ch. 3.5 - (a) Prove that sin1x + cos1x = /2. (b) Use part...Ch. 3.5 - Prove that ddx(cot1x)=11+x2.Ch. 3.5 - Prove that ddx(sec1x)=1xx21.Ch. 3.5 - Prove that ddx(csc1x)=1xx21.Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - 1629 Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - Find the derivative of the function. Simplify...Ch. 3.5 - 3031 Find the derivative of the function. Find the...Ch. 3.5 - Find the derivative of the function. Find the...Ch. 3.5 - Find y if tan1(xy) = 1 + x2y.Ch. 3.5 - If g(x)=xsin1(x/4)+16x2, find g(2).Ch. 3.5 - Find an equation of the tangent line to the curve...Ch. 3.5 - Prob. 35ECh. 3.5 - Find the limit. limxarccos(1+x21+2x2)Ch. 3.5 - Find the limit. limxarctan(ex)Ch. 3.5 - Prob. 38ECh. 3.5 - A ladder 10 ft long leans against a vertical wall....Ch. 3.5 - A lighthouse is located on a small island, 3 km...Ch. 3.5 - Some authors define y = sec1x sec y = x and y ...Ch. 3.5 - (a) Sketch the graph of the function f(x) =...Ch. 3.6 - Find the numerical value of each expression. 1....Ch. 3.6 - Find the numerical value of each expression. (a)...Ch. 3.6 - 1-6 Find the numerical value of each...Ch. 3.6 - Find the numerical value of each expression. 4....Ch. 3.6 - Find the numerical value of each expression. 5....Ch. 3.6 - Find the numerical value of each expression. 6....Ch. 3.6 - Prove the identity. 7. sinh(x) = sinh x (This...Ch. 3.6 - Prove the identity. 8. cosh(x) = cosh x (This...Ch. 3.6 - Prove the identity. 9. cosh x + sinh x = exCh. 3.6 - Prove the identity. 10. cosh x sinh r = exCh. 3.6 - Prove the identity. 11. sinh(x + y) = sinh x cosh...Ch. 3.6 - Prove the identity. 12. cosh(x + y) = cosh x cosh...Ch. 3.6 - Prove the identity. 15. sinh 2x = 2 sinh x cosh xCh. 3.6 - Prove the identity. 18. 1+tanhx1tanhx=e2xCh. 3.6 - Prove the identity. 19. (cosh x + sinh x)n = cosh...Ch. 3.6 - If x=1213 find the values of the other hyperbolic...Ch. 3.6 - If cosh=53 and x 0. find the values of the other...Ch. 3.6 - (a) Use the graphs of sinh, cosh, and tanh in...Ch. 3.6 - Use the definitions of the hyperbolic functions to...Ch. 3.6 - Prove the formulas given in Table 1 for the...Ch. 3.6 - Give an alternative solution 10 Example 3 by...Ch. 3.6 - Prove Equation 4.Ch. 3.6 - Prove Formula 5 using (a) the method of Example 3...Ch. 3.6 - For each of I he following functions (i) give a...Ch. 3.6 - Prove the formulas given in Table 6 for the...Ch. 3.6 - Find the derivative. Simplify where possible. f(x)...Ch. 3.6 - Find the derivative. Simplify where possible. f(x)...Ch. 3.6 - Find the derivative. Simplify where possible. g(x)...Ch. 3.6 - Find the derivative. Simplify where possible. h(x)...Ch. 3.6 - Find the derivative. Simplify where possible. f(t)...Ch. 3.6 - Find the derivative. Simplify where possible. f(t)...Ch. 3.6 - Find the derivative. Simplify where possible. y =...Ch. 3.6 - Find the derivative. Simplify where possible. 37....Ch. 3.6 - Find the derivative. Simplify where possible....Ch. 3.6 - 26-41 Find the derivative. Simplify where...Ch. 3.6 - Find the derivative. Simplify where possible. 40....Ch. 3.6 - 30-45 Find the derivative. Simplify where...Ch. 3.6 - Find the derivative. Simplify where possible. 42....Ch. 3.6 - Find the derivative. Simplify where possible. 43....Ch. 3.6 - Find the derivative. Simplify where possible. 44....Ch. 3.6 - Find the derivative. Simplify where possible. 45....Ch. 3.6 - Show that ddx1+tanhx1tanhx4=12ex/2.Ch. 3.6 - Show that ddx arctan(tanh x) = sech 2x.Ch. 3.6 - The Gateway Arch in St. Louis was designed by Eero...Ch. 3.6 - If a water wave with length L. moves with velocity...Ch. 3.6 - A flexible cable always hangs in the shape of a...Ch. 3.6 - Prob. 47ECh. 3.6 - Using principles from physics it can be shown that...Ch. 3.6 - A cable with linear density = 2 kg/m is strung...Ch. 3.6 - Evaluate limxsinhxex.Ch. 3.6 - (a) Show that any function of the form y = A sinh...Ch. 3.6 - If x = ln( sec + tan ), show that sec = cosh x.Ch. 3.6 - 57. At what point of the curve y = cosh x does the...Ch. 3.6 - Show that if a 0 and b 0, then there exist...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 5ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 7ECh. 3.7 - Prob. 8ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 12ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - 33. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - 34. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - Prob. 17ECh. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - 49. Find the limit. Use l’Hospital’s Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 37ECh. 3.7 - 1-38 Find the limit. Use lHospitals Rule where...Ch. 3.7 - Prob. 39ECh. 3.7 - Prob. 40ECh. 3.7 - Prob. 41ECh. 3.7 - Prob. 42ECh. 3.7 - Prob. 43ECh. 3.7 - Prob. 44ECh. 3.7 - If an object with mass m is dropped from rest, one...Ch. 3.7 - If an initial amount A0 of money is invested at an...Ch. 3.7 - If an electrostatic field E acts on a liquid or a...Ch. 3.7 - 82. A metal cable has radius r and is covered by...Ch. 3.7 - Prob. 49ECh. 3.7 - The figure shows a sector of a circle with central...Ch. 3.7 - Evaluate limx[xx2ln(1+xx)].Ch. 3.7 - 86. Suppose f is a positive function. If and ,...Ch. 3.7 - If f is continuous, f(2) = 0, and f(2) = 7,...Ch. 3.7 - For what values of a and b is the following...Ch. 3.7 - If f is continuous, use lHospitals Rule to show...Ch. 3.7 - If f is continuous, show that...Ch. 3.7 - Let f(x)={e1/x2ifx00ifx=0 (a) Use the definition...Ch. 3.7 - Let f(x)={xxifx01ifx=0 (a) Show that f is...Ch. 3 - Prob. 1RCCCh. 3 - (a) How is the inverse sine function f(x) = sin1 x...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - The graph of g is given. (a) Why is g one-to-one?...Ch. 3 - 1112 Find the exact value of each expression. 11....Ch. 3 - 1112 Find the exact value of each expression. 12....Ch. 3 - 1316 Solve the equation for x. 13. (a) ex = 5 (b)...Ch. 3 - 1316 Solve the equation for x. 14. (a) eex=2 (b)...Ch. 3 - 1316 Solve the equation for x. 15. (a) ln(x + 1) +...Ch. 3 - 1316 Solve the equation for x. 16. (a) ln(1 + ex)...Ch. 3 - (a) Express e as a limit. (b) What is the value of...Ch. 3 - (a) What are the domain and range of the natural...Ch. 3 - (a) Write a differential equation that expresses...Ch. 3 - State the derivative of each function. (a) y = ex...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - 1-50 Calculate y'. 8. xey = y sin x Ch. 3 - Calculate y'. 9. y = ln(x ln x)Ch. 3 - Calculate y'. 10. y = emx' cos nxCh. 3 - Calculate y'. 12. y = (arcsin 2x)2Ch. 3 - Calculate y'. 13. y=e1/xx2Ch. 3 - Calculate y'. 14. y = ln sec xCh. 3 - Write the definitions of the hyperbolic functions...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - The graph of f is shown. Is f one-to-one? Explain.Ch. 3 - Suppose f is one-to-one, f(7) = 3, and f(7) = 8....Ch. 3 - Find the inverse function of f(x)=x+12x+1.Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - 59 Sketch a rough graph of the function without...Ch. 3 - Let a 1. For large values of x, which of the...Ch. 3 - 1743 Differentiate. 22. y = x cos1xCh. 3 - 1743 Differentiate. 23. f(t) = t2 ln tCh. 3 - 1743 Differentiate. 24. g(t)=et1+etCh. 3 - 1743 Differentiate. 29. h() = etan 2Ch. 3 - 1743 Differentiate. 36. y = sin1(ex)Ch. 3 - Show that ddx(12tan1x+14ln(x+1)2x2+1)=1(1+x)(1+x2)Ch. 3 - 4548 Find f in terms of g. 45. f(x)=eg(x)Ch. 3 - 4548 Find f in terms of g. 46. f(x) = g(ex)Ch. 3 - 4548 Find f in terms of g. 47. f(x) = ln |g(x)|Ch. 3 - 4548 Find f in terms of g. 48. f(x) = g(ln x)Ch. 3 - 4950 Find f(n)(x). 49. f(x) = 2xCh. 3 - 4950 Find f(n)(x). 50. f(x) = ln(2x)Ch. 3 - Find an equation of the tangent to the curve y = x...Ch. 3 - A bacteria culture contains 200 cells initially...Ch. 3 - Cobalt-60 has a half-life of 5.24 years. (a) Find...Ch. 3 - Let C(t) be the concentration of a drug in the...Ch. 3 - A cup of hot chocolate has temperature 80C in a...Ch. 3 - 6176 Evaluate the limit. 61. limx0+tan1(1/x)Ch. 3 - 6176 Evaluate the limit. 62. limxexx2Ch. 3 - 6176 Evaluate the limit. 63. limx3e2/(x3)Ch. 3 - 6176 Evaluate the limit. 64. limxarctan(x3x)Ch. 3 - 6176 Evaluate the limit. 65. limx0+ln(sinhx)Ch. 3 - Prob. 66RECh. 3 - 6176 Evaluate the limit. 67. limx1+2x12xCh. 3 - 6176 Evaluate the limit. 68. limx(1+4x)xCh. 3 - 6176 Evaluate the limit. 69. limx0ex1tanxCh. 3 - 6176 Evaluate the limit. 70. limx0tan4xx+sin2xCh. 3 - Prob. 71RECh. 3 - Prob. 72RECh. 3 - 6176 Evaluate the limit. 73. limx(x2x3)e2xCh. 3 - 6176 Evaluate the limit. 74. limx(x)cscxCh. 3 - Prob. 75RECh. 3 - 6176 Evaluate the limit. 76. limx(/2)(tanx)cosxCh. 3 - If f(x) = ln x + tan1x, find (f1)(/4).Ch. 3 - Show that cosarctan[sin(arccotx)]=x2+1x2+2Ch. 3 - Calculate y'. 17. y=arctanCh. 3 - Calculate y'. 21. y = 3x ln xCh. 3 - Calculate y'. 27. y = log5(1 + 2x)Ch. 3 - Calculate y'. 28. y = (cos x)xCh. 3 - Calculate y'. 29. y=lnsinx12sin2xCh. 3 - Calculate y'. 30. y=(x2+1)4(2x+1)3(3x1)5Ch. 3 - Calculate y'. 31. y = x tan1(4x)Ch. 3 - Calculate y'. 32. y = ecos x + cos(ex)Ch. 3 - Calculate y'. 34. y = 10tanCh. 3 - Calculate y'. 38. y=arctan(arcsinx)Ch. 3 - Calculate y'. 41. y=x+1(2x)5(x+3)7Ch. 3 - Calculate y'. 43. y = x sinh(x2)Ch. 3 - Calculate y'. 45. y = ln( cosh 3x)Ch. 3 - 1-50 Calculate y'. 47. y = cosh–1(sinh x) Ch. 3 - Calculate y'. 48. y=xtanh1xCh. 3 - Calculate y'. 49. y=cos(etan3x)Ch. 3 - Use mathematical induction (page 72) to show that...Ch. 3 - If f(x) = xesin x find f(x). Graph f and f on the...Ch. 3 - At what point on the curve y = [ln(x + 4)]2 is the...Ch. 3 - (a) Find an equation of the tangent to the curve y...Ch. 3 - The function C(t) = K(eat ebt), where a, b, and K...Ch. 3 - (a) What does lHospitals Rule say? (b) How can you...
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY