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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
107E108E109E110E111E112E113E114E115EThe equation x y2 = 0 implicitly defines what two functions?Use implicit differentiation to find dydx for x y2 = 3.If a function is defined explicitly in the form y = f(x), which coordinates are needed to find the slope of a tangent linethe x-coordinate, the y-coordinate, or both?For some equations, such as x2 + y2 = l or x y2 = 0, it possible to solve for y and then calculate dydx. Even in these cases, explain why implicit differentiation is usually a more efficient method for calculating the derivative.Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly.Why are both the x-coordinate and the y-coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function?Identify and correct the error in the following argument. Suppose y2 + 2y = 2x3 7. Differentiating both sides with respect to x to find dydx, we have 2y+2dydx=6x2, which implies that dydx=3x2y.Calculate dydx using implicit differentiation. 5x=y2Calculate dydx using implicit differentiation. 6. 3x+4y3=7Calculate dydx using implicit differentiation. 7. sin y + 2 = xCalculate dydx using implicit differentiation. 8.eyex=C, where C is constantConsider the curve defined by 2x y + y3 = 0 (see figure). a.Find the coordinates of the y-intercepts of the curve. b.Verify that dydx=213y2. c.Find the slope of the curve at each point where x = 0.Find the slope of the curve x2 + y3 = 2 at each point where y = 1 (see figure).Consider the curve x=y3. Use implicit differentiation to verify that dydx=13y2 and then find d2ydx2.Consider the curve x=ey. Use implicit differentiation to verify that dydx=ey and then find d2ydx2.Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 5. x4 + y4 = 2; (1, 1)Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 6. x = ey; (2, ln 2)Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 7. y2 = 4x; (1, 2)Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 8. y2 + 3x = 8; (1,5)Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 9. sin y = 5x4 5; (1, ).Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 10. x2y=0;(4,1);Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 11. cosy=x;(0,2)Implicit differentiation Carry out the following steps. a. Use implicit differentiation to find dydx. b. Find the slope of the curve at the given point. 12. tan xy = x + y; (0, 0)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 21.xy=7;(1,7)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 22.xy2+1=1;(10,3)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 23.x3+y43=2;(1,1)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 24.x2/3+y2/3=2;(1,1)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 25.xy3+y=10;(1,8)Implicit differentiation Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 26.(x+y)2/3=y;(4,4)Implicit differentiation Use implicit differentiation to find dydx. 27.sinx+siny=yImplicit differentiation Use implicit differentiation to find dydx. 28.y=xeyImplicit differentiation Use implicit differentiation to find dydx. 15. x + y = cos yImplicit differentiation Use implicit differentiation to find dydx. 16. x+2y=yImplicit differentiation Use implicit differentiation to find dydx. 13. sin xy = x + yImplicit differentiation Use implicit differentiation to find dydx. 14. exy = 2yImplicit differentiation Use implicit differentiation to find dydx. 17. cos y2 + x = eyImplicit differentiation Use implicit differentiation to find dydx. 18. y=x+1y1Implicit differentiation Use implicit differentiation to find dydx. 19. x3=x+yxyImplicit differentiation Use implicit differentiation to find dydx. 20. (xy + 1)3 = x y2 + 8Implicit differentiation Use implicit differentiation to find dydx. 21. 6x3 + 7y3 = 13xyImplicit differentiation Use implicit differentiation to find dydx. 22. sin x cos y = sin x + cos yImplicit differentiation Use implicit differentiation to find dydx. 23. x4+y2=5x+2y3Implicit differentiation Use implicit differentiation to find dydx. 24. x+y2=sinyCobb-Douglas production function The output of an economic system Q, subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function Q = cLaKb. When a + b = 1, the case is called constant returns to scale. Suppose Q = 1280, a=13, b=23, and c = 40. a. Find the rate of change of capital with respect to labor, dK/dL. b. Evaluate the derivative in part (a) with L = 8 and K = 64.Surface area of a cone The lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A=rr2+h2. a. Find dr/dh for a cone with a lateral surface area of A = 1500. b. Evaluate this derivative when r = 30 and h = 40.Volume of a spherical cap Imagine slicing through a sphere with a plane (sheet of paper). The smaller piece produced is called a spherical cap. Its volume is V = h2(3r h)/3, where r is the radius of the sphere and h is the thickness of the cap. a. Find dr/dh for a sphere with a volume of 5/3. b. Evaluate this derivative when r = 2 and h = 1.Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V = 2(b + a)(b a)2/4. a. Find db/da for a torus with a volume of 642. b. Evaluate this derivative when a = 6 and b = 10.Tangent lines Carry out the following steps. a.Verify that the given point lies on the curve. b.Determine an equation of the line tangent to the curve at the given point. 45.sin y + 5x = y2; (0, 0)Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. 28. x3 + y 3 = 2xy; (1, 1)Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. 25. x2 + xy + y2 = 7; (2, 1)Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. 26. x4 x2y + y4 = 1; (1, 1)Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. 29. cos(xy)+siny=2;(2,4)Tangent lines Carry out the following steps. a. Verify that the given point lies on the curve. b. Determine an equation of the line tangent to the curve at the given point. 30. (x2+y2)2=254xy2;(1,2)Second derivatives Find d2ydx2. 31. x + y2 = 1Second derivatives Find d2ydx2. 32. 2x2 + y2 = 4Second derivatives Find d2ydx2. 33. x + y = sin ySecond derivatives Find d2ydx2. 34. x4 + y4 = 64Second derivatives Find d2ydx2. 35. e2y + x = ySecond derivatives Find d2ydx2 36. sin x + x2y = 10Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. For any equation containing the variables x and y, the derivative dy/dx can be found by first using algebra to rewrite the equation in the form y = f(x). b. For the equation of a circle of radius r, x2 + y2 = r2, we have dydx=xy, for y 0 and any real number r 0. c. If x = 1, then by implicit differentiation, 1 = 0. d. If xy = l, then y' = 1/x.Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 58.xy5/2+x3/2y=12;(4,1)Carry out the following steps. a.Use implicit differentiation to find dydx. b.Find the slope of the curve at the given point. 59.xy+x3/2y1/2=2;(1,1)Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. 52. x + y3 y = 1; x = 1Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. 53. x + y2 y = 1; x = 1Multiple tangent lines Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of x. b. Graph the tangent lines on the given graph. 54. 4x3 = y2(4 x); x = 2 (cissoid of Diocles)Witch of Agnesi Let y(x2 + 4) = 8 (see figure). a. Use implicit differentiation to find dydx. b. Find equations of all lines tangent to the curve y(x2 + 4) = 8 when y = 1. c. Solve the equation y(x2 + 4) = 8 for y to find an explicit expression for y and then calculate dydx. d. Verify that the results of parts (a) and (c) are consistent.Vertical tangent lines a. Determine the points at which the curve x + y3 y = 1 has a vertical tangent line (see Exercise 52). b. Does the curve have any horizontal tangent lines? Explain.Vertical tangent lines a. Determine the points where the curve x + y2 y = 1 has a vertical tangent line (see Exercise 53). b. Does the curve have any horizontal tangent lines? Explain.Tangent lines for ellipses Find the equations of the vertical and horizontal tangent lines of the following ellipses. 58. x2 + 4y2 + 2xy = 12Tangent lines for ellipses Find the equations of the vertical and horizontal tangent lines of the following ellipses. 59. 9x2 + y2 36x + 6y + 36 = 068EIdentifying functions from an equation The following equations implicitly define one or more functions. a. Find dydx using implicit differentiation. b. Solve the given equation for y to identify the implicitly defined functions y = f1(x), y = f2(x), . c. Use the functions found in part (b) to graph the given equation. 61. x + y3 xy = 1 (Hint: Rewrite as y3 l = xy x and then factor both sides.)70E71E72ENormal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. 73. Exercise 45 45.siny+5x=y2;(0,0)Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. 68. Exercise 28 28. x3 + y3 = 2xy; (1, 1)Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. 65. Exercise 25 25. x2 + xy + y2 = 7; (2, 1)Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. 66. Exercise 26 26. x4 x2y + y4 = 1; (1, 1)77ENormal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. 70. Exercise 30 30. (x2+y2)2=254xy2;(1,2)79EVisualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 65-70.) b. Graph the tangent and normal lines on the given graph 72. x4 = 2x2 + 2y2; (x0, y0) = (2, 2) (kampyle of Eudoxus)Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 65-70.) b. Graph the tangent and normal lines on the given graph 73. (x2 = y2 2x)2 = 2(x2 + y2); (x0, y0) = (2, 2) (limaon of Pascal)82EOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx2 form orthogonal trajectories with the family of ellipses x2 + 2y2 = k, where c and k are constants (see figure). Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. 79. y = mx; x2 + y2 = a2, where m and a are constantsOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx2 form orthogonal trajectories with the family of ellipses x2 + 2y2 = k, where c and k are constants (see figure). Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. 80. y = cx2; x2 + 2y2 = k, where c and k are constantsOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx2 form orthogonal trajectories with the family of ellipses x2 + 2y2 = k, where c and k are constants (see figure). Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. 81. xy = a; x2 y2 = b, where a and b are constantsFinding slope Find the slope of the curve 5x10y=sinx at the point (4 , ).A challenging derivative Find dydx, where (x2 + y2)(x2 + y2 + x) = 8xy2.88EA challenging derivative Find d2ydx2, where y+xy=1.Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. 86. y2 3xy = 2Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. 87. x2(3y2 2y3) = 4Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. 88. x2(y 2) ey = 0Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. 89. x(1 y2) + y3 = 0Quick Check 1
Simplify e2 ln x. Express 5x using the base e.
Find ddx(lnxp), where x 0 and p is a real number in two ways: (1) using the Chain Rule and (2) by first using a property of logarithms.3QC4QCShow that the derivative computed in Example 7b can be expressed in base 2 as T'(n) = log2 (en).Use x = ey to explain why ddx(lnx)=1x, for x 0.2EShow that ddx(lnkx)=ddx(lnx), where x 0 and k is a positive real number.State the derivative rule for the exponential function f(x) = bx. How does it differ from the derivative formula for ex?State the derivative rule for the logarithmic function f(x) = logb x . How does it differ from the derivative formula for ln x?Explain why bx = ex ln bSimplify the expression exln(x2+1).8EFind ddx(lnx2+1).Evaluate ddx(xe+ex)Express the function f(x)=f(x)h(x) in terms of the natural logarithm and natural exponential functions (base e)12EFind ddx(ln(xex)) without using the Chain Rule and the Product Rule.Find ddx(lnx101) without using the Chain Rule.Derivatives involving ln x Find the following derivatives. 9.ddx(ln7x)Derivatives involving ln x Find the following derivatives. 10.ddx(x2lnx)Derivatives involving ln x Find the following derivatives. 11.ddx(lnx2)Derivatives involving ln x Find the following derivatives. 12. ddx(ln2x8)Derivatives involving ln x Find the following derivatives. 13. ddx(lnsinx)Derivatives Find the derivative of the following functions. 20.y=1+lnx34x3Derivatives Find the derivative of the following functions. 21.y=ln(x4+1)Derivatives Find the derivative of the following functions. 22.y=lnx4+x2Derivatives involving ln x Find the following derivatives. 15. ddx(ln(x+1x1))Derivatives Find the derivative of the following functions. 24.y=xlnxxDerivatives involving ln x Find the following derivatives. 17. ddx((x2+1)lnx)Derivatives involving ln x Find the following derivatives. 18. ddx(lnx21)Derivatives Find the derivative of the following functions. 27.y=x2(1lnx2)Derivatives Find the derivative of the following functions. 28.y=3x2lnxx3Derivatives involving ln x Find the following derivatives. 19. ddx(ln(lnx))Derivatives Find the derivative of the following functions. 30.y=ln(cos2x)Derivatives involving ln x Find the following derivatives. 21. ddx(lnxlnx+1)Derivatives involving ln x Find the following derivatives. 22. ddx(ln(ex+ex))Derivatives Find the derivative of the following functions. 33.y=xeDerivatives Find the derivative of the following functions. 34.y=exxeGeneral Power Rule Use the General Power Rule where appropriate to find the derivative of the following functions. 44. f(x) = (2x + 1)General Power Rule Use the General Power Rule where appropriate to find the derivative of the following functions. 40. y = ln (x3 + 1)Derivatives of bx Find the derivatives of the following functions. 23. y = 8xDerivatives of bx Find the derivatives of the following functions. 24. y = 53tDerivatives of bx Find the derivatives of the following functions. 25. y = 5 4xDerivatives of bx Find the derivatives of the following functions. 26. y = 4x sin xDerivatives Find the derivative of the following functions. 41.y=23+sinxDerivatives Find the derivative of the following functions. 42.y=10ln2xDerivatives of bx Find the derivatives of the following functions. 27. y = x3 3xDerivatives of bx Find the derivatives of the following functions. 28. P=401+2tDerivatives of bx Find the derivatives of the following functions. 29. A = 250(1.045)4rDerivatives Find the derivative of the following functions. 46.y=10x(ln10x1)General Power Rule Use the General Power Rule where appropriate to find the derivative of the following functions. 43. f(x)=2x2x+1General Power Rule Use the General Power Rule where appropriate to find the derivative of the following functions. 38. s(t) = cos 2tDerivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 45. f(x) = xcos x; a = /2Derivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 46. g(x) = xln x; a = eDerivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 47. h(x)=xx;a=4Derivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 48. f(x) = (x2 + 1)x; a = 1Derivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 49. f(x) = (sin x)ln x; a = /2Derivatives of Tower Functions (or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 50. f(x) = (tan x)x 1; a = /4Derivatives of lower function(or gh) Find the derivative of each function and evaluate the derivative at the given value of a. 55.f(x)=(4sinx2)cosx;a=Magnitude of an earthquake The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.) a. Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve. b. Compute dE/dM and evaluate it for M = 3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)Exponential model The following table shows the time of useful consciousness at various altitudes in the situation where a pressurized airplane suddenly loses pressure. The change in pressure drastically reduces available oxygen, and hypoxia sets in. The upper value of each time interval is roughly modeled by T = 10 20.274a, where T measures time in minutes and a is the altitude over 22,000 in thousands of feet (a = 0 corresponds to 22,000 ft). Altitude (in ft) Time of Useful Consciousness 22,000 5 to 10 min 25,000 3 to 5 min 28,000 2.5 to 3 min 30,000 1 to 2 min 35,000 30 to 60 s 40,000 15 to 20 s 45,000 9 to 15 s a. A Learjet flying at 38,000 ft (a = 16) suddenly loses pressure when the seal on a window fails. According to this model, how long do the pilot and passengers have to deploy oxygen masks before they become incapacitated? b. What is the average rate of change of T with respect to a over the interval from 24,000 to 30,000 ft (include units)? c. Find the instantaneous rate of change dT/da, compute it at 30,000 ft. and interpret its meaning.Diagnostic scanning Iodine-123 is a radioactive isotope used in medicine to test the function of the thyroid gland. If a 350-microcurie (Ci) dose of iodine-123 is administered to a patient, the quantity Q left in the body after t hours is approximately Q=350(12)t/13.1. a. How long does it take for the level of iodine-123 to drop to Ci? b. Find the rate of change of the quantity of iodine-123 at 12 hr, 1 day, and 2 days. What do your answers say about the rate at which iodine decreases as time increases?Find an equation of the line tangent to y = xsin x at the point x = 1.Determine whether the graph of y=xx has any horizontal tangent lines.The graph of y = (x2)x has two horizontal tangent lines. Find equations for both of them.The graph of y = xln x has one horizontal tangent line. Find an equation for it.Derivatives of logarithmic functions Calculate the derivative of the following functions. 55. y = 4 log3(x2 1)Derivatives of logarithmic functions Calculate the derivative of the following functions. 56. y = log10xDerivatives of logarithmic functions Calculate the derivative of the following functions. 57. y = (cos x) ln (cos2 x)Derivatives of logarithmic functions Calculate the derivative of the following functions. 58. y = logg |tan x|Derivatives of logarithmic functions Calculate the derivative of the following functions. 59. y=1log4xDerivatives of logarithmic functions Calculate the derivative of the following functions. 60. y = log2 (log2 x)Derivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 77. f(x) = ln(3x + 1)4Derivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 78. f(x)=ln2x(x2+1)3Derivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 79. f(x)=ln10xDerivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 80. f(x)=log28x+1Derivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 81. f(x)=ln(2x1)(x+2)3(14x)2Derivatives of logarithmic functions Use the properties of logarithms to simplify the following functions before computing f(x). 82. f(x) = ln (sec4 x tan2 x)General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. 85. ddx(x10x)General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. 86. ddx(2x)2xLogarithmic differentiation Use logarithmic differentiation to evaluate f(x). 61. f(x)=(x+1)10(2x4)8Logarithmic differentiation Use logarithmic differentiation to evaluate f(x). 62. f(x) = x2 cos xLogarithmic differentiation Use logarithmic differentiation to evaluate f(x). 63. f(x) = xln xLogarithmic differentiation Use logarithmic differentiation to evaluate f(x). 64. f(x)=tan10x(5x+3)6Logarithmic differentiation Use logarithmic differentiation to evaluate f(x). 65.f(x)=(x+1)3/2(x4)5/2(5x+3)2/3Logarithmic differentiation Use logarithmic differentiation to evaluate f(x). 66. f(x)=x8cos3xx1Logarithmic differentiation Use logarithmic differentiation to evaluate f(x). 67. f(x) = (sin x)tan x 6168. Logarithmic differentiation Use logarithmic differentiation to evaluate f(x).General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. 90. ddx(1+x2)sinxLogarithmic differentiation Use logarithmic differentiation to evaluate f(x). 85.f(x)(1+1x)xLogarithmic differentiation Use logarithmic differentiation to evaluate f(x). 86.f(x)x(x20)Explain why or why not Determine whether the following statements are true and give an explanation or counterexample a.The derivative of log29 is 1/(9 ln 2). b.In (x + 1) + ln (x 1) = ln (x2 1), for all x c.The exponential function 2x + 1 can be written in base e as e2 ln(x +1) d.ddx(2x)=x2x1 e.ddx(x2)=2x21 f.(4x1)lnx=xln(4x+1)Higher-order derivatives Find the following higher-order derivatives
88.
Higher-order derivatives Find the following higher-order derivatives. 71. d2dx2(log10x)Higher-order derivatives Find the following higher-order derivatives. 72. dndxn(2x)Higher-order derivatives Find the following higher-order derivatives. 73. d3dx3(x2lnx)Derivatives by different methods Calculate the derivative of the following functions (i) using the fact that bx = ex ln b and (ii) by using logarithmic differentiation. Verify that both answers are the same. 74. y = (x2 + l)xDerivatives by different methods Calculate the derivative of the following functions (i) using the fact that bx = ex ln b and (ii) by using logarithmic differentiation. Verify that both answers are the same. 75. y = 3xDerivatives by different methods Calculate the derivative of the following functions (i) using the fact that bx = ex ln b and (ii) using logarithmic differentiation verify that both answers are the same 94.y=(4x+1)lnxTangent lines Find the equation of the line tangent to y = 2sin x at x = /2. Graph the function and the tangent line.Horizontal tangents The graph of y = cos x ln cos2 x has seven horizontal tangent lines on the interval [0, 2]. Find the approximate x-coordinates of all points at which these tangent lines occur.Logistic growth Scientists often use the logistic growth function P(t)=P0KP0+(KP0)er0tto model population growth, where P0 is the initial population at time t = 0, K is the carrying capacity, and r0 is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. 93. Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t)=400,00050+7950e0.5t, where t is measured in years. a. Graph P using a graphing utility. Experiment with different windows until you produce an S-shaped curve characteristic of the logistic model. What window works well for this function? b. How long does it take the population to reach 5000 fish? How long does it take the population to reach 90% of the carrying capacity? c. How fast (in fish per year) is the population growing at t = 0? At t = 5? d. Graph P and use the graph to estimate the year in which the population is growing fastest.Logistic growth Scientists often use the logistic growth function P(t)=P0KP0+(KP0)er0tto model population growth, where P0 is the initial population at time t = 0, K is the carrying capacity, and r0 is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. 94. World population (part 1) The population of the world reached 6 billion in 1999 (t = 0). Assume Earths carrying capacity is 15 billion and the base growth rate is r0 = 0.025 per year. a. Write a logistic growth function for the worlds population (in billions) and graph your equation on the interval 0 t 200 using a graphing utility. b. What will the population be in the year 2020? When will it reach 12 billion?World population (part 2) The relative growth rate r of a function f measures the rate of change of the function compared to its value at a particular point. It is computed as r(t) = f(t)/f(t). a.Confirm that the relative growth rate in 1999 (t = 0) for the logistic model in Exercise 98 is r(0) = P(0)/P(0) = 0.015. This means the worlds population was growing at 1.5% per year in 1999. b.Compute the relative growth rate of the worlds population in 2010 and 2020. What appears to be happening to the relative growth rate as time increases? c.Evaluate limt(t)=limtP(t)P(t) where P(t) is the logistic growth function from Exercise 98. What does your answer say about populations that follow a logistic growth pattern?Logistic growth Scientists often use the logistic growth function P(t)=P0KP0+(KP0)er0tto model population growth, where P0 is the initial population at time t = 0, K is the carrying capacity, and r0 is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. 96. Population crash The logistic model can be used for situations in which the initial population P0 is above the carrying capacity K. For example, consider a deer population of 1500 on an island where a fire has reduced the carrying capacity to 1000 deer. a. Assuming a base growth rate of r0 = 0.1 and an initial population of P(0) = 1500, write a logistic growth function for the deer population and graph it. Based on the graph, what happens to the deer population in the long run? b. How fast (in deer per year) is the population declining immediately after the fire at t = 0? c. How long does it take the deer population to decline to 1200 deer?Savings plan Beginning at age 30, a self-employed plumber saves 250 per month in a retirement account until he reaches age 65. The account offers 6% interest, compounded monthly. The balance in the account after t years is given by A(t) = 50,000(1.00512t 1). a. Compute the balance in the account after 5, 15, 25, and 35 years. What is the average rate of change in the value of the account over the intervals [5, 15], [15, 25], and [25, 35]? b. Suppose the plumber started saving at age 25 instead of age 30. Find the balance at age 65 (after 40 years of investing). c. Use the derivative dA/dt to explain the surprising result in part (b) and to explain this advice: Start saving for retirement as early as possible.Tangency question It is easily verified that the graphs of y = x2 and y = ex have no points of intersection (for x 0), and the graphs of y = x3 and y = ex have two points of intersection. It follows that for some real number 2 p 3, the graphs of y = xp and y = ex have exactly one point of intersection (for x 0). Using analytical and/or graphical methods, determine p and the coordinates of the single point of intersection.Tangency question It is easily verified that the graphs y=1.1x and y = x have two points of intersection, and the graphs of y = 2x and y = z have no point of intersection. It follows that for some real number 1.1 p 2. the graphs of y = px and y = x have exactly one point of intersection. Using analytical and/or graphical methods determine p and the coordinates of the single point of intersection.Triple intersection Graph the functions f(x) = x3, g(x) = 3x, and h(x) = xx and find their common intersection point (exactly).Calculating limits exactly Use the definition of the derivative to evaluate the following limits. 101. limxelnx1xeCalculating limits exactly Use the definition of the derivative to evaluate the following limits. 102. limh0ln(e8+h)8hCalculating limits exactly Use the definition of the derivative to evaluate the following limits. 103. limh0(3+h)3+h27hCalculating limits exactly Use the definition of the derivative to evaluate the following limits. 104. limx25x25x2Derivative of u(x)v(x) Use logarithmic differentiation to prove that ddx(u(x)v(x))=u(x)v(x)(dvdxlnu(x)+v(x)u(x)dudx).Tangent lines and exponentials. Assume b is given with b 0 and b 1. Find the y-coordinate of the point on the curve y = bx at which the tangent line passes through the origin. (Source: The College Mathematics Journal, 28, Mar 1997)Is f(x) = sin1x an even or odd function? Is f(x) an even or odd function?How do the slopes of the lines tangent to the graph of y = tan1x behave as x ?Summarize how the derivatives of inverse trigonometric functions are related to the derivatives of the corresponding inverse cofunctions (for example, inverse tangent and inverse cotangent).Example 3 makes the claim that d/ds = 0.0024 rad/ft is equivalent to 0.138/ft. Verify this claim.Sketch the graphs of y = sin x and y = sin1x. Then verify that Theorem 3.21 holds at the point (0. 0)State the derivative formulas for sin1 x, tan1 x, and sec1 x.What is the slope of the line tangent to the graph of y = sin1 x at x = 0?What is the slope of the line tangent to the graph of y = tan1 x at x = 2?How are the derivatives of sin1 x and cos1 x related?Suppose f is a one-to-one function with f(2) = 8 and f(2) = 4. What is the value of (f1)(8)?Explain how to find (f1)(y0), given that y0 = f(x0).Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. 51. a. (f1)(4) b. (f1)(6) c. (f1)(1) d. f(1)Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. 52. a. f(f(0)) b. (f1)(0) c. (f1)(1) d. (f1)(f(4))If f is a one-to-one function with f(3) = 8 and f(3) = 7, find the equation of the line tangent to y=f1(x) at x = 8The line tangent to the graph of the one-to-one function y=f(x) at x = 3 is y=5x+1. Find f1(16) and (f1)(16).Find the slope of the curve y = sin1x at (12,6)without calculating the derivative of sin1x.12EDerivatives of inverse sine Evaluate the derivatives of the following functions. 7. f(x) = sin1 2xDerivatives of inverse sine Evaluate the derivatives of the following functions. 8. f(x) = x sin1 xDerivatives of inverse sine Evaluate the derivatives of the following functions. 9. f(w) = cos (sin1 2w)Derivatives of inverse sine Evaluate the derivatives of the following functions. 10. f(x) = sin1 (ln x)Derivatives of inverse sine Evaluate the derivatives of the following functions. 11. f(x) = sin1 (e2x)Derivatives of inverse sine Evaluate the derivatives of the following functions. 12. f(x) = sin1 (esin x)Derivatives Evaluate the derivatives of the following functions. 13. f(x) = tan1 10xEvaluate the derivative of the following functions. 20.f(x)=2xtan1xln(1+x2)21EDerivatives Evaluate the derivatives of the following functions. 16. g(z) = tan1 (1/z)Derivatives Evaluate the derivatives of the following functions. 17. f(z)=cot1zDerivatives Evaluate the derivatives of the following functions. 18. f(x)=sec1xEvaluate the derivative of the following functions. 25.f(x)=x2+2x3cot1xln(1+x2)26EEvaluate the derivative of the following functions. 27.f(w)=w2tan1w2Evaluate the derivative of the following functions. 28.f(t)=ln(sin1t2)Derivatives Evaluate the derivatives of the following functions. 19. f(x) = cos1 (1/x)30EDerivatives Evaluate the derivatives of the following functions. 21. f(u) = csc1 (2u +1)Derivatives Evaluate the derivatives of the following functions. 22. f(t) = ln (tan1 t)Derivatives Evaluate the derivatives of the following functions. 23. f(y) = cot1 (1/(y2 + 1))Derivatives Evaluate the derivatives of the following functions. 24. f(w) = sin (sec1 2w)Derivatives Evaluate the derivatives of the following functions. 25. f(x) = sec1 (ln x)Derivatives Evaluate the derivatives of the following functions. 26. f(x) = tan1 (e4x)Derivatives Evaluate the derivatives of the following functions. 27. f(x) = csc1 (tan ex)Derivatives Evaluate the derivatives of the following functions. 28. f(x) = sin (tan1 (ln x))Derivatives Evaluate the derivatives of the following functions. 29. f(s) = cot1 (eg)Derivatives Evaluate the derivatives of the following functions. 30. f(x) = 1/tan1 (x2 + 4)Tangent lines Find an equation of the line tangent to the graph of f at the given point. 31. f(x) = tan1 2x; (l/2, /4)Tangent lines Find an equation of the line tangent to the graph of f at the given point. 32. f(x) = sin1 (x/4); (2, /6)Tangent lines Find an equation of the line tangent to the graph of f at the given point. 33. f(x)=cos1x2;(1/2,/3)Tangent lines Find an equation of the line tangent to the graph of f at the given point. 34. f(x) = sec1 (ex); (ln 2, /3)Angular size A boat sails directly toward a 150-meter skyscraper that stands on the edge of a harbor. The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer (see figure). a. What is the rate of change of the angular size d/dx when the boat is x = 500 m from the building? b. Graph d/dx as a function of x and determine the point at which the angular size changes most rapidly.46EDerivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f1. 37. f(x) = 3x + 4; (16, 4)Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f1. 38. f(x)=12x+8;(10,4)Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse evaluate the derivative of the inverse at the given point. 49.f(x)=ln(5x+e);(1,0)Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f1. 40. f(x) = x2 + 1, for x 0; (5, 2)Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f1. 41. f(x) = tan x; (1, /4)Derivatives of inverse functions at a point Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find f1. 42. f(x) = x2 2x 3, for x 1; (12, 3)Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse evaluate the derivative of the inverse at the given point. 53.f(x)=x;(2,4)Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse evaluate the derivative of the inverse at the given point. 54.f(x)=4e10x;(4,0)Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse evaluate the derivative of the inverse at the given point. 55.f(x)=(x+2)2;(36,4)Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse evaluate the derivative of the inverse at the given point. 56.f(x)=log103x;(0,13)Find (f1)(3), where f(3)=x3+x+1.Suppose the slope of the curve y = f(x) at (7, 0) is 2/3. Find the slope of the curve y = f1(x) at (4, 7).Suppose the slope of the curve y = f1(x) at (4. 7) is 4/5. Find f(7).60EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. ddx(sin1x+cos1x)=0. b. ddx(tan1x)=sec2x. c. The lines tangent to the graph of y = sin1 x on the interval [1, 1] have a minimum slope of 1. d. The lines tangent to the graph of y = sin x on the interval [/2, /2] have a maximum slope of 1. e. If f(x) = 1/x, then (f1)(x) = 1/x2.62EGraphing f and f a. Graph f with a graphing utility. b. Compute and graph f. c. Verify that the zeros of f correspond to points at which f has a horizontal tangent line. 55. f(x) = (x2 1) sin1 x on [1, 1]64E65E66EDerivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 67.f(x)=3x468E69E70EDerivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 71.f(x)=e3x+1Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 72.f(x)=ln(5x+4)Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 73.f(x)=1012x6Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 74.f(x)=log10(2x+6)Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function. 75.f(x)=x+2