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All Textbook Solutions for Calculus: Early Transcendentals

The displacement of a particle on a vibrating string is given by the equation s(t)=10+14sin(10t) where sis measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.If the equation of motion of a particle is given by s = A cos(t + ), the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by 0.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function B(t)=4.0+0.35sin(2t5.4) (a) Find the rate of change of the brightness after t days. (b) Find, correct to two decimal places, the rate of increase after one day.In Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Philadelphia on the t th day of the year: L(t)=12+2.8sin[2365(t80)] Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May2l.83E Under certain circumstance a rumor spreads according to the equation p(t)=11+aekt 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 equation for p(t).] (a) Find limt p(t). (b) Find the rate of spread of the rumor. (c) Graph p for the case a= 10, k = 0.5 with 1 measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function C(t) = 0.0225te0.0467t where t is measured in minutes after consumption and C is measured in mg/mL. (a) How rapidly was the BAC increasing after 10 minutes? (b) How rapidly was it decreasing half an hour later? Source: Adapted from P. Wilkinson et al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmauutics5 (1977): 207- 24.In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function P(t) = (1436.53) (1.01395)t where t = 0 corresponds to the year 1900 and P(t) is measured in millions. According to this model, what was the rate of increase of world population in 1920? In 1950? In2000?87E Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V(t) and its radius is r(t). (a) What do the derivatives dV/dr and dV/dt represent? (b) Express dV/dt in tenns of dr/dt.89E 90E Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.94E (a) If n is a positive integer, prove that ddx(sinnxcosnx)=nsinn1xcos(n+1)x (b) Find a formula for the derivative of y = cosnx cos nx that is similar to the one in part (a).96E Use the Chain Rule to show that if is measured in degrees, then dd(sin)=180cos (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: the differentiation formulas would not be as simple if we used degree measure.)(a) Write |x|=x2 and use the Chain Rule to show that ddx|x|=x|x| (b) If f(x) = | sin x |. find f(x) and sketch the graphs of f and f. Where is f not differentiable? (c) If g(x) = sin | x |, find g(x) and sketch the graphs of g and g. Where is g not differentiable?99E 100E (a) Find y by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 1. 9x2 y2 = 1 (a) Find y by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 2. 2x2 + x + xy = 1 (a) Find y by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 3. x+y=1 (a) Find y by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a). 4. 2x1v=4 Find dy/dx by implicit differentiation. 5. x2 4xy + y2 = 4 Find dy/dx by implicit differentiation. 6. 2x2 + xy y2 = 2 Find dy/dx by implicit differentiation. 7. x4 + x2y2 + y3 = 5 Find dy/dx by implicit differentiation. 8. x3 xy2 + y3 = 1 Find dy/dx by implicit differentiation. 9. x2x+y=y2+1 Find dy/dx by implicit differentiation. 10. xey = x y Find dy/dx by implicit differentiation. 11. y cos x = x2 + y2 Find dy/dx by implicit differentiation. 12. cos(xy) = 1 + sin y Find dy/dx by implicit differentiation. 13. x+y=x4+y4 Find dy/dx by implicit differentiation. 14. ey sin x = x + xy Find dy/dx by implicit differentiation. 15. ex/y =x y Find dy/dx by implicit differentiation. 16. xy=x2+y2 Find dy/dx by implicit differentiation. 17. tan1(x2y) = x + xy2 Find dy/dx by implicit differentiation. 18. x sin y + y sin x = 1 Find dy/dx by implicit differentiation. 19. sin(xy) = cos(x + y)Find dy/dx by implicit differentiation. 20. tan(xy)=y1+x2 If f(x) + x2 [f(x)]3 = 10 and f(1) = 2, find f(1).If g(x) + x sin g(x) = x2, find g(0).Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. 23. x4y2 x3y + 2xy3 = 0 Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. 24. y sec x = x tan y Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 25. y sin 2x = x cos 2y, (/2, /4)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 26. sin(x + y) = 2x 2y, (, )Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 27. x2 xy y2 = 1, (2, 1) (hyperbola)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 28. x2 + 2xy + 4y2 = 12, (2, 1) (ellipse)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 29. x2 + y2 = (2x2 + 2y2 x)2, (0,12), (cardioid)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 30. x2/3+y2/3=4,(33,1), (astroid)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 31. 2(x2 + y2)2 = 25(x2 y2), (3, 1), (lemniscate)Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 32. y2(y2 4) = x2(x2 5), (0. 2), (devils curve)(a) The curve with equation y2 = 5x4 x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)(a) The curve with equation y2 = x3 + 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, 2). (b) At what points does this curve have horizontal tangents? (c) Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen.Find y by implicit differentiation. 35. x2 + 4y2 = 4 Find y by implicit differentiation. 36. x2 + xy + y2 = 3 Find y by implicit differentiation. 37. sin y + cos x =1 Find y by implicit differentiation. 38. x3 y3 = 7 If xy + ey = e, find the value of y at the point where x = 0.If x2 + xy + y3 = 1, find the value of y at the point where x = 1.Find the points on the lemniscate in Exercise 31 where the tangent is horizontal.Show by implicit differentiation that the tangent to the ellipse a2a2+y2b2=1 at the point (x0, y0) is x0xa2+y0yb2=1 Find an equation of the tangent line to the hyperbola x2a2y2b2=1 at the point (x0, y0).Show that the sum of the x-and y-intercepts of any tangent line to the curve x+y=c is equal to c.Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.The Power Rule can be proved using implicit differentiation for the case where n is a rational number, n = p/q, and y = f(x) = xn is assumed beforehand to be a differentiable function. If y = xp/q, then yq = xp. Use implicit differential ion to show that y=pqx(p/q)1 Find the derivative of the function. Simplify where possible. 49. y = (tan1x)2 Find the derivative of the function. Simplify where possible. 50. y = tan1(x2)Find the derivative of the function. Simplify where possible. 51. y = sin1(2x + 1)52E Find the derivative of the function. Simplify where possible. 53. F(x) = x sec1(x3)Find the derivative of the function. Simplify where possible. 54. y=tan1(x1+x2)Find the derivative of the function. Simplify where possible. 55. h(t) = cot1(t) + cot1(1/t)56E Find the derivative of the function. Simplify where possible. 57. y=xsin1x+1x2 Find the derivative of the function. Simplify where possible. 58. y = cos1(sin1t)59E Find the derivative of the function. Simplify where possible. 60. y=arctan1x1+x Find f(x). Check that your answer is reasonable by comparing the graphs of f and f. 61. f(x)=1x2arcsinx 62E Prove the formula for (d/dx)(cos1x) by the same method as for (d/ dx)(sin1).(a) One way of defining sec1x is to say that y=sec1xsecy=xand0y/2ory3/2 Show that, with this definition, ddx(sec1x)=1x21x (b) Another way of defining sec1x that is sometimes used is to say that y=sec1xsecy=x and 0 y , y /2. Show that, with this definition, ddx(sec1x)=1|x|x21 Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. 65. x2 + y2 = r2, ax + by = 0 66E Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. 67. y = cx2, x2 + 2y2 = k 68E Show that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2/A2 y2/ B2 = 1 are orthogonal trajectories if A2 a2 and a2 b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).70E (a) The van der Waals equation for n moles of a gas is (p+n2aV2)(Vnb)=nRT where P is the pressure. V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas. If T remains constant, use implicit differentiation to find dV/dP. (b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of V = 10 Land a pressure of P = 2.5 atm. Use a = 3.592 L2-atm/ mole2 and b = 0.04267 L/mole.The equation x2 xy + y2 = 3 re presents a "rotated ellipse, that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.(a) Where does the normal line to the ellipse x2 xy + y2 = 3 at the point (1, 1) intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line.Find all points on the curve x2y2 + xy = 2 where the slope of the tangent line is 1.Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3).(a) Suppose f is a one-to-one differentiable function and its inverse function f1 is also differentiable. Use implicit differentiation to show that (f1)(x)=1f(f1(x)) provided that the denominator is not 0. (b) If f(4) = 5 and f(4)=23, find (f1)(5).78E The Bessel function of order 0, y = J(x), satisfies the differential equation xy + y + xy = 0 for all values of x and its value at 0 is J(0) = 1. (a) Find J(0). (b) Use implicit differentiation to find J(0).The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 5. If the point (5. 0) is on the edge of the shadow, how far above the x-axis is the lamp located?Explain why the natural logarithmic function y = ln x is used much more frequently in calculus than the other logarithmic functions y = logb x.Differentiate the function. f(x) = x ln x x Differentiate the function. f(x ) = sin(ln x)Differentiate the function. f(x) = ln(sin2x)Differentiate the function. f(x)=ln1x Differentiate the function. y=1lnx Differentiate the function. f(x) = log10(1 + cos x)Differentiate the function. f(x)log10x Differentiate the function. g(x) = ln(xe2x)Differentiate the function. g(t)=1+lnt Differentiate the function. F(t) =(ln t)2 sin t Differentiate the function. h(x)=ln(x+x21)Differentiate the function. G(y)=ln(2y+1)5y2+1 Differentiate the function. p(v)=lnv1v Differentiate the function. F(s) = ln ln s Differentiate the function. y = ln |1 + t t3|Differentiate the function. T(z) = 2z log2z Differentiate the function. y = ln(csc x cot x)Differentiate the function. y = ln(ex + xex)Differentiate the function. H(z)=a2z2a2+z2 Differentiate the function. y = tan[ln(ax + b)]Differentiate the function. y = log2 (x log5 x)Find y and y. y=xlnx Find y and y. y=lnx1+lnx Find y and y. y = ln |sec x|Find y and y. y = ln(l + ln x)Differentiate f and find the domain of f. f(x)=x1ln(x1)Differentiate f and find the domain of f. f(x)=2+lnx Differentiate f and find the domain of f. f(x) = ln(x2 2x)Differentiate f and find the domain of f. f(x) ln ln ln x If f(x) = ln(x + ln x), find f(1).If f(x) = cos(ln x2), find f(1).Find an equation of the tangent line to the curve at the given point. y = ln(x2 3x + 1), (3. 0)Find an equation of the tangent line to the curve at the given point. y = x2 ln x (1,0)If f(x) = sin x + ln x, find f(x). Check that your answer is reasonable by comparing the graphs off and f.Find equations of the tangent lines to the curve y = (ln x)/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.Let f(x) = cx + ln(cos x). For what value of c is f(/4) = 6?Let f(x) = logb (3x2 2). For what value of b is f(1) = 3?Use logarithmic differentiation to find the derivative of the function. y = (x2 + 2)2(x4 + 4)4 Use logarithmic differentiation to find the derivative of the function. y=excos2xx2+x+1 Use logarithmic differentiation to find the derivative of the function. y=x1x4+1 Use logarithmic differentiation to find the derivative of the function. y=xex2x(x+1)2/3 Use logarithmic differentiation to find the derivative of the function. y = xx Use logarithmic differentiation to find the derivative of the function. y = xcos.x Use logarithmic differentiation to find the derivative of the function. y = xsin x Use logarithmic differentiation to find the derivative of the function. y=(x)x Use logarithmic differentiation to find the derivative of the function. y = (cos x)x Use logarithmic differentiation to find the derivative of the function. y = (sin x)ln x Use logarithmic differentiation to find the derivative of the function. y = (tan x)1/x Use logarithmic differentiation to find the derivative of the function. y = (ln x)cos x Find y if y = ln(x2 + y2).Find y if xy = yx.Find a formula for f(n)(x) if f(x) = ln(x 1).Find d9dx9(x8lnx).Use the definition of derivative to prove that limx0ln(1+x)x=1 Show that limn(1+xn)n=exfor any x 0.A particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t) = t3 8t2 + 24t A particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t)=9tt2+9 A particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t) = sin(t/2)A particle moves according to a law of motion s = f(t), t 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. (h) Graph the position, velocity, and acceleration functions for 0 t 6. (i) When is the particle speeding up? When is it slowing down? FIGURE 2 f(t) = t2et Graphs of the velocity functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) (b)Graphs of the position functions of two particles are shown, where 1 is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) (b)The height (in meters) of a projectile shot vertically upward from a point 2m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t 4.9t2 after t seconds. (a) Find the velocity after 2 sand after 4 s. (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit the ground? (e) With what velocity does it hit the ground?If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s = 80t 16t2. (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96ft above the ground on its way up? On its way down?If a rock is thrown vertically upward from the surface of Mars with velocity 15 m/s, its height after t seconds is h = 15t 1.8612t2 (a) What is the velocity of the rock after 2 s? (b) What is the velocity of the rock when its height is 25m on its way up? On its way down?A particle moves with position function s = t4 4t3 20t2 + 20t t 0 (a) At what time does the particle have a velocity of 20 m/s? (b) At what time is the acceleration 0? What is the significance of this value of t?(a) A company makes computer chips from square wafers of silicon. It wants to keep the side length o f a wafer very close to 15 mm and it wants to know how the area A(x) of a wafer changes when the side length x changes. Find A(l5) and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount x. How can you approximate the resulting change in area A if x his small?(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b).(a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r = 2. (c) Show that the rate of change of the area of a circle with respect to its radius {at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount r. How can you approximate the resulting change in area A if r is small?A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after {a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude?A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4r2) with respect to the radius r when r is (a) 1 ft, (b) 2ft, and (c) 3ft. What conclusion can you make?(a) The volume of a growing spherical cell is V=43r3, where the radius r is measured in micrometers (1 m = 106 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8m (ii) 5 to 6m (iii) 5 to 5.1m (b) Find the instantaneous rate of change of V with respect to r when r = 5m. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c).The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x2 kg. Find the linear density {see Example 2) when x is (a) 1m, (b) 2m, and (c) 3m. Where is the density the highest? The lowest? EXAMPLE 2 If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as V=5000(1140t)20t40 Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water Bowing out the fastest? The slowest? Summarize your findings.The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Q(t) = t3 2t2 + 6t + 2. Find the current when (a) t = 0.5 sand (b) t = 1 s. [See Example 3. The unit of current is an ampere (1 A = 1 C/s).) At what time is the current lowest? EXAMPLE 3 FIGURE6 Newtons Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is F=GmMr2 where G is the gravitational constant and r is the distance between the bodies. (a) Find dF/dr and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 N/km when r = 20,000 km. How fast does this force change when r = 10,000 km?21E Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at 6:45 AM. This helps explain the following model for the water depth D (in meters) as a function of the time t (in hours after midnight) on that day: D(t) = 7 + 5 cos[0.503(t - 6.75)] How fast was the tide rising (or falling) at the following times? (a) 3:00 AM (b) 6:00 AM (c) 9:00AM (d) Noon Boyles Law states that when a sample of gas is compressed al a constant temperature, the product of the pressure and the volume remains constant: PV = C. (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given by = 1/P. EXAMPLE 5 If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value [A] = [B] = a moles/ L, then [C] = a2kt/(akt + 1) where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x = [C], then dxdt=k(ax)2 c) What happens to the concentration as t ? (d) What happens to the rate of reaction as t 4 ? (e) What do the results of parts (c) and (d) mean in practical terms? EXAMPLE 4 In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number n of bacteria after t hours and use it to estimate the rate of growth of the bacteria population after 2.5 hours. EXAMPLE 6 The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n=f(t)=a1+be0.7t where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?28E 29E 30E 31E The cost function for a certain commodity is C(q) = 84 + 0.16q 0.0006q2 + 0.000003q3 (a) Find and interpret C(100). (b) Compare C(100) with the cost of producing the 101st item.33E If R denotes the reaction of the body to some stimulus of strength x, the sensitivity S is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula R=40+24x0.41.4x0.4 has been used to model the dependence of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both R and S as functions of x. Comment on the values of R and S at low levels of brightness. Is this what you would expect?Patients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c I, is given by the equation t=ln(3c+9c28c2) Calculate the derivative of t with respect to c and interpret it.Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f(r)=2Dr, where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate r and explain its meaning.The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where n is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at a certain instant, P = 8.0 atm and is increasing at a rate of 0.10 atm/min and V = 10 Land is decreasing at a rate of 0.15 L/min. Find the rate of change of T with respect to time at that instant if n = 10 mol.In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation dPdt=r0(1P(t)Pc)P(t)P(t) where r0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain (called the carrying capacity), and is the percentage of the population that is harvested. (a) What value of dP/dt corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if is raised to 5%?In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W(t), and caribou, given by C(t), in northern Canada. The interaction has been modeled by the equations dCdt=aCbCWdWdt=cW+dCW (a) What values of dC/dt and dW/dt correspond to stable populations? (b) How would the statement The caribou go extinct be represented mathematically? (c) Suppose that a = 0.05, b = 0.001, c = 0.05, and d = 0.0001. Find all population pairs (C, W) that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. (a) Find the 1elative growth rate. (b) Find an expression for the number of cells after t hours. (c) Find the number of cells after 6 hours. (d) Find the rate of growth after 6 hours. (c) When will the population reach a million cells?A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a) Find an expression for the number of bacteria after t hour. (b) Find the number of bacteria after 3 hours. (c) Find the rate of growth after 3 hours. (d) When will the population reach 10.000?A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative growth rate? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find an expression for the number of bacteria after t hours. (d) Find the number of cells after 4.5 hours. (e) Find the rate of growth after 4.5 hours. (f) When will the population reach 50,000?The table gives estimates of the world population, in millions, from 1750 to 2000. (a) Use the exponential model and the population figures for 1750 and 800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. (c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.The table gives the population of Indonesia, in millions, for the second half of the 20th century. (a) Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and 1960 to predict the population in 1980. Compare with the actual figure. (b) Use the census figures for 1960 and 1980 to predict the population in 2000. Compare with the actual population. (c) Use the census figures for 1980 and 2000 to predict the population in 2010 and compare with the actual population of 243 million. (d) Use the model in pan (c) to predict the population in 2020. Do you think the prediction will be too high or too low? Why?Experiments show that if the chemical reaction N2O2NO2+12O2 takes place at 45C, the rate of reaction of dinitrogen pent-oxide is proportional to its concentration as follows: d[N2O5]dt=0.0005[N2O5] (See Example 3.7.4.) (a) Find an expression for the concentration [N2O5] after t seconds if the initial concentration is C. (b) How long will the reaction take to reduce the concentration of N2O5, to 90% of its original value?Strontium-90 has a half-life of 28 days. (a) A sample has a mass of 50 mg initially. Find a formula for the mass remaining after t days. (b) Find the mass remaining after 40 days. (c) How long does it take the sample to decay to a mass of 2 mg? (d) Sketch the graph of the mass function.The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after t years. (b) How much of the sample remains after 100 years? (c) After how long will only 1 mg remain?A sample oflritium-3 decayed to 94.5% of its original amount after a year. (a) What is the half-life of lritium-3? (b) How long would it take the sample to decay to 20% of its original amount?Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore the level of radioactivity must also decay exponentially. A discovery revealed a parchment fragment that had about 74% as much 14C radioactivity as does plant material on the earth today. Estimate the age of the parchment.Dinosaur fossils are too old to be reliably dated using carbon-14. (See Exercise 11.) Suppose we had a 68-millionyear-old dinosaur fossil. What fraction of the living dinosaur's 14C would be remaining today? Suppose the minimum detectable amount is 0.1 %. What is the maximum age of a fossil that we could date using 14C?13E A curve passes through the point (0, 5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?A roast turkey is taken from an oven when its temperature has reached 185F and is placed on a table in a room where the temperature is 75F. (a) If the temperature of the turkey is 150F after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to 100F?In a murder investigation, the temperature of the corpse was 32.5C at 1:30 PM and 30.3C an hour later. Normal body temperature is 37.0C and the temperature of the surroundings was 20.0C. When did the murder take place?When a cold drink is taken from a refrigerator, its temperature is 5C. After 25 minutes in a 20C room its temperature has increased to 10C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature be 15C?A freshly brewed cup of coffee has temperature 95C in a 20C room. When its temperature is 70C, it is cooling at a rate of 1C per minute. When does this occur?19E (a) If 1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly. (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose 1000 is borrowed and the interest is compounded continuously. If A(t) is the amount due after t years, where 0 t 3, graph A(1) for each of the interest rates 6%, 8%, and I 0% on a common screen.(a) If 3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly. (iv) weekly, (v) daily, and (vi) continuously. (b) If A(t) is the amount of the investment at time 1 for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(r).(a) How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b) What is the equivalent annual interest rate?1E (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in tenns of dr/dt. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30 m?Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/ min. How fast is the height of the water increasing?The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?The radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 8 cm?The area of a triangle with sides of lengths a and band contained angle is A=12absin (a) If a = 2 cm, b = 3 cm, and increases at a rate of 0.2 rad/min, how fast is the area increasing when = /3? (b) If a = 2 cm, b increases at a rate of 1.5 cm/ min, and increases at a rate of 0.2 rad/min, how fast is the area increasing when b = 3 cm and 0 = /3? (c) If a increases at a rate of 2.5 cm/ min, b increases at a rate of 1.5 cm/ min, and 0 increases at a rate of 0.2 rad/min, how fast is the area increasing when a = 2 cm, b = 3 cm, and 0 = 7r/ 3?Suppose y=2x+1, where x and y are functions of t. (a) If dx/dt = 3, find dy/dt when x = 4. (b) If dy/dt = 5, find dx/dt when x = 12.Suppose 4x2 + 9y2 = 36, where x and y are functions of t. (a) Ifdy/dt=13, find dx/dt when x = 2 and y=235 (b) If dx/dt = 3, find dy/dt when x = 2 and y=235 If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4, find dz/dr when (x, y, z) = (2, 2. 1).A particle is moving along a hyperbola xy = 8. As it reaches the point (4, 2), they-coordinate is decreasing at a rate of 3 cm/s. How fast is the x-coordinate of the point c hanging at that instant?(a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time 1. (d) Write an equation that relates the quantities. (e) Finish solving the problem. A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.(a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time 1. (d) Write an equation that relates the quantities. (e) Finish solving the problem. If a snowball melts so that its surface area decreases at a rate of 1 cm2/min. find the rate at which the diameter decreases when the diameter is 10 cm.(a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time 1. (d) Write an equation that relates the quantities. (e) Finish solving the problem. A street light is mounted at the top of a 15-ft-tall pole. A man 6ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 n from the pole?(a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time 1. (d) Write an equation that relates the quantities. (e) Finish solving the problem. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 kin/h. How fast is the distance between the ships changing at 4:00 PM?Two cars start moving from the same point. One travels south at60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?A spotlight on the ground shines on a wall 12m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s. how fast is the length of his shadow on the building decreasing when he is 4 m from the building?A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 fl due east of P. At what rate are the people moving apart 15 min after the woman starts walking?A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment?The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/ min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate or 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00PM?24E Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find t he rate at which water is being pumped in to the tank.A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height o f 1ft. If the trough is being filled with water at a rate of 12 ft3/min, how fast is the water level rising when the water is 6 inches deep?A water trough is 10m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m 3/min, how fast is the water level rising when the water is 30 cm deep?A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the s hallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 ft3/ min, how fast is the water level rising when the depth at the deepest point is 5 ft?Gravel is being dumped from a conveyor belt at a rate of 30ft 3 min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high?A kite 100ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?The sides of an equilateral triangle are increasing at a rate of 10 cm/min. At what rate is the area of the triangle increasing when the sides are 30 cm long?How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values o f y? EXAMPLE 2 FIGURE 1 FIGURE 2 35E A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm3. The volume of the portion of a sphere with radius r from the bottom to a height h is V=(rh213h3) as we will show in Chapter 6.)Boyles Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant?When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPa/ min. At what rate is the volume increasing at this instant?If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (), is given by 1R=1R1+1R2 If R1 and R2 arc increasing at rates of 0.3 0/s and 0.2 /s, respectively, how fast is R changing when R1 = 80 and R2 = 100?Brain weight B as a function of body weight Win fish has been modeled by the power function B = 0.007W2/3, where R and W are measured in grams. A model for body weight as a function of body length L. (measured in centimeters) is W = 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species' brain growing when the average length was 18 cm?Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60?Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 600 ft / s when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the cameras angle of elevation changing at that same moment?A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is /3, this angle is decreasing at a rate of /6 rad/min. How fast is the plane traveling at that time?A Ferris wheel with a radius of 10m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?A plane flying with a constant speed of 300 km/h passes over a around radar station at an altitude of 1 km and climbs at an angle of 30. At what rate is the distance from the plane to the radar station increasing a minute later?Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast i the distance between the people changing after 15 minutes?A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 on from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o' clock?Find the linearization L(x) of the function at n. 1. f(x) = x3 x2 + 3, a = 2 Find the linearization L(x) of the function at n. 2. f(x) = sin x, a = /6 Find the linearization L(x) of the function at n. 3. f(x)=x,a=4 Find the linearization L(x) of the function at n. 4. f(x) = 2x, a = 0 Find the linear approximation of the function f(x)=1x at a = 0 and use it to approximate the numbers 0.9 and 0.99. Illustrate by graphing f and the tangent line.Find the linear approximation of the function g(x)=1+x3 at a = 0 and use it to approximate the numbers 0.953 and 1.13. Illustrate by graphing g and the tangent line.7E 8E Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 9. 1+2x41+12x 10E Find the differential of each function. 11. (a) y = xe4x (b) y=1t4 Find the differential of each function. 12. (a) y=1+2u1+3u (b) y = 2 sin 2 Find the differential of each function. 13. (a) y=tant (b) y=1v21+v2 Find the differential of each function. 14. (a) y = ln(sin ) (b) y=ex1ex (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 15. y = ex/10, x = 0, dx = 0.1 (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 16. y=cosx,x=13,dx=0.02 (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 17. y=3+x2,x=1,dx=0.1 18E Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 19. y = x2 4x. x = 3, x = 0.5 Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 20. y = x x3, x = 0, x = 0.3 Compute y and dy for the given values of x and dx = x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and y. 21. y=x2,x=3,x=0.8 22E Use a linear approximation (or differentials) to estimate the given number. 23. (1.999)4 Use a linear approximation (or differentials) to estimate the given number. 24. 1/4.002 Use a linear approximation (or differentials) to estimate the given number. 25. 10013 Use a linear approximation (or differentials) to estimate the given number. 26. 100.5 Use a linear approximation (or differentials) to estimate the given number. 27. e0.1 Use a linear approximation (or differentials) to estimate the given number. 28. cos 29 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 29. sec 0.08 1 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 30. 4.022.005 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 31. 19.980.1002 32E The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube.The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error?The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error?Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness r. (b) What is the error involved in using the formula from part (a)?

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