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All Textbook Solutions for Single Variable Calculus: Early Transcendentals

The equation y" + y' 2y = x2 is called a differential equation because it involves an unknown function y and its derivatives y' and y". Find constants A, B. and C such that the function y = Ax2 + Bx + C satisfies this equation. (Differential equations will he studied in detail in Chapter 9.)Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (2, 6) and (2, 0).70E 71E At what numbers is the following function g differentiable? g(x){2xifx02xx2if0x22xifx2 Give a formula for g' and sketch the graphs of g and g'.73E 74E Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 3x 2.76E 77E 78E What is the value of c such that the line y = 2x + 3 is tangent to the parabola y = cx2?80E 81E A tangent line is drawn to the hyperbola xy = c at a point P. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola.83E 84E 85E 86E Find the derivative of f(x) = (1 + 2x2)(x x2) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?Find the derivative o f the function F(x)=x45x3+xx2 in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?3E Differentiate. g(x)=(x+22)ex Differentiate. y=xex Differentiate. y=ex1ex 7E Differentiate. G(x)=x222x+1 Differentiate. H(u)=(uu)(u+u)10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E Differentiate. F(t)=AtBt2+Ct3 25E Differentiate. f(x)=ax+bcx+d 27E 28E 29E 30E 31E 32E 33E Find equations of the tangent line and normal line to the given curve at the specified point. y=2xx2+1,(1,1)35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2 47E 48E If f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x). (a) Find u'(l). (b) Find v'(5).50E 51E If f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x2f(x) (b) y=f(x)x2 (c) y=x2f(x) (d) y=1+xf(x)x 53E Find equations of the tangent lines to the curve y=x1x+1 that are parallel to the line x 2y = 2.55E 56E 57E A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f(p). Then the total revenue earned with selling price p is R(p) = pf(p). (a) What does it mean to say that f(20) = 10,000 and f'(20) = 350? (b) Assuming the values in part (a), find R'(20) and interpret your answer.59E The biomass B(t) of a fish population is the total mass of the members of the population at time t. It is the product of the number of individuals N(t) in the population and the average mass M(t) of a fish at time t. In the case of guppies, breeding occurs continually. Suppose that at time t = 4 weeks the population is 820 guppies and is growing at a rate of 50 guppies per week, while the average mass is 1.2 g and is increasing at a rate of 0.14 g/week. At what rate is the biomass increasing when t = 4?(a) Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h + fgh'. (b) Taking f = g = h in part (a), show that dx[f(x)]3=3[f(x)]2f(x) (c) Use part (b) to differentiate y = e3x.(a) If F(x) = f(x) g(x), where f and g have derivatives of all orders, show that F" = f"g + 2f'g' + fg". (b) Find similar formulas for F"' and F(4). (c) Guess a formula for F(n).Find expressions for the first five derivatives of f(x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n)(x) and prove it using mathematical induction.64E 1E Differentiate. f(x) = x cos x + 2 tan x Differentiate. f(x) = ex cos x Differentiate. y = 2 sec x csc x Differentiate. y = sec tan 6E 7E 8E 9E 10E Differentiate f()=sin1+cos 12E 13E Differentiate. y=sint1+tant Differentiate. f() = cos sin Differentiate. f(t) = tet cot t 17E 18E 19E Prove, using the definition of derivative. that if f(x) = cos x, then f'(x) = sin x.21E Find an equation of the tangent line to the curve at the given point. y = ex cos x, (0, 1)23E 24E 25E 26E 27E 28E If H() = sin , find H'() and H"( ).If f(t) = sec t, find f"(/4).31E 32E For what values of x does the graph of f have a horizontal tangent? f(x) = x + 2 sin x 34E 35E An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3 sin t, t 0, where s is measured in centimeters and t in seconds. (Take the positive direction to be downward.) (a) Find the velocity and acceleration at time t. (b) Graph the velocity and acceleration functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest?37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E Find the given derivative by finding the first few derivatives and observing the pattern that occurs. d99dx99(sinx)Find the given derivative by finding the first few derivatives and observing the pattern that occurs. d35dx35(xsinx)53E 54E Differentiate each trigonometric identity to obtain a new (or familiar) identity. (a) tanx=sinxcosx (b) secx=1cosx (c) sinx+cosx=1+cotxcscx A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A() is the area of the semicircle and B() is the area of the triangle, find. lim0+A()B()The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle . Find lim0+sd 58E Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y=1+4x3 Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y = (2x3 + 5)4 3E Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx. y = sin( cot x)5E 6E Find the derivative of the function. F(x) = (5x6 + 2x3)4 Find the derivative of the function. F (x) = (1 + x + x2)99 Find the derivative of the function. f(x)=5x+1 10E 11E 12E 13E Find the derivative of the function. f(t) = t sin t 15E 16E Find the derivative of the function. f(x) = (2x 3)4(x2 + x + 1)5 Find the derivative of the function. g(x) = (x2 + 1)3(x2 + 2)6 19E Find the derivative of the function. F(t) = (3t 1)4(2t + 1)3 21E Find the derivative of the function. y=(x+1x)5 23E Find the derivative of the function. f(t)2t3 25E 26E 27E Find the derivative of the function. f(z) = ez/(z1)29E 30E 31E 32E 33E 34E 35E Find the derivative of the function. y = x2 e1/x 37E 38E 39E 40E 41E 42E Find the derivative of the function. g(x) = (2 rarx + n)P 44E 45E 46E 47E 48E 49E Find y and y. y=eex Find an equation of the tangent line to the curve at the given point. y = 2x, (0. 1)Find an equation of the tangent line to the curve at the given point. y=1+x3,(2,3)53E 54E 55E 56E 57E 58E 59E At what point on the curve y=1+2x is the tangent line perpendicular to the line 6x + 2y = 1?61E 62E A table of values for f, g, f, and g is given. (a) If h(x) = f(g(x)), find h(1). (b) If H(x) = g(f(x)), find H(1).Let f and g be the functions in Exercise 63. (a) If F(x) = f(f(x)), find F(2). (b) If G(x) = g(g(x)), find G(3).65E 66E 67E Suppose f is differentiable on and is a real number. Let F(x) = f(x) and G(x) = [f(x)]. Find expressions for (a) F(x) and (b) G(x).Suppose f is differentiable on . Let F(x) = f(ex) and G(x) = ef(x). Find expressions for (a) F(x) and (b) G(x).70E 71E 72E 73E 74E 75E 76E 77E Find the 1000th derivative of f(x) = xex.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?81E 82E The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is s(t) = 2e1.5t sin 2t where s is measured in centimeters and t in seconds. Find the velocity after t seconds and graph both the position and velocity functions for 0 1 2.84E 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 88E 89E The table gives the US population from 1790 to 1860. (a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines. (c) Use the exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b). (d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,558,000. Can you explain the discrepancy?91E 92E 93E 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 97E 98E 99E If y = f(u) and u = g(x), where f and g possess third derivatives, find a formula for d3y/dx3 similar to the one given in Exercise 99.(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 2E 3E 4E Find dy/dx by implicit differentiation. 5. x2 4xy + y2 = 4 6E 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 14E 15E 16E 17E 18E Find dy/dx by implicit differentiation. 19. sin(xy) = cos(x + y)20E 21E 22E 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 25E Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 26. sin(x + y) = 2x 2y, (, )27E 28E 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)30E 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)32E (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.)34E Find y by implicit differentiation. 35. x2 + 4y2 = 4 36E 37E 38E 39E If x2 + xy + y3 = 1, find the value of y at the point where x = 1.41E 42E 43E 44E Find an equation of the tangent line to the hyperbola x2a2y2b2=1 at the point (x0, y0).46E Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.48E 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 53E Find the derivative of the function. Simplify where possible. 54. y=tan1(x1+x2)55E 56E Find the derivative of the function. Simplify where possible. 57. y=xsin1x+1x2 58E 59E

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