Explanation Given: The function f ( x ) = x e x . Derivative rule: (1) Product Rule: d d x [ f 1 ( x ) f 2 ( x ) ] = f 1 ( x ) d d x [ f 2 ( x ) ] + f 2 ( x ) d d x [ f 1 ( x ) ] (2) Power Rule: d d x ( x n ) = n x n − 1 (3) Sum rule : d d x ( f + g ) = d d x ( f ) + d d x ( g ) (4) Constant multiple rule: d d x ( c f ) = c d d x ( f ) (5) Natural exponential function: d d x ( e x ) = e x Calculation: The first derivative of the function f ( x ) = x e x is f ′ ( x ) , which is obtained as follows, f ′ ( x ) = d d x ( f ( x ) ) = d d x ( x e x ) = d d x ( x 1 2 e x ) Apply the product rule (1), f ′ ( x ) = ( x 1 2 ) d d x ( e x ) + e x d d x ( x 1 2 ) Apply the derivative rules (2),(5) and simplify the terms. f ′ ( x ) = ( x 1 2 ) ( e x ) + e x ( 1 2 x 1 2 − 1 ) = ( x 1 2 ) ( e x ) + e x ( 1 2 x 1 2 − 2 2 ) = e x x 1 2 + 1 2 e x x − 1 2 = e x ( x 1 2 + 1 2 x − 1 2 ) Therefore, the first derivative of the function f ( x ) = x e x is f ′ ( x ) = e x ( x 1 2 + 1 2 x − 1 2 )

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 3 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 3.1 - (a) How is the number e defined? (b) Use a...Ch. 3.1 - (a) Sketch, by hand, the graph of the function...Ch. 3.1 - Differentiate the function. f(x) = 240 Ch. 3.1 - Prob. 4E Ch. 3.1 - Differentiate the function. f(x) = 5.2x + 2.3 Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Differentiate the function. f(t) = 1.4t5 2.5t2+...Ch. 3.1 - Prob. 9E Ch. 3.1 - Differentiate the function. H(u) = (3u 1)(u + 2)

Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Prob. 14E Ch. 3.1 - Prob. 15E Ch. 3.1 - Prob. 16E Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Differentiate the function. S(R) = 4R2 Ch. 3.1 - Prob. 21E Ch. 3.1 - Prob. 22E Ch. 3.1 - Prob. 23E Ch. 3.1 - Prob. 24E Ch. 3.1 - Prob. 25E Ch. 3.1 - Differentiate the function. k(r) = er + re Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Find an equation of the tangent line to the curve...Ch. 3.1 - Prob. 34E Ch. 3.1 - Prob. 35E Ch. 3.1 - Prob. 36E Ch. 3.1 - Find equations of the tangent line and normal line...Ch. 3.1 - Prob. 38E Ch. 3.1 - Prob. 39E Ch. 3.1 - Prob. 40E Ch. 3.1 - Prob. 41E Ch. 3.1 - Prob. 42E Ch. 3.1 - Prob. 43E Ch. 3.1 - Prob. 44E Ch. 3.1 - Find the first and second derivatives of the...Ch. 3.1 - Prob. 46E Ch. 3.1 - Prob. 47E Ch. 3.1 - Prob. 48E Ch. 3.1 - The equation of motion of a particle is s = t3 ...Ch. 3.1 - The equation of motion of a particle is s = t4 ...Ch. 3.1 - Prob. 51E Ch. 3.1 - The number of tree species S in a given area A in...Ch. 3.1 - Prob. 53E Ch. 3.1 - Prob. 55E Ch. 3.1 - Prob. 56E Ch. 3.1 - Show that the curve y = 2ex + 3x + 5x3 has no...Ch. 3.1 - Prob. 58E Ch. 3.1 - Prob. 59E Ch. 3.1 - Prob. 60E Ch. 3.1 - Prob. 61E Ch. 3.1 - Prob. 62E Ch. 3.1 - Prob. 63E Ch. 3.1 - Prob. 64E Ch. 3.1 - Prob. 65E Ch. 3.1 - Find the nth derivative of each function by...Ch. 3.1 - Prob. 67E Ch. 3.1 - Prob. 68E Ch. 3.1 - Find a cubic function y = ax3 + bx2 + cx + d whose...Ch. 3.1 - Prob. 70E Ch. 3.1 - Prob. 71E Ch. 3.1 - Prob. 72E Ch. 3.1 - Prob. 73E Ch. 3.1 - Prob. 74E Ch. 3.1 - Prob. 75E Ch. 3.1 - Suppose the curve y = x4 + ax3 + bx2 + cx + d has...Ch. 3.1 - Prob. 77E Ch. 3.1 - Prob. 78E Ch. 3.1 - Prob. 79E Ch. 3.1 - Prob. 80E Ch. 3.1 - Prob. 81E Ch. 3.1 - A tangent line is drawn to the hyperbola xy = c at...Ch. 3.1 - Evaluate limx1x10001x1.Ch. 3.1 - Prob. 84E Ch. 3.1 - Prob. 85E Ch. 3.1 - Prob. 86E Ch. 3.2 - Prob. 1E Ch. 3.2 - Prob. 2E Ch. 3.2 - Prob. 3E Ch. 3.2 - Prob. 4E Ch. 3.2 - Prob. 5E Ch. 3.2 - Prob. 6E Ch. 3.2 - Prob. 7E Ch. 3.2 - Prob. 8E Ch. 3.2 - Prob. 9E Ch. 3.2 - Prob. 10E Ch. 3.2 - Prob. 11E Ch. 3.2 - Prob. 12E Ch. 3.2 - Prob. 13E Ch. 3.2 - Prob. 14E Ch. 3.2 - Differentiate. y=t3+3tt24t+3 Ch. 3.2 - Prob. 16E Ch. 3.2 - Differentiate. y=ep(p+pp)Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - Prob. 21E Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - Differentiate. f(x)=xx+cx Ch. 3.2 - Prob. 26E Ch. 3.2 - Find f'(x) and f"(x). f(x) = (x3 + 1)ex Ch. 3.2 - Prob. 28E Ch. 3.2 - Prob. 29E Ch. 3.2 - Prob. 30E Ch. 3.2 - Prob. 31E Ch. 3.2 - Prob. 32E Ch. 3.2 - Prob. 33E Ch. 3.2 - Prob. 34E Ch. 3.2 - (a) The curve y = 1/(1 + x2) is called a witch of...Ch. 3.2 - Prob. 36E Ch. 3.2 - Prob. 37E Ch. 3.2 - Prob. 38E Ch. 3.2 - Prob. 39E Ch. 3.2 - Prob. 40E Ch. 3.2 - Prob. 41E Ch. 3.2 - Prob. 42E Ch. 3.2 - Prob. 43E Ch. 3.2 - Suppose that f(4) = 2, g(4) = 5, f'(4) = 6. and...Ch. 3.2 - If f(x) = exg(x), where g(0) = 2 and g'(0) = 5,...Ch. 3.2 - If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2 Ch. 3.2 - Prob. 47E Ch. 3.2 - Prob. 48E Ch. 3.2 - If f and g are the functions whose graphs are...Ch. 3.2 - Prob. 50E Ch. 3.2 - Prob. 51E Ch. 3.2 - Prob. 52E Ch. 3.2 - How many tangent lines to the curve y = x/(x + 1)...Ch. 3.2 - Prob. 54E Ch. 3.2 - Prob. 55E Ch. 3.2 - Prob. 56E Ch. 3.2 - In this exercise we estimate the rate at which the...Ch. 3.2 - A manufacturer produces bolts of a fabric with a...Ch. 3.2 - Prob. 59E Ch. 3.2 - The biomass B(t) of a fish population is the total...Ch. 3.2 - (a) Use the Product Rule twice to prove that if f,...Ch. 3.2 - Prob. 62E Ch. 3.2 - Prob. 63E Ch. 3.2 - Prob. 64E Ch. 3.3 - Differentiate. f(x) = x2 sin x Ch. 3.3 - Prob. 2E Ch. 3.3 - Prob. 3E Ch. 3.3 - Prob. 4E Ch. 3.3 - Differentiate. y = sec tan Ch. 3.3 - Prob. 6E Ch. 3.3 - Differentiate. y = c cos t + t2 sin t Ch. 3.3 - Differentiate. f(t)=cottet Ch. 3.3 - Prob. 9E Ch. 3.3 - Prob. 10E Ch. 3.3 - Prob. 11E Ch. 3.3 - Prob. 12E Ch. 3.3 - Prob. 13E Ch. 3.3 - Prob. 14E Ch. 3.3 - Prob. 15E Ch. 3.3 - Differentiate. f(t) = tet cot t Ch. 3.3 - Prove that ddx(cscx)=cscxcotx.Ch. 3.3 - Prob. 18E Ch. 3.3 - Prob. 19E Ch. 3.3 - Prob. 20E Ch. 3.3 - Prob. 21E Ch. 3.3 - Prob. 22E Ch. 3.3 - Prob. 23E Ch. 3.3 - Prob. 24E Ch. 3.3 - Prob. 25E Ch. 3.3 - Prob. 26E Ch. 3.3 - Prob. 27E Ch. 3.3 - Prob. 28E Ch. 3.3 - Prob. 29E Ch. 3.3 - Prob. 30E Ch. 3.3 - Prob. 31E Ch. 3.3 - Prob. 32E Ch. 3.3 - Prob. 33E Ch. 3.3 - For what values of x does the graph of f have a...Ch. 3.3 - Prob. 35E Ch. 3.3 - Prob. 36E Ch. 3.3 - A ladder 10 ft long rests against a vertical wall....Ch. 3.3 - Prob. 38E Ch. 3.3 - Prob. 39E Ch. 3.3 - Prob. 40E Ch. 3.3 - Prob. 41E Ch. 3.3 - Prob. 42E Ch. 3.3 - Prob. 43E Ch. 3.3 - Find the limit. limx0sin3xsin5xx2 Ch. 3.3 - Prob. 45E Ch. 3.3 - Prob. 46E Ch. 3.3 - Prob. 47E Ch. 3.3 - Find the limit. limx0sin(x2)x Ch. 3.3 - Prob. 49E Ch. 3.3 - Prob. 50E Ch. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Find the given derivative by finding the first few...Ch. 3.3 - Prob. 53E Ch. 3.3 - Prob. 54E Ch. 3.3 - Prob. 55E Ch. 3.3 - A semicircle with diameter PQ sits on an isosceles...Ch. 3.3 - Prob. 57E Ch. 3.3 - Prob. 58E Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 2E Ch. 3.4 - Write the composite function in the form f(g(x))....Ch. 3.4 - Prob. 4E Ch. 3.4 - Prob. 5E Ch. 3.4 - Prob. 6E Ch. 3.4 - Prob. 7E Ch. 3.4 - Prob. 8E Ch. 3.4 - Prob. 9E Ch. 3.4 - Prob. 10E Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.4 - Prob. 13E Ch. 3.4 - Prob. 14E Ch. 3.4 - Prob. 15E Ch. 3.4 - Prob. 16E Ch. 3.4 - Prob. 17E Ch. 3.4 - Prob. 18E Ch. 3.4 - Prob. 19E Ch. 3.4 - Prob. 20E Ch. 3.4 - Prob. 21E Ch. 3.4 - Prob. 22E Ch. 3.4 - Find the derivative of the function. y = e tan Ch. 3.4 - Prob. 24E Ch. 3.4 - Prob. 25E Ch. 3.4 - Prob. 26E Ch. 3.4 - Prob. 27E Ch. 3.4 - Find the derivative of the function. f(z) =...Ch. 3.4 - Prob. 29E Ch. 3.4 - Prob. 30E Ch. 3.4 - Prob. 31E Ch. 3.4 - Prob. 32E Ch. 3.4 - Prob. 33E Ch. 3.4 - Prob. 34E Ch. 3.4 - Prob. 35E Ch. 3.4 - Prob. 36E Ch. 3.4 - Prob. 37E Ch. 3.4 - Prob. 38E Ch. 3.4 - Prob. 39E Ch. 3.4 - Prob. 40E Ch. 3.4 - Prob. 41E Ch. 3.4 - Prob. 42E Ch. 3.4 - Prob. 43E Ch. 3.4 - Prob. 44E Ch. 3.4 - Prob. 45E Ch. 3.4 - Prob. 46E Ch. 3.4 - Prob. 47E Ch. 3.4 - Prob. 48E Ch. 3.4 - Prob. 49E Ch. 3.4 - Prob. 50E Ch. 3.4 - Prob. 51E Ch. 3.4 - Prob. 52E Ch. 3.4 - Prob. 53E Ch. 3.4 - Prob. 54E Ch. 3.4 - Prob. 55E Ch. 3.4 - Prob. 56E Ch. 3.4 - Prob. 57E Ch. 3.4 - Prob. 58E Ch. 3.4 - Prob. 59E Ch. 3.4 - Prob. 60E Ch. 3.4 - Prob. 61E Ch. 3.4 - Prob. 62E Ch. 3.4 - Prob. 63E Ch. 3.4 - Prob. 64E Ch. 3.4 - If f and g are the functions whose graphs are...Ch. 3.4 - Prob. 66E Ch. 3.4 - Prob. 67E Ch. 3.4 - Prob. 68E Ch. 3.4 - Prob. 69E Ch. 3.4 - Prob. 70E Ch. 3.4 - Prob. 71E Ch. 3.4 - Prob. 72E Ch. 3.4 - Prob. 73E Ch. 3.4 - Prob. 74E Ch. 3.4 - Prob. 75E Ch. 3.4 - Prob. 76E Ch. 3.4 - Prob. 77E Ch. 3.4 - Find the 1000th derivative of f(x) = xex.Ch. 3.4 - Prob. 79E Ch. 3.4 - Prob. 80E Ch. 3.4 - A Cepheid variable star is a star whose brightness...Ch. 3.4 - Prob. 82E Ch. 3.4 - Prob. 83E Ch. 3.4 - Prob. 84E Ch. 3.4 - Prob. 85E Ch. 3.4 - Prob. 86E Ch. 3.4 - A particle moves along a straight line with...Ch. 3.4 - Prob. 88E Ch. 3.4 - The flash unit on a camera operates by storing...Ch. 3.4 - Prob. 90E Ch. 3.4 - Use the Chain Rule to prove the following. (a) The...Ch. 3.4 - Prob. 94E Ch. 3.4 - Prob. 95E Ch. 3.4 - Prob. 96E Ch. 3.4 - Prob. 97E Ch. 3.4 - Prob. 98E Ch. 3.4 - Prob. 99E Ch. 3.4 - Prob. 100E Ch. 3.5 - (a) Find y by implicit differentiation. (b) Solve...Ch. 3.5 - Prob. 2E Ch. 3.5 - Prob. 3E Ch. 3.5 - Prob. 4E Ch. 3.5 - Prob. 5E Ch. 3.5 - Prob. 6E Ch. 3.5 - Prob. 7E Ch. 3.5 - Prob. 8E Ch. 3.5 - Prob. 9E Ch. 3.5 - Prob. 10E Ch. 3.5 - Find dy/dx by implicit differentiation. 11. y cos...Ch. 3.5 - Prob. 12E Ch. 3.5 - Prob. 13E Ch. 3.5 - Prob. 14E Ch. 3.5 - Prob. 15E Ch. 3.5 - Prob. 16E Ch. 3.5 - Prob. 17E Ch. 3.5 - Prob. 18E Ch. 3.5 - Prob. 19E Ch. 3.5 - Prob. 20E Ch. 3.5 - Prob. 21E Ch. 3.5 - Prob. 22E Ch. 3.5 - Prob. 23E Ch. 3.5 - Regard y as the independent variable and x as the...Ch. 3.5 - Prob. 25E Ch. 3.5 - Use implicit differentiation to find an equation...Ch. 3.5 - Prob. 27E Ch. 3.5 - Prob. 28E Ch. 3.5 - Prob. 29E Ch. 3.5 - Prob. 30E Ch. 3.5 - Prob. 31E Ch. 3.5 - Prob. 32E Ch. 3.5 - (a) The curve with equation y2 = 5x4 x2 is called...Ch. 3.5 - (a) The curve with equation y2 = x3 + 3x2 is...Ch. 3.5 - Prob. 35E Ch. 3.5 - Prob. 36E Ch. 3.5 - Prob. 37E Ch. 3.5 - Prob. 38E Ch. 3.5 - Prob. 39E Ch. 3.5 - If x2 + xy + y3 = 1, find the value of y at the...Ch. 3.5 - Prob. 43E Ch. 3.5 - Show by implicit differentiation that the tangent...Ch. 3.5 - Prob. 45E Ch. 3.5 - Prob. 46E Ch. 3.5 - Show, using implicit differentiation, that any...Ch. 3.5 - Prob. 48E Ch. 3.5 - Prob. 49E Ch. 3.5 - Prob. 50E Ch. 3.5 - Prob. 51E Ch. 3.5 - Prob. 52E Ch. 3.5 - Prob. 53E Ch. 3.5 - Prob. 54E Ch. 3.5 - Prob. 55E Ch. 3.5 - Prob. 56E Ch. 3.5 - Prob. 57E Ch. 3.5 - Prob. 58E Ch. 3.5 - Prob. 59E Ch. 3.5 - Prob. 60E Ch. 3.5 - Prob. 61E Ch. 3.5 - Prob. 62E Ch. 3.5 - Prob. 63E Ch. 3.5 - Prob. 64E Ch. 3.5 - Prob. 65E Ch. 3.5 - Prob. 66E Ch. 3.5 - Prob. 67E Ch. 3.5 - Prob. 68E Ch. 3.5 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 3.5 - Prob. 70E Ch. 3.5 - Prob. 71E Ch. 3.5 - The equation x2 xy + y2 = 3 re presents a...Ch. 3.5 - Prob. 74E Ch. 3.5 - Prob. 75E Ch. 3.5 - Prob. 76E Ch. 3.5 - (a) Suppose f is a one-to-one differentiable...Ch. 3.5 - Prob. 78E Ch. 3.5 - The Bessel function of order 0, y = J(x),...Ch. 3.5 - Prob. 80E Ch. 3.6 - Explain why the natural logarithmic function y =...Ch. 3.6 - Prob. 2E Ch. 3.6 - Prob. 3E Ch. 3.6 - Differentiate the function. f(x) = ln(sin2x)Ch. 3.6 - Differentiate the function. f(x)=ln1x Ch. 3.6 - Prob. 6E Ch. 3.6 - Differentiate the function. f(x) = log10(1 + cos...Ch. 3.6 - Prob. 8E Ch. 3.6 - Prob. 9E Ch. 3.6 - Prob. 10E Ch. 3.6 - Prob. 11E Ch. 3.6 - Prob. 12E Ch. 3.6 - Prob. 13E Ch. 3.6 - Prob. 14E Ch. 3.6 - Prob. 15E Ch. 3.6 - Prob. 16E Ch. 3.6 - Differentiate the function. T(z) = 2z log2z Ch. 3.6 - Differentiate the function. y = ln(csc x cot x)Ch. 3.6 - Differentiate the function. y = ln(ex + xex)Ch. 3.6 - Prob. 20E Ch. 3.6 - Differentiate the function. y = tan[ln(ax + b)]Ch. 3.6 - Differentiate the function. y = log2 (x log5 x)Ch. 3.6 - Prob. 23E Ch. 3.6 - Prob. 24E Ch. 3.6 - Prob. 25E Ch. 3.6 - Prob. 26E Ch. 3.6 - Prob. 27E Ch. 3.6 - Prob. 28E Ch. 3.6 - Prob. 29E Ch. 3.6 - Prob. 30E Ch. 3.6 - Prob. 31E Ch. 3.6 - Prob. 32E Ch. 3.6 - Prob. 33E Ch. 3.6 - Find an equation of the tangent line to the curve...Ch. 3.6 - If f(x) = sin x + ln x, find f(x). Check that your...Ch. 3.6 - Prob. 36E Ch. 3.6 - Prob. 37E Ch. 3.6 - Let f(x) = logb (3x2 2). For what value of b is...Ch. 3.6 - Prob. 39E Ch. 3.6 - Prob. 40E Ch. 3.6 - Prob. 41E Ch. 3.6 - Prob. 42E Ch. 3.6 - Prob. 43E Ch. 3.6 - Prob. 44E Ch. 3.6 - Prob. 45E Ch. 3.6 - Prob. 46E Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 48E Ch. 3.6 - Use logarithmic differentiation to find the...Ch. 3.6 - Prob. 50E Ch. 3.6 - Prob. 51E Ch. 3.6 - Prob. 52E Ch. 3.6 - Prob. 53E Ch. 3.6 - Prob. 54E Ch. 3.6 - Prob. 55E Ch. 3.6 - Prob. 56E Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - A particle moves according to a law of motion s =...Ch. 3.7 - Prob. 4E Ch. 3.7 - Graphs of the velocity functions of two particles...Ch. 3.7 - Graphs of the position functions of two particles...Ch. 3.7 - The height (in meters) of a projectile shot...Ch. 3.7 - Prob. 8E Ch. 3.7 - Prob. 9E Ch. 3.7 - Prob. 10E Ch. 3.7 - Prob. 11E Ch. 3.7 - Prob. 12E Ch. 3.7 - Prob. 13E Ch. 3.7 - A stone is dropped into a lake, creating a...Ch. 3.7 - A spherical balloon is being inflated. Find the...Ch. 3.7 - Prob. 16E Ch. 3.7 - The mass of the part of a metal rod that lies...Ch. 3.7 - If a tank holds 5000 gallons of water, which...Ch. 3.7 - Prob. 19E Ch. 3.7 - Prob. 20E Ch. 3.7 - Prob. 21E Ch. 3.7 - Prob. 22E Ch. 3.7 - Boyles Law states that when a sample of gas is...Ch. 3.7 - If, in Example 4, one molecule of the product C is...Ch. 3.7 - Prob. 25E Ch. 3.7 - The number of yeast cells in a laboratory culture...Ch. 3.7 - The table shows how the average age of first...Ch. 3.7 - Refer to the law of laminar flow given in Example...Ch. 3.7 - Prob. 30E Ch. 3.7 - Prob. 31E Ch. 3.7 - Prob. 32E Ch. 3.7 - If p(x) is the total value of the production when...Ch. 3.7 - Prob. 34E Ch. 3.7 - Prob. 35E Ch. 3.7 - Prob. 36E Ch. 3.7 - Prob. 37E Ch. 3.7 - Prob. 38E Ch. 3.7 - In the study of ecosystems, predator-prey models...Ch. 3.8 - A population of protozoa develops with a constant...Ch. 3.8 - A common inhabitant of human intestines is the...Ch. 3.8 - A bacteria culture initially contains 100 cells...Ch. 3.8 - A bacteria culture grows with constant relative...Ch. 3.8 - Prob. 5E Ch. 3.8 - Prob. 6E Ch. 3.8 - Prob. 7E Ch. 3.8 - Strontium-90 has a half-life of 28 days. (a) A...Ch. 3.8 - The half-life of cesium-137 is 30 years. Suppose...Ch. 3.8 - Prob. 10E Ch. 3.8 - Scientists can determine the age of ancient...Ch. 3.8 - Dinosaur fossils are too old to be reliably dated...Ch. 3.8 - Prob. 13E Ch. 3.8 - Prob. 14E Ch. 3.8 - Prob. 15E Ch. 3.8 - Prob. 16E Ch. 3.8 - Prob. 17E Ch. 3.8 - Prob. 18E Ch. 3.8 - Prob. 19E Ch. 3.8 - (a) If 1000 is borrowed at 8% interest, find the...Ch. 3.8 - (a) If 3000 is invested at 5% interest, find the...Ch. 3.8 - Prob. 22E Ch. 3.9 - Prob. 1E Ch. 3.9 - (a) If A is the area of a circle with radius r and...Ch. 3.9 - Each side of a square is increasing at a rate of 6...Ch. 3.9 - The length of a rectangle is increasing at a rate...Ch. 3.9 - Prob. 5E Ch. 3.9 - The radius of a sphere is increasing at a rate of...Ch. 3.9 - Prob. 7E Ch. 3.9 - The area of a triangle with sides of lengths a and...Ch. 3.9 - Prob. 9E Ch. 3.9 - Suppose 4x2 + 9y2 = 36, where x and y are...Ch. 3.9 - Prob. 11E Ch. 3.9 - Prob. 12E Ch. 3.9 - Prob. 13E Ch. 3.9 - Prob. 14E Ch. 3.9 - Prob. 15E Ch. 3.9 - Prob. 16E Ch. 3.9 - Prob. 17E Ch. 3.9 - A spotlight on the ground shines on a wall 12m...Ch. 3.9 - Prob. 19E Ch. 3.9 - Prob. 20E Ch. 3.9 - Prob. 21E Ch. 3.9 - Prob. 22E Ch. 3.9 - Prob. 23E Ch. 3.9 - Prob. 24E Ch. 3.9 - Prob. 25E Ch. 3.9 - Prob. 26E Ch. 3.9 - A water trough is 10m long and a cross-section has...Ch. 3.9 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 3.9 - Prob. 29E Ch. 3.9 - A kite 100ft above the ground moves horizontally...Ch. 3.9 - Prob. 31E Ch. 3.9 - Prob. 32E Ch. 3.9 - Prob. 33E Ch. 3.9 - According to the model we used to solve Example 2,...Ch. 3.9 - Prob. 35E Ch. 3.9 - Prob. 36E Ch. 3.9 - Prob. 37E Ch. 3.9 - When air expands adiabatically (without gaining or...Ch. 3.9 - Prob. 39E Ch. 3.9 - Prob. 40E Ch. 3.9 - Prob. 41E Ch. 3.9 - Two carts, A and B, are connected by a rope 39 ft...Ch. 3.9 - A television camera is positioned 4000 ft from the...Ch. 3.9 - A lighthouse is located on a small island 3 km...Ch. 3.9 - Prob. 45E Ch. 3.9 - Prob. 46E Ch. 3.9 - Prob. 47E Ch. 3.9 - Prob. 48E Ch. 3.9 - A runner sprints around a circular track of radius...Ch. 3.9 - Prob. 50E Ch. 3.10 - Prob. 1E Ch. 3.10 - Prob. 2E Ch. 3.10 - Find the linearization L(x) of the function at n....Ch. 3.10 - Prob. 4E Ch. 3.10 - Find the linear approximation of the function...Ch. 3.10 - Find the linear approximation of the function...Ch. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Verify the given linear approximation at a = 0....Ch. 3.10 - Find the differential of each function. 11. (a) y...Ch. 3.10 - Find the differential of each function. 12. (a)...Ch. 3.10 - Find the differential of each function. 13. (a)...Ch. 3.10 - Find the differential of each function. 14. (a) y...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - (a) Find the differential dy and (b) evaluate dy...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Compute y and dy for the given values of x and dx...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Use a linear approximation (or differentials) to...Ch. 3.10 - Prob. 28E Ch. 3.10 - Explain, in terms of linear approximations or...Ch. 3.10 - Prob. 30E Ch. 3.10 - Explain, in terms of linear approximations or...Ch. 3.10 - Prob. 32E Ch. 3.10 - The edge of a cube was found to be 30 cm with a...Ch. 3.10 - The radius of a circular disk is given as 24 cm...Ch. 3.10 - Prob. 35E Ch. 3.10 - Use differentials to estimate the amount of paint...Ch. 3.10 - Prob. 37E Ch. 3.10 - Prob. 38E Ch. 3.10 - If a current I passes through a resistor with...Ch. 3.10 - When blood flows along a blood vessel, the flux F...Ch. 3.10 - Prob. 41E Ch. 3.10 - Prob. 42E Ch. 3.10 - Suppose that the only information we have about a...Ch. 3.10 - Prob. 44E Ch. 3.11 - Prob. 1E Ch. 3.11 - Prob. 2E Ch. 3.11 - Find the numerical value of each expression. 3....Ch. 3.11 - Prob. 4E Ch. 3.11 - Prob. 5E Ch. 3.11 - Prob. 6E Ch. 3.11 - Prob. 7E Ch. 3.11 - Prob. 8E Ch. 3.11 - Prob. 9E Ch. 3.11 - Prob. 10E Ch. 3.11 - Prob. 11E Ch. 3.11 - Prob. 12E Ch. 3.11 - Prove the identity. 13. coth2x 1 = csch2x Ch. 3.11 - Prob. 14E Ch. 3.11 - Prob. 15E Ch. 3.11 - Prob. 16E Ch. 3.11 - Prob. 17E Ch. 3.11 - Prob. 18E Ch. 3.11 - Prob. 19E Ch. 3.11 - Prob. 20E Ch. 3.11 - Prob. 21E Ch. 3.11 - Prob. 22E Ch. 3.11 - Use the definitions of the hyperbolic functions to...Ch. 3.11 - Prob. 24E Ch. 3.11 - Prob. 25E Ch. 3.11 - Prob. 26E Ch. 3.11 - Prob. 27E Ch. 3.11 - Prob. 28E Ch. 3.11 - Prob. 29E Ch. 3.11 - Prob. 30E Ch. 3.11 - Prob. 31E Ch. 3.11 - Prob. 32E Ch. 3.11 - Prob. 33E Ch. 3.11 - Prob. 34E Ch. 3.11 - Prob. 35E Ch. 3.11 - Prob. 36E Ch. 3.11 - Prob. 37E Ch. 3.11 - Prob. 38E Ch. 3.11 - Prob. 39E Ch. 3.11 - Prob. 40E Ch. 3.11 - Prob. 41E Ch. 3.11 - Prob. 42E Ch. 3.11 - Prob. 43E Ch. 3.11 - Prob. 44E Ch. 3.11 - Prob. 45E Ch. 3.11 - Prob. 46E Ch. 3.11 - Show that ddx arctan(tanh x) = sech 2x.Ch. 3.11 - Prob. 48E Ch. 3.11 - Prob. 49E Ch. 3.11 - A flexible cable always hangs in the shape of a...Ch. 3.11 - Prob. 51E Ch. 3.11 - Prob. 52E Ch. 3.11 - Prob. 53E Ch. 3.11 - Prob. 54E Ch. 3.11 - Prob. 55E Ch. 3.11 - Prob. 56E Ch. 3.11 - Prob. 57E Ch. 3.11 - Prob. 58E Ch. 3 - State each differentiation rule both in symbols...Ch. 3 - Prob. 2RCC Ch. 3 - Prob. 3RCC Ch. 3 - Prob. 4RCC Ch. 3 - Give several examples of how the derivative can be...Ch. 3 - Prob. 6RCC Ch. 3 - Prob. 7RCC Ch. 3 - Prob. 1RQ Ch. 3 - Prob. 2RQ Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 4RQ Ch. 3 - Prob. 5RQ Ch. 3 - Prob. 6RQ Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 8RQ Ch. 3 - Prob. 9RQ Ch. 3 - Prob. 10RQ Ch. 3 - Prob. 11RQ Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 13RQ Ch. 3 - Prob. 14RQ Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Prob. 1RE Ch. 3 - Prob. 2RE Ch. 3 - Prob. 3RE Ch. 3 - Prob. 4RE Ch. 3 - Prob. 5RE Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Prob. 8RE Ch. 3 - Prob. 9RE Ch. 3 - Prob. 10RE Ch. 3 - Prob. 11RE Ch. 3 - Prob. 12RE Ch. 3 - Prob. 13RE Ch. 3 - Prob. 14RE Ch. 3 - Prob. 15RE Ch. 3 - Prob. 16RE Ch. 3 - Prob. 17RE Ch. 3 - Prob. 18RE Ch. 3 - Prob. 19RE Ch. 3 - Prob. 20RE Ch. 3 - Prob. 21RE Ch. 3 - Prob. 22RE Ch. 3 - Prob. 23RE Ch. 3 - Prob. 24RE Ch. 3 - Prob. 25RE Ch. 3 - Prob. 26RE Ch. 3 - Prob. 27RE Ch. 3 - Prob. 28RE Ch. 3 - Prob. 29RE Ch. 3 - Prob. 30RE Ch. 3 - Prob. 31RE Ch. 3 - Prob. 32RE Ch. 3 - Prob. 33RE Ch. 3 - Prob. 34RE Ch. 3 - Prob. 35RE Ch. 3 - Prob. 36RE Ch. 3 - Prob. 37RE Ch. 3 - Prob. 38RE Ch. 3 - Prob. 39RE Ch. 3 - Prob. 40RE Ch. 3 - Prob. 41RE Ch. 3 - Prob. 42RE Ch. 3 - Prob. 43RE Ch. 3 - Prob. 44RE Ch. 3 - Prob. 45RE Ch. 3 - Prob. 46RE Ch. 3 - Prob. 47RE Ch. 3 - Prob. 48RE Ch. 3 - Prob. 49RE Ch. 3 - Prob. 50RE Ch. 3 - Prob. 51RE Ch. 3 - Prob. 52RE Ch. 3 - Prob. 53RE Ch. 3 - Prob. 54RE Ch. 3 - Use mathematical induction (page 72) to show that...Ch. 3 - Prob. 56RE Ch. 3 - Prob. 57RE Ch. 3 - Prob. 58RE Ch. 3 - Prob. 59RE Ch. 3 - Prob. 60RE Ch. 3 - Prob. 61RE Ch. 3 - Prob. 62RE Ch. 3 - Prob. 63RE Ch. 3 - (a) If f(x) = 4x tan x, /2 x /2, find f and f....Ch. 3 - Prob. 65RE Ch. 3 - Prob. 66RE Ch. 3 - Prob. 67RE Ch. 3 - Prob. 68RE Ch. 3 - Prob. 69RE Ch. 3 - If f and g are the functions whose graphs are...Ch. 3 - Prob. 71RE Ch. 3 - Prob. 72RE Ch. 3 - Prob. 73RE Ch. 3 - Prob. 74RE Ch. 3 - Prob. 75RE Ch. 3 - Prob. 76RE Ch. 3 - Prob. 77RE Ch. 3 - Prob. 78RE Ch. 3 - Prob. 79RE Ch. 3 - Prob. 80RE Ch. 3 - Prob. 81RE Ch. 3 - Prob. 82RE Ch. 3 - Prob. 83RE Ch. 3 - Prob. 84RE Ch. 3 - Find a parabola y = ax2 + bx + c that passes...Ch. 3 - Prob. 86RE Ch. 3 - Prob. 87RE Ch. 3 - Prob. 88RE Ch. 3 - A particle moves on a vertical line so that its...Ch. 3 - The volume of a right circular cone is V=13r2h,...Ch. 3 - Prob. 91RE Ch. 3 - Prob. 92RE Ch. 3 - Prob. 93RE Ch. 3 - Prob. 94RE Ch. 3 - Prob. 95RE Ch. 3 - Prob. 96RE Ch. 3 - Prob. 97RE Ch. 3 - Prob. 98RE Ch. 3 - A balloon is rising at a constant speed of 5 ft/s....Ch. 3 - Prob. 100RE Ch. 3 - The angle of elevation of the sun is decreasing at...Ch. 3 - Prob. 102RE Ch. 3 - Prob. 103RE Ch. 3 - Prob. 104RE Ch. 3 - Prob. 105RE Ch. 3 - Prob. 106RE Ch. 3 - Express the limit as a derivative and evaluate....Ch. 3 - Prob. 108RE Ch. 3 - Prob. 109RE Ch. 3 - Prob. 110RE Ch. 3 - Prob. 111RE Ch. 3 - Show that the length of the portion of any tangent...Ch. 3 - Prob. 1P Ch. 3 - Prob. 2P Ch. 3 - Prob. 3P Ch. 3 - Prob. 4P Ch. 3 - Prob. 5P Ch. 3 - Find the values of the constants a and b such that...Ch. 3 - Show that sin-1(tanh x) = tan1(sinh x).Ch. 3 - A car is traveling at night along a highway shaped...Ch. 3 - Prob. 9P Ch. 3 - Prob. 10P Ch. 3 - Prob. 11P Ch. 3 - Find all values of r such that the parabolas y =...Ch. 3 - Prob. 13P Ch. 3 - Prob. 14P Ch. 3 - The figure shows a rotating wheel with radius 40...Ch. 3 - Prob. 16P Ch. 3 - Prob. 17P Ch. 3 - Prob. 18P Ch. 3 - Prob. 19P Ch. 3 - Prob. 20P Ch. 3 - Prob. 21P Ch. 3 - Let P(x1, y1) be a point on the parabola y2 = 4px...Ch. 3 - Prob. 23P Ch. 3 - Prob. 24P Ch. 3 - Prob. 25P Ch. 3 - Prob. 27P Ch. 3 - Prob. 28P Ch. 3 - Prob. 29P Ch. 3 - Prob. 30P Ch. 3 - Find the two points on the curve y = x4 2x2 x...Ch. 3 - Prob. 32P Ch. 3 - A lattice point in the plane is a point with...Ch. 3 - Prob. 34P Ch. 3 - Prob. 35P

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The derivatives f ′ ( x ) and f ″ ( x ) .

Solution Summary: The author explains the derivatives of f(x), and the power rule.

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To find: The derivatives f′(x) and f″(x).

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Chapter 3 Solutions