Explanation Given: The function is y = ln sin x − 1 2 sin 2 x . Result used: Chain Rule If g is differentiable at t and f is differentiable at g ( t ) , then the composite function F = f ∘ g defined by F ( t ) = f ( g ( t ) ) is differentiable at x and F ′ is given by the product F ′ ( t ) = f ′ ( g ( t ) ) ⋅ g ′ ( t ) (1) Calculation: Obtain the derivative of y . y ′ = d d x ( y ) = d d x ( ln sin x − 1 2 sin 2 x ) = d d x ( ln sin x ) − d d x ( 1 2 sin 2 x ) = d d x ( ln sin x ) − 1 2 ( 2 ( sin x ) 2 − 1 ⋅ d d x ( sin x ) ) (2) Obtain the derivative d d x ( ln sin x ) by using the chain rule as shown in equation (1), Let h ( x ) = sin x and f ( u ) = ln u where u = h ( x ) . Apply the chain rule as shown in equation (1), d d x ( ln sin x ) = f ′ ( h ( x ) ) ⋅ h ′ ( x ) (3) The derivative f ′ ( h ( x ) ) is computed as follows, f ′ ( h ( x ) ) = f ′ ( u ) [ ∵ u = h ( x ) ] = d d u ( f ( u

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

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The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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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 derivative of y .

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