Single Variable Calculus
Single Variable Calculus
8th Edition
ISBN: 9781305266636
Author: James Stewart
Publisher: Cengage Learning
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Chapter 2.5, Problem 89E

If y = f(u) and u = g(x), where f and g are twice differentiable functions, show that

d 2 y d x 2 = d 2 y d u 2 ( d u d x ) 2 + d y d u d 2 u d x 2

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Chapter 2 Solutions

Single Variable Calculus

Ch. 2.1 - A curve has equation y = f(x). (a) Write an...Ch. 2.1 - Graph the curve y = sin x in the viewing...Ch. 2.1 - (a) Find the slope of the tangent line to the...Ch. 2.1 - (a) Find the slope of the tangent Line to the...Ch. 2.1 - Prob. 5ECh. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - Find an equation of the tangent line to the curve...Ch. 2.1 - Prob. 8ECh. 2.1 - Prob. 9ECh. 2.1 - Prob. 10E
Ch. 2.1 - (a) A particle starts by moving to the right along...Ch. 2.1 - Prob. 12ECh. 2.1 - Prob. 13ECh. 2.1 - If a rock is thrown upward on the planet Mars with...Ch. 2.1 - Prob. 15ECh. 2.1 - The displacement (in feet) of a particle moving in...Ch. 2.1 - Prob. 17ECh. 2.1 - Prob. 18ECh. 2.1 - Prob. 19ECh. 2.1 - Prob. 20ECh. 2.1 - If an equation of the tangent line to the curve y...Ch. 2.1 - Prob. 22ECh. 2.1 - Sketch the graph of a function f for which f(0) =...Ch. 2.1 - Sketch the graph of a function g for which g(0) =...Ch. 2.1 - Sketch the graph of a function g that is...Ch. 2.1 - Sketch the graph of a function f where the domain...Ch. 2.1 - Prob. 27ECh. 2.1 - Prob. 28ECh. 2.1 - (a) If F(x) = 5x/(1 + x2), find F(2) and use it to...Ch. 2.1 - (a) If G(x) = 4x2 x3, find G(a) and use it to...Ch. 2.1 - Find f(a). 31. f(t) = 3x2 4x + 1Ch. 2.1 - Prob. 32ECh. 2.1 - Prob. 33ECh. 2.1 - Prob. 34ECh. 2.1 - Prob. 35ECh. 2.1 - Find f(a). 36. f(x)=41xCh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 38ECh. 2.1 - Each limit represents the derivative of some...Ch. 2.1 - Prob. 40ECh. 2.1 - Prob. 41ECh. 2.1 - Prob. 42ECh. 2.1 - Prob. 43ECh. 2.1 - Prob. 44ECh. 2.1 - Prob. 45ECh. 2.1 - A roast turkey is taken from an oven when its...Ch. 2.1 - Researchers measured the average blood alcohol...Ch. 2.1 - Prob. 48ECh. 2.1 - Prob. 49ECh. 2.1 - Prob. 50ECh. 2.1 - The cost (in dollars) of producing x units of a...Ch. 2.1 - If a cylindrical tank holds 100.000 gallons of...Ch. 2.1 - Prob. 53ECh. 2.1 - The number of bacteria after t hours in a...Ch. 2.1 - Let H(t) be the daily cost (in dollar) to heat an...Ch. 2.1 - Prob. 56ECh. 2.1 - The quantity of oxygen that can dissolve in water...Ch. 2.1 - The graph shows the influence of the temperature T...Ch. 2.1 - Prob. 59ECh. 2.1 - Determine whether f(0) exists. 60....Ch. 2.1 - Prob. 61ECh. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Use the given graph to estimate the value of each...Ch. 2.2 - Match the graph of each function in (a)(d) with...Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Prob. 7ECh. 2.2 - Prob. 8ECh. 2.2 - Prob. 9ECh. 2.2 - Prob. 10ECh. 2.2 - Trace or copy the graph of the given function f....Ch. 2.2 - Prob. 12ECh. 2.2 - A rechargeable battery is plugged into a charger....Ch. 2.2 - The graph (from the US Department of Energy) shows...Ch. 2.2 - Prob. 15ECh. 2.2 - Prob. 16ECh. 2.2 - Prob. 17ECh. 2.2 - Let f(x) = x3. (a) Estimate the values of f(0),...Ch. 2.2 - Prob. 19ECh. 2.2 - Prob. 20ECh. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Prob. 22ECh. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Prob. 24ECh. 2.2 - Prob. 25ECh. 2.2 - Prob. 26ECh. 2.2 - Prob. 27ECh. 2.2 - Prob. 28ECh. 2.2 - Find the derivative of the function using the...Ch. 2.2 - Prob. 30ECh. 2.2 - Prob. 31ECh. 2.2 - (a) If f(x) = x + 1/x, find f(x). (b) Check to see...Ch. 2.2 - The unemployment rate U(t) varies with time. The...Ch. 2.2 - Prob. 34ECh. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Let P represent the percentage of a citys...Ch. 2.2 - Suppose N is the number of people in the United...Ch. 2.2 - The graph of f is given. State, with reasons, the...Ch. 2.2 - Prob. 40ECh. 2.2 - Prob. 41ECh. 2.2 - Prob. 42ECh. 2.2 - Prob. 43ECh. 2.2 - Prob. 44ECh. 2.2 - Prob. 45ECh. 2.2 - Prob. 46ECh. 2.2 - The figure shows the graphs of f, f, and f ....Ch. 2.2 - The figure shows graphs of f, f and f. Identify...Ch. 2.2 - The figure shows the graphs of three functions....Ch. 2.2 - Prob. 50ECh. 2.2 - Use the definition of a derivative to find f(x)...Ch. 2.2 - Prob. 52ECh. 2.2 - If f(x) = 2x2 x3, find f(x), f(x), f(x), and...Ch. 2.2 - (a) The graph of a position function of a car is...Ch. 2.2 - Let f(x)=x3. (a) If a 0, use Equation 2.1.5 to...Ch. 2.2 - Prob. 56ECh. 2.2 - Show that the function f(x) = |x 6| is not...Ch. 2.2 - Prob. 58ECh. 2.2 - (a) Sketch the graph of the function f(x) = x|x|....Ch. 2.2 - Prob. 60ECh. 2.2 - Prob. 61ECh. 2.2 - Prob. 62ECh. 2.2 - Nick starts jogging and runs faster and faster for...Ch. 2.2 - When you turn on a hot-water faucet, the...Ch. 2.2 - Let be the tangent line to the parabola y = x2 at...Ch. 2.3 - Differentiate the function. 1. f(x) = 240Ch. 2.3 - Differentiate the function. 2. f(x) = 2Ch. 2.3 - Differentiate the function. 3. f(x) = 5.2x + 2.3Ch. 2.3 - Prob. 4ECh. 2.3 - Prob. 5ECh. 2.3 - Differentiate the function. 6. f(t) = 1.4t5 2.5t2...Ch. 2.3 - Prob. 7ECh. 2.3 - Prob. 8ECh. 2.3 - Prob. 9ECh. 2.3 - Prob. 10ECh. 2.3 - Differentiate the function. 11. F(r)=5r3Ch. 2.3 - Prob. 12ECh. 2.3 - Prob. 13ECh. 2.3 - Prob. 14ECh. 2.3 - Prob. 15ECh. 2.3 - Differentiate the function. 16. S(R) = 4R2Ch. 2.3 - Prob. 17ECh. 2.3 - Prob. 18ECh. 2.3 - Prob. 19ECh. 2.3 - Prob. 20ECh. 2.3 - Prob. 21ECh. 2.3 - Differentiate the function. 22. D(t)=1+16t2(4t)3Ch. 2.3 - Prob. 23ECh. 2.3 - Find the derivative of the function F(x)=x45x3+xx2...Ch. 2.3 - Differentiate. 25. f(x) = (5x2 2)(x3 + 3x)Ch. 2.3 - Prob. 26ECh. 2.3 - Prob. 27ECh. 2.3 - Prob. 28ECh. 2.3 - Prob. 29ECh. 2.3 - Prob. 30ECh. 2.3 - Prob. 31ECh. 2.3 - Prob. 32ECh. 2.3 - Prob. 33ECh. 2.3 - Prob. 34ECh. 2.3 - Differentiate. 35. y=sss2Ch. 2.3 - Prob. 36ECh. 2.3 - Prob. 37ECh. 2.3 - Prob. 38ECh. 2.3 - Prob. 39ECh. 2.3 - Differentiate. 40. A(v) = v2/3(2v2 + 1 v2)Ch. 2.3 - Prob. 41ECh. 2.3 - Prob. 42ECh. 2.3 - Prob. 43ECh. 2.3 - Prob. 44ECh. 2.3 - The general polynomial of degree n has the form...Ch. 2.3 - Prob. 46ECh. 2.3 - Prob. 47ECh. 2.3 - Prob. 48ECh. 2.3 - Prob. 49ECh. 2.3 - Prob. 50ECh. 2.3 - Prob. 51ECh. 2.3 - Find an equation of the tangent line to the curve...Ch. 2.3 - Prob. 53ECh. 2.3 - (a) The curve y = x/(1 + x2) is called a...Ch. 2.3 - Prob. 55ECh. 2.3 - Prob. 56ECh. 2.3 - Prob. 57ECh. 2.3 - Prob. 58ECh. 2.3 - Prob. 59ECh. 2.3 - Prob. 60ECh. 2.3 - Prob. 61ECh. 2.3 - Prob. 62ECh. 2.3 - Prob. 63ECh. 2.3 - Prob. 64ECh. 2.3 - Biologists have proposed a cubic polynomial to...Ch. 2.3 - Prob. 66ECh. 2.3 - Prob. 67ECh. 2.3 - Prob. 68ECh. 2.3 - Prob. 69ECh. 2.3 - Prob. 70ECh. 2.3 - If f(x)=xg(x), where g(4) = 8 and g(4) = 7, find...Ch. 2.3 - Prob. 72ECh. 2.3 - Prob. 73ECh. 2.3 - Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F...Ch. 2.3 - Prob. 75ECh. 2.3 - Prob. 76ECh. 2.3 - Prob. 77ECh. 2.3 - For what values of x does the graph of f(x) = x3 +...Ch. 2.3 - Prob. 79ECh. 2.3 - Prob. 80ECh. 2.3 - Find equations of both lines that are tangent to...Ch. 2.3 - Prob. 82ECh. 2.3 - Prob. 83ECh. 2.3 - Where does the normal line to the parabola y = x2 ...Ch. 2.3 - Prob. 85ECh. 2.3 - (a) Find equations of both lines through the point...Ch. 2.3 - Prob. 87ECh. 2.3 - Find the nth derivative of each function by...Ch. 2.3 - Find a second-degree polynomial P such that P(2) =...Ch. 2.3 - The equation y + y 2y = x3 is called a...Ch. 2.3 - Prob. 91ECh. 2.3 - Find a parabola with equation y = ax2 + bx + c...Ch. 2.3 - In this exercise we estimate the rate at which the...Ch. 2.3 - Prob. 94ECh. 2.3 - Prob. 95ECh. 2.3 - Prob. 96ECh. 2.3 - Prob. 97ECh. 2.3 - Prob. 98ECh. 2.3 - Prob. 99ECh. 2.3 - Prob. 100ECh. 2.3 - For what values of a and b is the line 2x + y = b...Ch. 2.3 - Prob. 102ECh. 2.3 - Find the value of c such that the line y=32x+6 is...Ch. 2.3 - Let f(x)={x2ifx2mx+bifx2 Find the values of m and...Ch. 2.3 - An easy proof of the Quotient Rule can he given if...Ch. 2.3 - Prob. 106ECh. 2.3 - Evaluate limx1x10001x1.Ch. 2.3 - Prob. 108ECh. 2.3 - Prob. 109ECh. 2.3 - Sketch the parabolas y = x2 and y = x2 2x + 2. Do...Ch. 2.4 - Differentiate. 1. f(x) = x2 sin xCh. 2.4 - Prob. 2ECh. 2.4 - Differentiate. 3. f(x) = 3 cot x 2 cos xCh. 2.4 - Prob. 4ECh. 2.4 - Differentiate. 5. y = sec tanCh. 2.4 - Differentiate. 6. g(t) = 4 sec t + tan tCh. 2.4 - Prob. 7ECh. 2.4 - Differentiate. 8. y = u(a cos u + b cot u)Ch. 2.4 - Prob. 9ECh. 2.4 - Prob. 10ECh. 2.4 - Differentiate. 11. f()=sin1+cosCh. 2.4 - Differentiate. 12. y=cosx1sinxCh. 2.4 - Prob. 13ECh. 2.4 - Prob. 14ECh. 2.4 - Differentiate. 15. f() = cos sinCh. 2.4 - Prob. 16ECh. 2.4 - Prob. 17ECh. 2.4 - Prob. 18ECh. 2.4 - Prob. 19ECh. 2.4 - Prob. 20ECh. 2.4 - Prob. 21ECh. 2.4 - Prob. 22ECh. 2.4 - Prob. 23ECh. 2.4 - Prob. 24ECh. 2.4 - Prob. 25ECh. 2.4 - Prob. 26ECh. 2.4 - (a) If f(x) = sec x x, find f(x). (b) Check to...Ch. 2.4 - (a) If f(x)=xsinx, find f(x). (b) Check to see...Ch. 2.4 - Prob. 29ECh. 2.4 - Prob. 30ECh. 2.4 - Prob. 31ECh. 2.4 - Prob. 32ECh. 2.4 - For what values of x does the graph of f(x) = x +...Ch. 2.4 - Prob. 34ECh. 2.4 - Prob. 35ECh. 2.4 - Prob. 36ECh. 2.4 - Prob. 37ECh. 2.4 - Prob. 38ECh. 2.4 - Prob. 39ECh. 2.4 - Prob. 40ECh. 2.4 - Prob. 41ECh. 2.4 - Prob. 42ECh. 2.4 - Prob. 43ECh. 2.4 - Prob. 44ECh. 2.4 - Prob. 45ECh. 2.4 - Find the limit. 46. limx0cscxsin(sinx)Ch. 2.4 - Prob. 47ECh. 2.4 - Prob. 48ECh. 2.4 - Prob. 49ECh. 2.4 - Prob. 50ECh. 2.4 - Find the given derivative by finding the first few...Ch. 2.4 - Find the given derivative by finding the first few...Ch. 2.4 - Prob. 53ECh. 2.4 - Prob. 54ECh. 2.4 - Prob. 55ECh. 2.4 - A semicircle with diameter PQ sits on an isosceles...Ch. 2.4 - Prob. 57ECh. 2.4 - Prob. 58ECh. 2.5 - Write the composite function in the form f(g(x))....Ch. 2.5 - Prob. 2ECh. 2.5 - Prob. 3ECh. 2.5 - Prob. 4ECh. 2.5 - Write the composite function in the form f(g(x))....Ch. 2.5 - Write the composite function in the form f(g(x))....Ch. 2.5 - Prob. 7ECh. 2.5 - Prob. 8ECh. 2.5 - Prob. 9ECh. 2.5 - Prob. 10ECh. 2.5 - Prob. 11ECh. 2.5 - Prob. 12ECh. 2.5 - Find the derivative of the function. 13. f() =...Ch. 2.5 - Prob. 14ECh. 2.5 - Prob. 15ECh. 2.5 - Find the derivative of the function. 16. f(t) = t...Ch. 2.5 - Prob. 17ECh. 2.5 - Prob. 18ECh. 2.5 - Prob. 19ECh. 2.5 - Prob. 20ECh. 2.5 - Prob. 21ECh. 2.5 - Prob. 22ECh. 2.5 - Prob. 23ECh. 2.5 - Find the derivative of the function. 24....Ch. 2.5 - Prob. 25ECh. 2.5 - Prob. 26ECh. 2.5 - Prob. 27ECh. 2.5 - Prob. 28ECh. 2.5 - Prob. 29ECh. 2.5 - Prob. 30ECh. 2.5 - Prob. 31ECh. 2.5 - Find the derivative of the function. 32. J() =...Ch. 2.5 - Prob. 33ECh. 2.5 - Prob. 34ECh. 2.5 - Prob. 35ECh. 2.5 - Find the derivative of the function. 36. y=xsin1xCh. 2.5 - Prob. 37ECh. 2.5 - Prob. 38ECh. 2.5 - Find the derivative of the function. 39. f(t) =...Ch. 2.5 - Prob. 40ECh. 2.5 - Prob. 41ECh. 2.5 - Prob. 42ECh. 2.5 - Find the derivative of the function. 43. g(x) =...Ch. 2.5 - Find the derivative of the function. 44. y =...Ch. 2.5 - Prob. 45ECh. 2.5 - Prob. 46ECh. 2.5 - Find y and y. 47. y = cos(sin 3)Ch. 2.5 - Prob. 48ECh. 2.5 - Prob. 49ECh. 2.5 - Prob. 50ECh. 2.5 - Prob. 51ECh. 2.5 - Prob. 52ECh. 2.5 - Prob. 53ECh. 2.5 - Prob. 54ECh. 2.5 - Prob. 55ECh. 2.5 - Prob. 56ECh. 2.5 - Prob. 57ECh. 2.5 - Prob. 58ECh. 2.5 - Prob. 59ECh. 2.5 - Prob. 60ECh. 2.5 - Prob. 61ECh. 2.5 - Prob. 62ECh. 2.5 - Prob. 63ECh. 2.5 - Prob. 64ECh. 2.5 - Prob. 65ECh. 2.5 - Prob. 66ECh. 2.5 - Prob. 67ECh. 2.5 - Prob. 68ECh. 2.5 - Prob. 69ECh. 2.5 - If g is a twice differentiable function and f(x) =...Ch. 2.5 - Prob. 71ECh. 2.5 - Prob. 72ECh. 2.5 - Find the given derivative by finding the first few...Ch. 2.5 - Find the given derivative by finding the first few...Ch. 2.5 - The displacement of a particle on a vibrating...Ch. 2.5 - Prob. 76ECh. 2.5 - A Cepheid variable star is a star whose brightness...Ch. 2.5 - In Example 1.3.4 we arrived at a model for the...Ch. 2.5 - Prob. 79ECh. 2.5 - Prob. 80ECh. 2.5 - Prob. 83ECh. 2.5 - Prob. 84ECh. 2.5 - Prob. 85ECh. 2.5 - Suppose y = f(x) is a curve that always lies above...Ch. 2.5 - Use the Chain Rule to show that if is measured in...Ch. 2.5 - Prob. 88ECh. 2.5 - If y = f(u) and u = g(x), where f and g are twice...Ch. 2.5 - Prob. 90ECh. 2.6 - (a) Find y by implicit differentiation. (b) Solve...Ch. 2.6 - (a) Find y by implicit differentiation. (b) Solve...Ch. 2.6 - (a) Find y by implicit differentiation. (b) Solve...Ch. 2.6 - Prob. 4ECh. 2.6 - Prob. 5ECh. 2.6 - Prob. 6ECh. 2.6 - Prob. 7ECh. 2.6 - Prob. 8ECh. 2.6 - Prob. 9ECh. 2.6 - Prob. 10ECh. 2.6 - Prob. 11ECh. 2.6 - Prob. 12ECh. 2.6 - Prob. 13ECh. 2.6 - Prob. 14ECh. 2.6 - Prob. 15ECh. 2.6 - Find dy/dx by implicit differentiation. 16....Ch. 2.6 - Prob. 17ECh. 2.6 - Find dy/dx by implicit differentiation. 18. x sin...Ch. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - Prob. 21ECh. 2.6 - Prob. 22ECh. 2.6 - Regard y as the independent variable and x as the...Ch. 2.6 - Prob. 24ECh. 2.6 - Prob. 25ECh. 2.6 - Prob. 26ECh. 2.6 - Prob. 27ECh. 2.6 - Prob. 28ECh. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Prob. 31ECh. 2.6 - Prob. 32ECh. 2.6 - (a) The curve with equation y2 = 5x4 x2 is called...Ch. 2.6 - (a) The curve with equation y2 = x3 + 3x2 is...Ch. 2.6 - Prob. 35ECh. 2.6 - Prob. 36ECh. 2.6 - Prob. 37ECh. 2.6 - Prob. 38ECh. 2.6 - If xy + y3 = 1, find the value of y at the point...Ch. 2.6 - If x2 + xy + y3 = 1, find the value of y at the...Ch. 2.6 - Find the points on the lemniscate in Exercise 31...Ch. 2.6 - Prob. 44ECh. 2.6 - Prob. 45ECh. 2.6 - Show that the sum of the x- and y-intercepts of...Ch. 2.6 - Prob. 47ECh. 2.6 - The Power Rule can be proved using implicit...Ch. 2.6 - Prob. 49ECh. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Prob. 52ECh. 2.6 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 2.6 - Prob. 54ECh. 2.6 - Prob. 55ECh. 2.6 - Prob. 56ECh. 2.6 - The equation x2 xy + y2 = 3 represents a rotated...Ch. 2.6 - (a) Where does the normal line to the ellipse x2 ...Ch. 2.6 - Prob. 59ECh. 2.6 - Prob. 60ECh. 2.6 - Prob. 61ECh. 2.6 - The figure shows a lamp located three units to the...Ch. 2.7 - A particle moves according to a law of motion s =...Ch. 2.7 - A particle moves according to a law of motion s =...Ch. 2.7 - A particle moves according to a law of motion s =...Ch. 2.7 - A particle moves according to a law of motion s =...Ch. 2.7 - Prob. 5ECh. 2.7 - Graphs of the position functions of two particles...Ch. 2.7 - The height (in meters) of a projectile shot...Ch. 2.7 - If a ball is thrown vertically upward with a...Ch. 2.7 - If a rock is thrown vertically upward from the...Ch. 2.7 - Prob. 10ECh. 2.7 - (a) A company makes computer chips from square...Ch. 2.7 - Prob. 12ECh. 2.7 - (a) Find the average rate of change of the area of...Ch. 2.7 - A stone is dropped into a lake, creating a...Ch. 2.7 - A spherical balloon is being inflated. Find the...Ch. 2.7 - Prob. 16ECh. 2.7 - The mass of the part of a metal rod that lies...Ch. 2.7 - If a tank holds 5000 gallons of water, which...Ch. 2.7 - The quantity of charge Q in coulombs (C) that has...Ch. 2.7 - Newtons Law of Gravitation says that the magnitude...Ch. 2.7 - The force F acting on a body with mass m and...Ch. 2.7 - Some of the highest tides in the world occur in...Ch. 2.7 - Boyles Law states that when a sample of gas is...Ch. 2.7 - Prob. 24ECh. 2.7 - The table gives the population of the world P(t),...Ch. 2.7 - The table shows how the average age of first...Ch. 2.7 - Prob. 27ECh. 2.7 - Prob. 28ECh. 2.7 - Prob. 29ECh. 2.7 - The cost function for a certain commodity is...Ch. 2.7 - If p(x) is the total value of the production when...Ch. 2.7 - If R denotes the reaction of the body to some...Ch. 2.7 - Prob. 33ECh. 2.7 - Prob. 34ECh. 2.7 - In the study of ecosystems, predator-prey models...Ch. 2.7 - Prob. 36ECh. 2.8 - Prob. 1ECh. 2.8 - (a) If A is the area of a circle with radius r and...Ch. 2.8 - Prob. 3ECh. 2.8 - Prob. 4ECh. 2.8 - Prob. 5ECh. 2.8 - The radius of a sphere is increasing at a rate of...Ch. 2.8 - Prob. 7ECh. 2.8 - The area of a triangle with sides of lengths a and...Ch. 2.8 - Prob. 9ECh. 2.8 - Prob. 10ECh. 2.8 - Prob. 11ECh. 2.8 - A particle is moving along a hyperbola xy = 8. As...Ch. 2.8 - (a) What quantities are given in the problem? (b)...Ch. 2.8 - Prob. 14ECh. 2.8 - (a) What quantities are given in the problem? (b)...Ch. 2.8 - Prob. 16ECh. 2.8 - Two cars start moving from the same point. One...Ch. 2.8 - A spotlight on the ground shines on a wall 12 m...Ch. 2.8 - A man starts walking north at 4 ft/s from a point...Ch. 2.8 - A baseball diamond is a square with side 90 ft. A...Ch. 2.8 - The altitude of a triangle is increasing at a rate...Ch. 2.8 - A boat is pulled into a dock by a rope attached to...Ch. 2.8 - At noon, ship A is 100 km west of ship B. Ship A...Ch. 2.8 - A particle moves along the curve y = 2 sin(x/2)....Ch. 2.8 - Prob. 25ECh. 2.8 - A trough is 10 ft long and its ends have the shape...Ch. 2.8 - A water trough is 10 m long and a cross-section...Ch. 2.8 - A swimming pool is 20 ft wide, 40 ft long, 3 ft...Ch. 2.8 - Gravel is being dumped from a conveyor belt at a...Ch. 2.8 - A kite 100 ft above the ground moves horizontally...Ch. 2.8 - The sides of an equilateral triangle are...Ch. 2.8 - How fast is the angle between the ladder and the...Ch. 2.8 - Prob. 33ECh. 2.8 - Prob. 34ECh. 2.8 - If the minute hand of a clock has length r (in...Ch. 2.8 - Prob. 36ECh. 2.8 - Boyles Law states that when a sample of gas is...Ch. 2.8 - When air expands adiabatically (without gaining or...Ch. 2.8 - Prob. 39ECh. 2.8 - Brain weight B as a function of body weight W in...Ch. 2.8 - Prob. 41ECh. 2.8 - Two carts, A and B, are connected by a rope 39 ft...Ch. 2.8 - A television camera is positioned 4000 ft from the...Ch. 2.8 - A lighthouse is located on a small island 3 km...Ch. 2.8 - A plane flies horizontally at an altitude of 5 km...Ch. 2.8 - A Ferris wheel with a radius of 10 m is rotating...Ch. 2.8 - A plane flying with a constant speed of 300 km/h...Ch. 2.8 - Two people start from the same point. One walks...Ch. 2.8 - Prob. 49ECh. 2.8 - The minute hand on a watch is 8 mm long and the...Ch. 2.9 - Find the linearization L(x) of the function at a....Ch. 2.9 - Find the linearization L(x) of the function at a....Ch. 2.9 - Prob. 3ECh. 2.9 - Prob. 4ECh. 2.9 - Prob. 5ECh. 2.9 - Prob. 6ECh. 2.9 - Prob. 7ECh. 2.9 - Prob. 8ECh. 2.9 - Prob. 9ECh. 2.9 - Prob. 10ECh. 2.9 - Prob. 11ECh. 2.9 - Prob. 12ECh. 2.9 - Prob. 13ECh. 2.9 - Prob. 14ECh. 2.9 - (a) Find the differential dy and (b) evaluate dy...Ch. 2.9 - (a) Find the differential dy and (b) evaluate dy...Ch. 2.9 - Prob. 17ECh. 2.9 - (a) Find the differential dy and (b) evaluate dy...Ch. 2.9 - Prob. 19ECh. 2.9 - Prob. 20ECh. 2.9 - Prob. 21ECh. 2.9 - Prob. 22ECh. 2.9 - Use a linear approximation (or differentials) to...Ch. 2.9 - Use a linear approximation (or differentials) to...Ch. 2.9 - Prob. 25ECh. 2.9 - Use a linear approximation (or differentials) to...Ch. 2.9 - Use a linear approximation (or differentials) to...Ch. 2.9 - Use a linear approximation (or differentials) to...Ch. 2.9 - Prob. 29ECh. 2.9 - Explain, in terms of linear approximations or...Ch. 2.9 - The edge of a cube was found to be 30 cm with a...Ch. 2.9 - Prob. 32ECh. 2.9 - Prob. 33ECh. 2.9 - Prob. 34ECh. 2.9 - (a) Use differentials to find a formula for the...Ch. 2.9 - Prob. 36ECh. 2.9 - Prob. 37ECh. 2.9 - When blood flows along a blood vessel, the flux F...Ch. 2.9 - Prob. 39ECh. 2.9 - Prob. 40ECh. 2.9 - Suppose that the only information we have about a...Ch. 2.9 - Suppose that we dont have a formula for g(x) but...Ch. 2 - Write an expression for the slope of the tangent...Ch. 2 - Prob. 2RCCCh. 2 - If y = f(x) and x changes from x1 to x2, write...Ch. 2 - Define the derivative f(a). Discuss two ways of...Ch. 2 - Prob. 5RCCCh. 2 - Prob. 6RCCCh. 2 - Prob. 7RCCCh. 2 - Prob. 8RCCCh. 2 - Prob. 9RCCCh. 2 - Prob. 10RCCCh. 2 - Prob. 11RCCCh. 2 - Prob. 12RCCCh. 2 - Prob. 1RQCh. 2 - Prob. 2RQCh. 2 - Prob. 3RQCh. 2 - Prob. 4RQCh. 2 - Determine whether the statement is true or false....Ch. 2 - Prob. 6RQCh. 2 - Prob. 7RQCh. 2 - Determine whether the statement is true or false....Ch. 2 - Prob. 9RQCh. 2 - Prob. 10RQCh. 2 - Prob. 11RQCh. 2 - Prob. 12RQCh. 2 - Determine whether the statement is true or false....Ch. 2 - Prob. 14RQCh. 2 - Determine whether the statement is true or false....Ch. 2 - The displacement (in meters) of an object moving...Ch. 2 - Prob. 2RECh. 2 - Prob. 3RECh. 2 - Prob. 4RECh. 2 - The figure shows the graphs of f, f, and f....Ch. 2 - Find a function f and a number a such that...Ch. 2 - Prob. 7RECh. 2 - Prob. 8RECh. 2 - Prob. 9RECh. 2 - Prob. 10RECh. 2 - Prob. 11RECh. 2 - Prob. 12RECh. 2 - Prob. 13RECh. 2 - Prob. 14RECh. 2 - Prob. 15RECh. 2 - Prob. 16RECh. 2 - Prob. 17RECh. 2 - Calculate y. 18. y=(x+1x2)7Ch. 2 - Prob. 19RECh. 2 - Prob. 20RECh. 2 - Prob. 21RECh. 2 - Prob. 22RECh. 2 - Prob. 23RECh. 2 - Prob. 24RECh. 2 - Calculate y. 25. y=sec21+tan2Ch. 2 - Prob. 26RECh. 2 - Prob. 27RECh. 2 - Prob. 28RECh. 2 - Prob. 29RECh. 2 - Prob. 30RECh. 2 - Prob. 31RECh. 2 - Prob. 32RECh. 2 - Prob. 33RECh. 2 - Prob. 34RECh. 2 - Prob. 35RECh. 2 - Prob. 36RECh. 2 - Prob. 37RECh. 2 - Prob. 38RECh. 2 - Prob. 39RECh. 2 - Prob. 40RECh. 2 - Prob. 41RECh. 2 - Prob. 42RECh. 2 - Prob. 43RECh. 2 - Prob. 44RECh. 2 - Prob. 45RECh. 2 - Prob. 46RECh. 2 - Prob. 47RECh. 2 - Prob. 48RECh. 2 - Prob. 49RECh. 2 - Find equations of the tangent line and normal line...Ch. 2 - Prob. 51RECh. 2 - Prob. 52RECh. 2 - Prob. 53RECh. 2 - Prob. 54RECh. 2 - Find a parabola y = ax2 + bx + c that passes...Ch. 2 - Prob. 56RECh. 2 - Prob. 57RECh. 2 - Prob. 58RECh. 2 - Prob. 59RECh. 2 - Prob. 60RECh. 2 - Prob. 61RECh. 2 - Prob. 62RECh. 2 - Prob. 63RECh. 2 - Prob. 64RECh. 2 - Prob. 65RECh. 2 - Prob. 66RECh. 2 - Prob. 67RECh. 2 - Find f in terms of g. 68. f(x)=g(tanx)Ch. 2 - Prob. 69RECh. 2 - Find h in terms of f and g. 70. h(x)=f(x)g(x)Ch. 2 - Prob. 71RECh. 2 - Prob. 72RECh. 2 - A particle moves on a vertical line so that its...Ch. 2 - Prob. 74RECh. 2 - Prob. 75RECh. 2 - The cost, in dollars, of producing x units of a...Ch. 2 - Prob. 77RECh. 2 - A paper cup has the shape of a cone with height 10...Ch. 2 - A balloon is rising at a constant speed of 5 ft/s....Ch. 2 - A waterskier skis over the ramp shown in the...Ch. 2 - The angle of elevation of the sun is decreasing at...Ch. 2 - Prob. 82RECh. 2 - Prob. 83RECh. 2 - Evaluate dy if y = x3 2x2 + 1, x = 2, and dx =...Ch. 2 - Prob. 85RECh. 2 - Prob. 86RECh. 2 - Prob. 87RECh. 2 - Prob. 88RECh. 2 - Prob. 89RECh. 2 - Suppose f is a differentiable function such that...Ch. 2 - Prob. 91RECh. 2 - Show that the length of the portion of any tangent...Ch. 2 - Find points P and Q on the parabola y = 1 x2 so...Ch. 2 - Prob. 2PCh. 2 - Show that the tangent lines to the parabola y =...Ch. 2 - Prob. 4PCh. 2 - If f(x)=limtxsectsecxtx, find the value of f(/4).Ch. 2 - Find the values of the constants a and b such that...Ch. 2 - Prob. 7PCh. 2 - If f is differentiable at a, where a 0, evaluate...Ch. 2 - Prob. 9PCh. 2 - Find all values of c such that the parabolas y =...Ch. 2 - How many lines are tangent to both of the circles...Ch. 2 - If f(x)=x46+x45+21+x, calculate f(46)(3). Express...Ch. 2 - The figure shows a rotating wheel with radius 40...Ch. 2 - Tangent lines T1 and T2 are drawn at two points P1...Ch. 2 - Let T and N be the tangent and normal lines to the...Ch. 2 - Prob. 16PCh. 2 - Prob. 17PCh. 2 - Let P(x1, y1) be a point on the parabola y2 = 4px...Ch. 2 - Prob. 19PCh. 2 - If f and g are differentiable functions with f(0)...Ch. 2 - Prob. 21PCh. 2 - Given an ellipse x2/a2 + y2/b2 = 1, where a b,...Ch. 2 - Find the two points on the curve y = x4 2x2 x...Ch. 2 - Suppose that three points on the parabola y = x2...Ch. 2 - Prob. 25PCh. 2 - Prob. 26PCh. 2 - Prob. 27P
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