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Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266636
Chapter 2.3, Problem 48E
Textbook Problem
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Find f′(x). Compare the graphs of f and f′ and use them to explain why your answer is reasonable.

48. f ( x ) = x + 1 x

To determine

To find: The derivative of f(x), f(x) and to compare their graphs.

Explanation of Solution

Given:

The function is f(x)=x+1x

Derivative rules:

(1) Derivative of constant function: ddx(c)=0

(2) Power Rule: ddx(xn)=nxn1

(3) Sum Rule: ddx(f+g)=ddx(f)+ddx(g)

(4) Difference Rule: ddx(fg)=ddx(f)ddx(g)

(5) Constant Multiple Rule: ddx(cf)=cddx(f)

Calculation:

The derivative of f(x) is f(x), which is obtained as follows.

f(x)=ddx(f(x)) =ddx(x+1x)=ddx(x+x1) 

Apply the sum rule (3),

 f(x)=ddx(x+x1

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Chapter 2 Solutions

Single Variable Calculus
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F(t) =...Ch. 2.5 - Find the derivative of the function. 21....Ch. 2.5 - Find the derivative of the function. 22. y=(x+1x)5Ch. 2.5 - Find the derivative of the function. 23. y=xx+1Ch. 2.5 - Find the derivative of the function. 24....Ch. 2.5 - Find the derivative of the function. 25. h() =...Ch. 2.5 - Find the derivative of the function. 26....Ch. 2.5 - Find the derivative of the function. 27....Ch. 2.5 - Find the derivative of the function. 28....Ch. 2.5 - Find the derivative of the function. 29....Ch. 2.5 - Find the derivative of the function. 30....Ch. 2.5 - Find the derivative of the function. 31. y =...Ch. 2.5 - Find the derivative of the function. 32. J() =...Ch. 2.5 - Find the derivative of the function. 33. y=sin1+x2Ch. 2.5 - Find the derivative of the function. 34....Ch. 2.5 - Find the derivative of the function. 35....Ch. 2.5 - Find the derivative of the function. 36. y=xsin1xCh. 2.5 - Find the derivative of the function. 37. y =...Ch. 2.5 - Find the derivative of the function. 38....Ch. 2.5 - Find the derivative of the function. 39. f(t) =...Ch. 2.5 - Find the derivative of the function. 40. g(u) =...Ch. 2.5 - Find the derivative of the function. 41. y=x+xCh. 2.5 - Find the derivative of the function. 42. y=x+x+xCh. 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 - Find the derivative of the function. 45....Ch. 2.5 - Find the derivative of the function. 46. y = [x +...Ch. 2.5 - Find y and y. 47. y = cos(sin 3)Ch. 2.5 - Find y and y. 48. y=1(1+tanx)2Ch. 2.5 - Find y and y. 49. y=1sectCh. 2.5 - Find y and y. 50. y=4xx+1Ch. 2.5 - Find an equation of the tangent line to the curve...Ch. 2.5 - Find an equation of the tangent line to the curve...Ch. 2.5 - Find an equation of the tangent line to the curve...Ch. 2.5 - Find an equation of the tangent line to the curve...Ch. 2.5 - (a) Find an equation of the tangent line to the...Ch. 2.5 - (a) The curve y=x/2x2 is called a bullet-nose...Ch. 2.5 - (a) If f(x)=x2x2, find f(x). (b) Check to see that...Ch. 2.5 - The function f(x) = sin(x + sin 2x), 0 x ,...Ch. 2.5 - Find all points on the graph of the function f(x)...Ch. 2.5 - At what point on the curve y=1+2x is the tangent...Ch. 2.5 - If F(x) =f(g(x)), where f(2) = 8, f(2) = 4, f(5) =...Ch. 2.5 - If h(x)=4+3f(x), where f(1) = 7 and f(1) = 4, find...Ch. 2.5 - A table of values for f, g, f and g is given. (a)...Ch. 2.5 - Let f and g be the functions in Exercise 63. (a)...Ch. 2.5 - If f and g are the functions whose graphs are...Ch. 2.5 - If f is the function whose graph is shown, let...Ch. 2.5 - If g(x)=f(x), where the graph of f is shown,...Ch. 2.5 - Suppose f is differentiable on and is a real...Ch. 2.5 - Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3,...Ch. 2.5 - If g is a twice differentiable function and f(x) =...Ch. 2.5 - If F(x) = f(3f(4f(x))), where f(0) = 0 and f(0) =...Ch. 2.5 - If F(x) = f(xf(xf(x))), where f(1) = 2, f(2) = 3,...Ch. 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 - If the equation of motion of a particle is given...Ch. 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 - A particle moves along a straight line with...Ch. 2.5 - Air is being pumped into a spherical weather...Ch. 2.5 - Use the Chain Rule to prove the following. (a) The...Ch. 2.5 - Use the Chain Rule and the Product Rule to give an...Ch. 2.5 - (a) If n is a positive integer, prove that...Ch. 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 - (a) Write x=x2 and use the Chain Rule to show that...Ch. 2.5 - If y = f(u) and u = g(x), where f and g are twice...Ch. 2.5 - If y = f(u) and u = g(x), where f and g possess...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 - (a) Find y by implicit differentiation. (b) Solve...Ch. 2.6 - (a) Find y by implicit differentiation. (b) Solve...Ch. 2.6 - Find dy/dx by implicit differentiation. 5. x2 4xy...Ch. 2.6 - Find dy/dx by implicit differentiation. 6. 2x2 +...Ch. 2.6 - Find dy/dx by implicit differentiation. 7. x4 +...Ch. 2.6 - Find dy/dx by implicit differentiation. 8. x3 xy2...Ch. 2.6 - Find dy/dx by implicit differentiation. 9....Ch. 2.6 - Find dy/dx by implicit differentiation. 10. y5 +...Ch. 2.6 - Find dy/dx by implicit differentiation. 11. y cos...Ch. 2.6 - Find dy/dx by implicit differentiation. 12....Ch. 2.6 - Find dy/dx by implicit differentiation. 13....Ch. 2.6 - Find dy/dx by implicit differentiation. 14. y...Ch. 2.6 - Find dy/dx by implicit differentiation. 15....Ch. 2.6 - Find dy/dx by implicit differentiation. 16....Ch. 2.6 - Find dy/dx by implicit differentiation. 17....Ch. 2.6 - Find dy/dx by implicit differentiation. 18. x sin...Ch. 2.6 - Find dy/dx by implicit differentiation. 19....Ch. 2.6 - Find dy/dx by implicit differentiation. 20....Ch. 2.6 - lf f(x) + x2[f(x)]3 = 10 and f(1) = 2, find f(1).Ch. 2.6 - If g(x) + x sin g(x) = x2, find g(0).Ch. 2.6 - Regard y as the independent variable and x as the...Ch. 2.6 - Regard y as the independent variable and x as the...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 2.6 - Use implicit differentiation to find an equation...Ch. 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 - Find y by implicit differentiation. 35. x2 + 4y2 =...Ch. 2.6 - Find y by implicit differentiation. 36. x2 + xy +...Ch. 2.6 - Find y by implicit differentiation. 37. sin y +...Ch. 2.6 - Find y by implicit differentiation. 38. x3 y3 = 7Ch. 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 - Show by implicit differentiation this the tangent...Ch. 2.6 - Find an equation of the tangent line to the...Ch. 2.6 - Show that the sum of the x- and y-intercepts of...Ch. 2.6 - Show, using implicit differentiation, that any...Ch. 2.6 - The Power Rule can be proved using implicit...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Two curves are orthogonal if their tangent lines...Ch. 2.6 - Show that the ellipse x2/a2 + y2/b2 = 1 and the...Ch. 2.6 - Find the value of the number a such that the...Ch. 2.6 - (a) The van tier Waals equation for n moles of a...Ch. 2.6 - (a) Use implicit differentiation to find y if...Ch. 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 - Find all points on the curve x2y2 + xy = 2 where...Ch. 2.6 - Find equations of both the tangent lines to the...Ch. 2.6 - The Bessel function of order 0, y = J(x),...Ch. 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 - Graphs of the velocity functions of two particles...Ch. 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 - A particle moves with position function...Ch. 2.7 - (a) A company makes computer chips from square...Ch. 2.7 - (a) Sodium chlorate crystals are easy to grow in...Ch. 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 - (a) The volume of a growing spherical cell is...Ch. 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 - If, in Example 4, one molecule of the product C is...Ch. 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 - Refer to the law of laminar flow given in Example...Ch. 2.7 - The frequency of vibrations of a vibrating violin...Ch. 2.7 - Suppose that the cost (in dollars) for a company...Ch. 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 - The gas law for an ideal gas at absolute...Ch. 2.7 - Invasive species often display a wave of advance...Ch. 2.7 - In the study of ecosystems, predator-prey models...Ch. 2.7 - In a fish farm, a population of fish is introduced...Ch. 2.8 - If V is the volume of a cube with edge length x...Ch. 2.8 - (a) If A is the area of a circle with radius r and...Ch. 2.8 - Each side of a square is increasing at a rate of 6...Ch. 2.8 - The length of a rectangle is increasing at a rate...Ch. 2.8 - A cylindrical tank with radius 5 m is being filled...Ch. 2.8 - The radius of a sphere is increasing at a rate of...Ch. 2.8 - The radius of a spherical ball is increasing at a...Ch. 2.8 - The area of a triangle with sides of lengths a and...Ch. 2.8 - Suppose y=2x+1, where x and y are functions of t....Ch. 2.8 - Suppose 4x2 + 9y2 = 36, where x and y are...Ch. 2.8 - If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4,...Ch. 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 - (a) What quantities are given in the problem? (b)...Ch. 2.8 - (a) What quantities are given in the problem? (b)...Ch. 2.8 - (a) What quantities are given in the problem? (b)...Ch. 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 - Water is leaking out of an inverted conical tank...Ch. 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 - The top of a ladder slides down a vertical wall at...Ch. 2.8 - According to the model we used to solve Example 2,...Ch. 2.8 - If the minute hand of a clock has length r (in...Ch. 2.8 - A faucet is filling a hemispherical basin of...Ch. 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 - If two resistors with resistances R1 and R2 are...Ch. 2.8 - Brain weight B as a function of body weight W in...Ch. 2.8 - Two sides of a triangle have lengths 12 m and 15...Ch. 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 - A runner sprints around a circular track of radius...Ch. 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 - 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 - Find the linear approximation of the function...Ch. 2.9 - Find the linear approximation of the function...Ch. 2.9 - Verify the given linear approximation at a = 0....Ch. 2.9 - Verify the given linear approximation at a = 0....Ch. 2.9 - Verify the given linear approximation at a = 0....Ch. 2.9 - Verify the given linear approximation at a = 0....Ch. 2.9 - Find the differential dy of each function. 11. (a)...Ch. 2.9 - Find the differential dy of each function. 12. (a)...Ch. 2.9 - Find the differential dy of each function. 13. (a)...Ch. 2.9 - Find the differential dy of each function. 14. (a)...Ch. 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 - (a) Find the differential dy and (b) evaluate dy...Ch. 2.9 - (a) Find the differential dy and (b) evaluate dy...Ch. 2.9 - Compute y and dy for the given values of x and dx...Ch. 2.9 - Compute y and dy for the given values of x and dx...Ch. 2.9 - Compute y and dy for the given values of x and dx...Ch. 2.9 - Compute y and dy for the given values of x and dx...Ch. 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 - 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 - Explain, in terms of linear approximations or...Ch. 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 - The radius of a circular disk is given as 24 cm...Ch. 2.9 - The circumference of a sphere was measured to be...Ch. 2.9 - Use differentials to estimate the amount of paint...Ch. 2.9 - (a) Use differentials to find a formula for the...Ch. 2.9 - One side of a right triangle is known to be 20 cm...Ch. 2.9 - If a current I passes through a resistor with...Ch. 2.9 - When blood flows along a blood vessel, the flux F...Ch. 2.9 - Establish the following rules for working with...Ch. 2.9 - On page 431 of Physics: Calculus, 2d ed., by...Ch. 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 - Suppose an object moves along a straight line with...Ch. 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 - (a) What does it mean for f to be differentiable...Ch. 2 - Describe several ways in which a function can fail...Ch. 2 - What are the second and third derivatives of a...Ch. 2 - State each differentiation rule both in symbols...Ch. 2 - State the derivative of each function. (a) y = xn...Ch. 2 - Explain how implicit differentiation works.Ch. 2 - Give several examples of how the derivative can be...Ch. 2 - (a) Write an expression for the linearization of f...Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - Determine whether the statement is true or false....Ch. 2 - The displacement (in meters) of an object moving...Ch. 2 - The graph of f is shown. State, with reasons, the...Ch. 2 - Trace or copy the graph of the function. Then...Ch. 2 - Trace or copy the graph of the function. Then...Ch. 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 - The total cost of repaying a student loan at an...Ch. 2 - The total fertility rate at time t, denoted by...Ch. 2 - Let P(t) be the percentage of Americans under the...Ch. 2 - Find f(x) from first principles, that is, directly...Ch. 2 - Find f(x) from first principles, that is, directly...Ch. 2 - (a) If f(x)=35x, use the definition of a...Ch. 2 - Calculate y. 13. y = (x2 + x3)4Ch. 2 - Calculate y. 14. y=1x1x35Ch. 2 - Calculate y. 15. y=x2x+2xCh. 2 - Calculate y. 16. y=tanx1+cosxCh. 2 - Calculate y. 17. y = x2 sin xCh. 2 - Calculate y. 18. y=(x+1x2)7Ch. 2 - Calculate y. 19. y=t41t4+1Ch. 2 - Calculate y. 20. y = sin(cos x)Ch. 2 - Calculate y. 21. y=tan1xCh. 2 - Calculate y. 22. y=1sin(xsinx)Ch. 2 - Calculate y. 23. xy4 + x2y = x + 3yCh. 2 - Calculate y. 24. y = sec(1 + x2)Ch. 2 - Calculate y. 25. y=sec21+tan2Ch. 2 - Calculate y. 26. x2 cos y + sin 2y = xyCh. 2 - Calculate y. 27. y = (1 x1)1Ch. 2 - Calculate y. 28. y=1/x+x3Ch. 2 - Calculate y. 29. sin(xy) = x2 yCh. 2 - Calculate y. 30. y=sinxCh. 2 - Calculate y. 31. y = cot(3x2 + 5)Ch. 2 - Calculate y. 32. y=(x+)4x4+4Ch. 2 - Calculate y. 33. y=xcosxCh. 2 - Calculate y. 34. y=sinmxxCh. 2 - Calculate y. 35. y = tan2(sin )Ch. 2 - Calculate y. 36. x tan y = y 1Ch. 2 - Calculate y. 37. y=xtanx5Ch. 2 - Calculate y. 38. y=(x1)(x4)(x2)(x3)Ch. 2 - Calculate y. 39. y=sin(tan1+x3)Ch. 2 - Calculate y. 40. y=sin2(cossinx)Ch. 2 - If f(x)=4t+1, find f(2).Ch. 2 - If g() = sin , find g(/6).Ch. 2 - Find y if x6 + y6 = 1.Ch. 2 - Find f(n)(x) if f(x) = 1/(2 x).Ch. 2 - Find the limit. 45. limx0secx1sinxCh. 2 - Find the limit. 46. limt0t3tan32tCh. 2 - Find an equation of the tangent to the curve at...Ch. 2 - Find an equation of the tangent to the curve at...Ch. 2 - Find equations of the tangent line and normal line...Ch. 2 - Find equations of the tangent line and normal line...Ch. 2 - (a) If f(x)=x5x, find f(x). (b) Find equations of...Ch. 2 - (a) If f(x) = 4x tan x, /2 x /2, find f and f....Ch. 2 - At what points on the curve y = sin x + cos x, 0 ...Ch. 2 - Find the points on the ellipse x2 + 2y2 = 1 where...Ch. 2 - Find a parabola y = ax2 + bx + c that passes...Ch. 2 - How many tangent lines to the curve y = x/(x + 1)...Ch. 2 - If f(x) = (x a)(x b)(x c), show that...Ch. 2 - (a) By differentiating the double-angle formula...Ch. 2 - Suppose that...Ch. 2 - If f and g are the functions whose graphs are...Ch. 2 - Find f in terms of g. 61. f(x) = x2g(x)Ch. 2 - Find f in terms of g. 62. f(x) = g(x2)Ch. 2 - Find f in terms of g. 63. f(x) = [g(x)]2Ch. 2 - Find f in terms of g. 64. f(x) = xag(xb)Ch. 2 - Find f in terms of g. 65. f(x) = g(g(x))Ch. 2 - Find f in terms of g. 66. f(x) = sin(g(x))Ch. 2 - Find f in terms of g. 67. f(x) = g(sin x)Ch. 2 - Find f in terms of g. 68. f(x)=g(tanx)Ch. 2 - Find h in terms of f and g. 69....Ch. 2 - Find h in terms of f and g. 70. h(x)=f(x)g(x)Ch. 2 - Find h in terms of f and g. 71. h(x) = f(g(sin...Ch. 2 - A particle moves along a horizontal line so that...Ch. 2 - A particle moves on a vertical line so that its...Ch. 2 - The volume of a right circular cone is V=13r2h,...Ch. 2 - The mass of part of a wire is x(1+x) kilograms,...Ch. 2 - The cost, in dollars, of producing x units of a...Ch. 2 - The volume of a cube is increasing at a rate of 10...Ch. 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 - (a) Find the linear approximation to f(x)=25x2...Ch. 2 - (a) Find the linearization of f(x)=1+3x3 at a = 0....Ch. 2 - Evaluate dy if y = x3 2x2 + 1, x = 2, and dx =...Ch. 2 - A window has the shape of a square surmounted by a...Ch. 2 - Express the limit as a derivative and evaluate....Ch. 2 - Express the limit as a derivative and evaluate....Ch. 2 - Express the limit as a derivative and evaluate....Ch. 2 - Evaluate limx01+tanx1+sinxx3.Ch. 2 - Suppose f is a differentiable function such that...Ch. 2 - Find f(x) if it is known that ddx[f(2x)]=x2Ch. 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 - Find the point where the curves y = x3 3x + 4 and...Ch. 2 - Show that the tangent lines to the parabola y =...Ch. 2 - Show that ddx(sin2x1+cotx+cos2x1+tanx)=cos2xCh. 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 - Prove that dndxn(sin4x+cos4x)=4n1cos(4x+n/2).Ch. 2 - If f is differentiable at a, where a 0, evaluate...Ch. 2 - The figure shows a circle with radius 1 inscribed...Ch. 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 - Evaluate limx0sin(3+x)2sin9x.Ch. 2 - (a) Use the identity for tan(x y) (see Equation...Ch. 2 - Let P(x1, y1) be a point on the parabola y2 = 4px...Ch. 2 - Suppose that we replace the parabolic mirror of...Ch. 2 - If f and g are differentiable functions with f(0)...Ch. 2 - Evaluate limx0sin(a+2x)2sin(a+x)+sinax2.Ch. 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 - A lattice point in the plane is a point with...Ch. 2 - A cone of radius r centimeters and height h...Ch. 2 - A container in the shape of an inverted cone has...

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