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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Chapter 3.4, Problem 43E

Textbook Problem

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Let *P* and *Q* be polynomials. Find

if the degree of *P* is (a) less than the degree of *Q* and (b) greater than the degree of *Q*.

(a)

To determine

**To find:** The value of

**Theorem used:**

If

**Calculation:**.

Let P and Q be the polynomials, such that:

Where

Consider the function

Take the limit of *x* approaches infinity.

(b)

To determine

**To find:** The value of

Single Variable Calculus

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Show that limx[f(x)x2]=0...Ch. 3.5 - Discuss the asymptotic behavior of f(x)=x4+1x in...Ch. 3.5 - Use the asymptotic behavior of f(x) = cos x + 1/x2...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Produce graphs of f that reveal all the important...Ch. 3.6 - Sketch the graph by hand using asymptotes and...Ch. 3.6 - Sketch the graph by hand using asymptotes and...Ch. 3.6 - In Example 4 we considered a member of the family...Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Describe how the graph of f varies as c varies....Ch. 3.6 - Investigate the family of curves given by the...Ch. 3.6 - (a) Investigate the family of polynomials given by...Ch. 3.6 - (a) Investigate the family of polynomials given by...Ch. 3.7 - Consider the following problem: Find two numbers...Ch. 3.7 - Find two numbers whose difference is 100 and whose...Ch. 3.7 - Find two positive numbers whose product is 100 and...Ch. 3.7 - The sum of two positive numbers is 16. What is the...Ch. 3.7 - What is the maximum vertical distance between the...Ch. 3.7 - What is the minimum vertical distance between the...Ch. 3.7 - Find the dimensions of a rectangle with perimeter...Ch. 3.7 - Find the dimensions of a rectangle with area 1000...Ch. 3.7 - A model used for the yield Y of an agricultural...Ch. 3.7 - The rate (in mg carbon/m3/h) at which...Ch. 3.7 - Consider the following problem: A farmer with 750...Ch. 3.7 - Consider the following problem: A box with an open...Ch. 3.7 - A farmer wants to fence in an area of 1.5 million...Ch. 3.7 - A box with a square base and open top must have a...Ch. 3.7 - If 1200 cm2 of material is available to make a box...Ch. 3.7 - A rectangular storage container with an open top...Ch. 3.7 - Do Exercise 16 assuming the container has a lid...Ch. 3.7 - A farmer wants to fence in a rectangular plot of...Ch. 3.7 - If the farmer in Exercise 18 wants to enclose 8000...Ch. 3.7 - (a) Show that of all the rectangles with a given...Ch. 3.7 - Find the point on the line y = 2x + 3 that is...Ch. 3.7 - Find the point on the curve y=x that is closest to...Ch. 3.7 - Find the points on the ellipse 4x2 + y2 = 4 that...Ch. 3.7 - Find, correct to two decimal places, the...Ch. 3.7 - Find the dimensions of the rectangle of largest...Ch. 3.7 - Find the area of the largest rectangle that can be...Ch. 3.7 - Find the dimensions of the rectangle of largest...Ch. 3.7 - Find the area of the largest trapezoid that can be...Ch. 3.7 - Find the dimensions of the isosceles triangle of...Ch. 3.7 - If the two equal sides of an isosceles triangle...Ch. 3.7 - A right circular cylinder is inscribed in a sphere...Ch. 3.7 - A right circular cylinder is inscribed in a cone...Ch. 3.7 - A right circular cylinder is inscribed in a sphere...Ch. 3.7 - A Norman window has the shape of a rectangle...Ch. 3.7 - The top and bottom margins of a poster are each 6...Ch. 3.7 - A poster is to have an area of 180 in2 with 1-inch...Ch. 3.7 - A piece of wire 10 m long is cut into two pieces....Ch. 3.7 - Solve Exercise 37 if one piece is bent into a...Ch. 3.7 - If you are offered one slice from a round pizza...Ch. 3.7 - A fence 8 ft tall runs parallel to a tall building...Ch. 3.7 - A cone-shaped drinking cup is made from a circular...Ch. 3.7 - A cone-shaped paper drinking cup is to be made to...Ch. 3.7 - A cone with height h is inscribed in a larger cone...Ch. 3.7 - An object with weight W is dragged along a...Ch. 3.7 - If a resistor of R ohms is connected across a...Ch. 3.7 - For a fish swimming at a speed v relative to the...Ch. 3.7 - In a beehive, each cell is a regular hexagonal...Ch. 3.7 - A boat leaves a dock at 2:00 pm and travels due...Ch. 3.7 - Solve the problem in Example 4 if the river is 5...Ch. 3.7 - A woman at a point A on the shore of a circular...Ch. 3.7 - An oil refinery is located on the north bank of a...Ch. 3.7 - Suppose the refinery in Exercise 51 is located 1...Ch. 3.7 - The illumination of an object by a light source is...Ch. 3.7 - Find an equation of the line through the point (3,...Ch. 3.7 - Let a and b be positive numbers. Find the length...Ch. 3.7 - At which points on the curve y = 1 + 40x3 3x5...Ch. 3.7 - What is the shortest possible length of the line...Ch. 3.7 - What is the smallest possible area of the triangle...Ch. 3.7 - (a) If C(x) is the cost of producing x units of a...Ch. 3.7 - (a) Show that if the profit P(x) is a maximum,...Ch. 3.7 - A baseball team plays in a stadium that holds...Ch. 3.7 - During the summer months Terry makes and sells...Ch. 3.7 - A retailer has been selling 1200 tablet computers...Ch. 3.7 - A company operates 16 oil wells in a designated...Ch. 3.7 - Show that of all the isosceles triangles with a...Ch. 3.7 - Consider the situation in Exercise 51 if the cost...Ch. 3.7 - Consider the tangent line to the ellipse...Ch. 3.7 - The frame for a kite is to be made from six pieces...Ch. 3.7 - A point P needs to be located somewhere on the...Ch. 3.7 - The graph shows the fuel consumption c of a car...Ch. 3.7 - Let v1 be the velocity of light in air and v2 the...Ch. 3.7 - Two vertical poles PQ and ST are secured by a rope...Ch. 3.7 - The upper right-hand corner of a piece of paper,...Ch. 3.7 - A steel pipe is being carried down a hallway 9 ft...Ch. 3.7 - An observer stands at a point P, one unit away...Ch. 3.7 - A rain gutter is to be constructed from a metal...Ch. 3.7 - Find the maximum area of a rectangle that can be...Ch. 3.7 - The blood vascular system consists of blood...Ch. 3.7 - Ornithologists have determined that some species...Ch. 3.7 - Two light sources of identical strength are placed...Ch. 3.8 - The figure shows the graph of a function f....Ch. 3.8 - Follow the instructions for Exercise 1(a) but use...Ch. 3.8 - Suppose the tangent line to the curve y = f(x) at...Ch. 3.8 - For each initial approximation, determine...Ch. 3.8 - For which of the initial approximations x1 = a, b,...Ch. 3.8 - Use Newtons method with the specified initial...Ch. 3.8 - Use Newtons method with the specified initial...Ch. 3.8 - Use Newtons method with the specified initial...Ch. 3.8 - Use Newtons method with initial approximation x1 =...Ch. 3.8 - Use Newtons method with initial approximation x1 =...Ch. 3.8 - Use Newtons method to approximate the given number...Ch. 3.8 - Use Newtons method to approximate the given number...Ch. 3.8 - (a) Explain how we know that the given equation...Ch. 3.8 - 2x5 + 9x4 7x3 11x = 0, [3, 4]Ch. 3.8 - Use Newtons method to approximate the indicated...Ch. 3.8 - Use Newtons method to approximate the indicated...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all solutions of the...Ch. 3.8 - Use Newtons method to find all the solutions of...Ch. 3.8 - Use Newtons method to find all the solutions of...Ch. 3.8 - Use Newtons method to find all the solutions of...Ch. 3.8 - Use Newtons method to find all the solutions of...Ch. 3.8 - (a) Apply Newtons method to the equation x2 a = 0...Ch. 3.8 - (a) Apply Newtons method to the equation 1/x a =...Ch. 3.8 - Explain why Newtons method doesnt work for finding...Ch. 3.8 - (a) Use Newtons method with x1 = 1 to find the...Ch. 3.8 - Explain why Newtons method fails when applied to...Ch. 3.8 - If f(x)={xifx0xifx0 then the root of the equation...Ch. 3.8 - (a) Use Newtons method to find the critical...Ch. 3.8 - Use Newtons method to find the absolute maximum...Ch. 3.8 - Use Newtons method to find the coordinates of the...Ch. 3.8 - Of the infinitely many lines that are tangent to...Ch. 3.8 - Use Newtons method to find the coordinates,...Ch. 3.8 - In the figure, the length of the chord AB is 4 cm...Ch. 3.8 - A car dealer sells a new car for 18,000. He also...Ch. 3.8 - The figure shows the sun located at the origin and...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the most general antiderivative of the...Ch. 3.9 - Find the antiderivative F of f that satisfies the...Ch. 3.9 - Find the antiderivative F of f that satisfies the...Ch. 3.9 - Find f. 23. f(x) = 20x3 12x2 + 6xCh. 3.9 - Find f. 24. f(x) = x6 4x4 + x + 1 2342 Find f.Ch. 3.9 - f(x)=4x3Ch. 3.9 - Find f. 26. f(x) = x2/3 + x2/3Ch. 3.9 - Find f. 27. f(t) = 12 + sin tCh. 3.9 - Find f. 28. f(t)=t2costCh. 3.9 - Find f. 29. f(x)=1+3x,f(4)=25Ch. 3.9 - Find f. 30. f(x) = 5x4 3x2 + 4, f(1) = 2Ch. 3.9 - Find f. 31. f(x)=x(6+5x),f(1)=10Ch. 3.9 - Find f. 32. f(t) = t + 1/t3, t 0, f(1) = 6Ch. 3.9 - Find f. 33. f(t) = sec t (sec t + tan t), /2 t ...Ch. 3.9 - Find f. 34. f(x)=(x+1)/x,f(1)=5Ch. 3.9 - Find f. 35. f(x) = 2 + 12x 12x2, f(0) = 4, f(0) =...Ch. 3.9 - Find f. 36. f(x) = 8x3 + 5, f(1) = 0, f(1) = 8Ch. 3.9 - Find f. 37. f() = sin + cos , f(0) = 3, f(0) = 4Ch. 3.9 - Find f. 38. f(t) = 4 6/t4, f(1) = 6, f(2) = 9, t ...Ch. 3.9 - Find f. 39. f(x) = 4 + 6x + 24x2, f(0) = 3, f(1) =...Ch. 3.9 - Find f. 40. f(x) = 20x3 + 12x2 + 4, f(0) = 8, f(1)...Ch. 3.9 - Find f. 41. f(t)=t3cost,f(0)=2,f(1)=2Ch. 3.9 - Find f. 42. f(x) = cos x, f(0) = 1, f(0) = 2, f(0)...Ch. 3.9 - Given that the graph of f passes through the point...Ch. 3.9 - Find a function f such that f(x) = x3 and the line...Ch. 3.9 - The graph of a function f is shown. Which graph is...Ch. 3.9 - The graph of a function f is shown. Which graph is...Ch. 3.9 - The graph of a function is shown in the figure....Ch. 3.9 - The graph of the velocity function of a particle...Ch. 3.9 - The graph of f is shown in the figure. Sketch the...Ch. 3.9 - (a) Use a graphing device to graph f(x)=2x3x. (b)...Ch. 3.9 - Draw a graph of f and use it to make a rough...Ch. 3.9 - Draw a graph of f and use it to make a rough...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A particle is moving with the given data. Find the...Ch. 3.9 - A stone is dropped from the upper observation deck...Ch. 3.9 - Show that for motion in a straight line with...Ch. 3.9 - An object is projected upward with initial...Ch. 3.9 - Two balls are thrown upward from the edge of the...Ch. 3.9 - A stone was dropped off a cliff and hit the ground...Ch. 3.9 - If a diver of mass m stands at the end of a diving...Ch. 3.9 - A company estimates that the marginal cost (in...Ch. 3.9 - The linear density of a rod of length 1 m is given...Ch. 3.9 - Since raindrops grow as they fall, their surface...Ch. 3.9 - A car is traveling at 50 mi/h when the brakes are...Ch. 3.9 - What constant acceleration is required to increase...Ch. 3.9 - A car braked with a constant deceleration of 16...Ch. 3.9 - A car is traveling at 100 km/h when the driver...Ch. 3.9 - A model rocket is fired vertically upward from...Ch. 3.9 - A high-speed bullet train accelerates and...Ch. 3 - Explain the difference between an absolute maximum...Ch. 3 - What does the Extreme Value Theorem say?Ch. 3 - (a) State Fermats Theorem. (b) Define a critical...Ch. 3 - Explain how the Closed Interval Method works.Ch. 3 - (a) State Rolles Theorem. (b) State the Mean Value...Ch. 3 - (a) State the Increasing/Decreasing Test. (b) What...Ch. 3 - (a) State the First Derivative Test. (b) State the...Ch. 3 - Explain the meaning of each of the following...Ch. 3 - If you have a graphing calculator or computer, why...Ch. 3 - (a) Given an initial approximation x1 to a root of...Ch. 3 - (a) What is an antiderivative of a function f? (b)...Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Determine whether the statement is true or false....Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the local and absolute extreme values of the...Ch. 3 - Find the limit. 7. limx3x4+x56x42x2+1Ch. 3 - Find the limit. 8. limtt3t+2(2t1)(t2+t+1)Ch. 3 - Find the limit. 9. limx4x2+13x1Ch. 3 - Find the limit. 10. limx(x2+x3)Ch. 3 - Find the limit. 11. limx(4x2+3x2x)Ch. 3 - Find the limit. 12. limxsin4xxCh. 3 - Sketch the graph of a function that satisfies the...Ch. 3 - Sketch the graph of a function that satisfies the...Ch. 3 - Sketch the graph of a function that satisfies the...Ch. 3 - The figure shows the graph of the derivative f of...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Use the guidelines of Section 3.5 to sketch the...Ch. 3 - Produce graphs of f that reveal all the important...Ch. 3 - Produce graphs of f that reveal all the important...Ch. 3 - Produce graphs of f that reveal all the important...Ch. 3 - Produce graphs of f that reveal all the important...Ch. 3 - Show that the equation 3x + 2 cos x + 5 = 0 has...Ch. 3 - Suppose that f is continuous on [0, 4], f(0) = 1,...Ch. 3 - By applying the Mean Value Theorem to the function...Ch. 3 - For what values of the constants a and b is (1, 3)...Ch. 3 - Let g(x) = f(x2), where f is twice differentiable...Ch. 3 - Find two positive integers such that the sum of...Ch. 3 - Show that the shortest distance from the point...Ch. 3 - Find the point on the hyperbola xy = 8 that is...Ch. 3 - Find the smallest possible area of an isosceles...Ch. 3 - Find the volume of the largest circular cone that...Ch. 3 - In ABC, D lies on AB, CD AB, |AD| = |BD| = 4 cm,...Ch. 3 - Solve Exercise 43 when |CD| = 2 cm. 43. In ABC, D...Ch. 3 - The velocity of a wave of length L in deep water...Ch. 3 - A metal storage tank with volume V is to be...Ch. 3 - A hockey team plays in an arena with a seating...Ch. 3 - A manufacturer determines that the cost of making...Ch. 3 - Use Newtons method to find the root of the...Ch. 3 - Use Newtons method to find all solutions of the...Ch. 3 - Use Newtons method to find the absolute maximum...Ch. 3 - Use the guidelines in Section 3.5 to sketch the...Ch. 3 - Find the most general antiderivative of the...Ch. 3 - Find the most general antiderivative of the...Ch. 3 - Find f. 55. f(t) = 2t 3 sin t, f(0) = 5Ch. 3 - Find f. 56. f(u)=u2+uu,f(1)=3Ch. 3 - Find f. 57. f(x) = 1 6x + 48x2, f(0) = 1, f(0) =...Ch. 3 - Find f. 58. f(x) = 5x3 + 6x2 + 2, f(0) = 3, f(l) =...Ch. 3 - A particle is moving according to the given data....Ch. 3 - A particle is moving according to the given data....Ch. 3 - Use a graphing device to draw a graph of the...Ch. 3 - Investigate the family of curves given by...Ch. 3 - A canister is dropped from a helicopter 500 m...Ch. 3 - In an automobile race along a straight road, car A...Ch. 3 - A rectangular beam will be cut from a cylindrical...Ch. 3 - If a projectile is fired with an initial velocity...Ch. 3 - A light is to be placed atop a pole of height h...Ch. 3 - Show that sinxcosx2 for all x.Ch. 3 - Show that x2y2(4 x2)(4 y2) 16 for all numbers x...Ch. 3 - Show that the inflection points of the curve y =...Ch. 3 - Find the point on the parabola y = 1 x2 at which...Ch. 3 - Find the highest and lowest points on the curve x2...Ch. 3 - Water is flowing at a constant rate into a...Ch. 3 - Find the absolute maximum value of the function...Ch. 3 - Find a function f such that f(1)=12, f(0) = 0, and...Ch. 3 - If P(a, a2) is any point on the parabola y = x2,...Ch. 3 - An isosceles triangle is circumscribed about the...Ch. 3 - The line y = mx + b intersects the parabola y = x2...Ch. 3 - Sketch the graph of a function f such that f(x) 0...Ch. 3 - Determine the values of the number a for which the...Ch. 3 - Sketch the region in the plane consisting of all...Ch. 3 - Let ABC be a triangle with BAC = 120 and |AB| ...Ch. 3 - A triangle with sides a, b, and c varies with time...Ch. 3 - ABCD is a square piece of paper with sides of...Ch. 3 - The speeds of sound c1 in an upper layer and c2 in...Ch. 3 - For what values of c is there a straight line that...Ch. 3 - One of the problems posed by the Marquis de...Ch. 3 - Assume that a snowball melts so that its volume...Ch. 3 - A hemispherical bubble is placed on a spherical...

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