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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises find the length and width of a rectangle that has the given area and a minimum perimeter.Area: A square centimeters
In Exercises apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. Let h(x) = ƒ(x) - g(x).] f(x) = g(x) = tan x y 6 4 2 =x g TU Зл 2
In Exercises find the point on the graph of the function that is closest to the given point. f(x) = √x – 8, (12, 0) -
In Exercises apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. Let h(x) = ƒ(x) - g(x).] f(x) = 3 - x 1 g(x) x² + 1 3 2 y 0.0 g 12 3 X
In Exercises apply Newton’s Method to approximate the -value(s) of the indicated point(s) of intersection of the two graphs. Continue the process until two successive approximations differ by less than 0.001. Let h(x) = ƒ(x) - g(x).] f(x) = x² g(x) = cos x y -π 3 نرا 2 T f co g I X
A rectangular page is to contain 30 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.
A farmer plans to fence a rectangular pasture adjacent to a river (see figure). The pasture must contain 245,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing?
Mechanic's Rule The Mechanic's Rule for approximating √a, a > 0, iswhere x1, is an approximation of √a.(a) Use Newton's Method and the function ƒ(x) = x² - a to derive the Mechanic's Rule.(b) Use the Mechanic's Rule to approximate √5 and √7 tothree decimal places. Xn+1 =
A rectangular page is to contain 36 square inches of print. The margins on each side are 1, 1/2 inches. Find the dimensions of the page such that the least amount of paper is used.
In Exercises apply Newton’s Method using the given initial guess, and explain why the method fails. y = 2x³ - 6x² + 6x − 1, x₁ = 1
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area when the total perimeter is 16 feet. 41 X2
A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (1, 2) (see figure).(a) Write the length L of the hypotenuse as a function of x.(b) Use a graphing utility to approximate x graphically such that the length of the hypotenuse is a minimum.(c) Find the
A rectangle is bounded by the x- and y-axes and the graph of y = (6 - x)/2 (see figure). What length and width should the rectangle have so that its area is a maximum? 5 4 2 1 -1 y y = 6-x 2 (x, y) 2 3 4 5 6 X
A rectangular solid (with a square base) has a surface area of 337.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.
In Exercises approximate the fixed point of the function to two decimal places. [A fixed point x0, of a function ƒ is a value of x such that ƒ(x0) = x0.] X 800 = (x)ƒ
(a) Use Newton's Method and the function ƒ(x) = xn - a to obtain a general rule for approximating x = n√a.(b) Use the general rule found in part (a) to approximate 4√6and 3√15 to three decimal places.
In Exercises apply Newton’s Method using the given initial guess, and explain why the method fails. y = x³ 2x2, x₁ = 0 --
In Exercises approximate the fixed point of the function to two decimal places. [A fixed point x0, of a function ƒ is a value of x such that ƒ(x0) = x0.] f(x) = cotx, 0 < x < T
A rectangle is bounded by the x-axis and the semicircle(see figure). What length and width should the rectangle have so that its area is a maximum? y = 25x²
Use Newton’s Method to show that the equationcan be used to approximate 1/a when x1, is an initial guess ofthe reciprocal of a. Note that this method of approximatingreciprocals uses only the operations of multiplication andsubtraction. ("xv - 7)"x = ¹+¹X
Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 6 (see figure).(a) Solve by writing the area as a function of h.(b) Solve by writing the area as a function of a.(c) Identify the type of triangle of maximum area. α 6 6
Find the dimensions of the largest rectangle that can be inscribed in a semicircle of radius (see Exercise 25).Data from in Exercise 25A rectangle is bounded by the x-axis and the semicircle(see figure). What length and width should the rectangle have so that its area is a maximum? y = 25x²
Use the result of Exercise 25 to approximate(a) 1/3 (b) 1/11 to three decimal places. Data from in Exercise 25Use Newton’s Method to show that the equationcan be used to approximate 1/a when x1, is an initial guess of the reciprocal of a. Note that this method of approximating
A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)
Rework Exercise 29 for a cylindrical package. (The cross section is circular.)The data from in Exercise 29A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package
For what value(s) will Newton’s Method fail to converge for the function shown in the graph? Explain your reasoning. -6-4-2 4 -2. -4 2 A 4
In your own words and using a sketch, describe Newton’s Method for approximating the zeros of a function.
A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.
Exercises present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution.Find the point on the graph of ƒ(x) = 4 - x² that is closest to the point (1, 0).
The perimeter of a rectangle is 20 feet. Of all possible dimensions, the maximum area is 25 square feet when its length and width are both 5 feet. Are there dimensions that yield a minimum area? Explain.
Exercises present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution.You are in a boat 2 miles from the nearest point on the coast (see figure). You are to go to a point Q that is 3 miles down the coast and 1
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 14 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.
Exercises present problems similar to exercises from the previous sections of this chapter. In each case, use Newton’s Method to approximate the solution.The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is given by When is the concentration the
An industrial tank of the shape described in Exercise 33 must have a volume of 4000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost.Data from in Exercise 33A solid is formed by adjoining two hemispheres
The total number of arrests T (in thousands) for all males ages 15 to 24 in 2010 is approximated by the modelfor 15 ≤ x ≤ 24, where x is the age in years (see figure Approximate the two ages that had total arrests of 300 thousand. T = 0.2988x+ 22.625x³+628.49x²7565.9x + 33,478
The sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The zeros of coincide with the zeros of P(x). f(x) = p(x) q(x)
Twenty feet of wire is to be used to form two figures. In each of the following cases, how much wire should be used for each figure so that the total enclosed area is maximum?(a) Equilateral triangle and square(b) Square and regular pentagon(c) Regular pentagon and regular hexagon(d) Regular
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the coefficients of a polynomial function are all positive, then the polynomial has no positive zeros.
A wooden beam has a rectangular cross section of height h and width w (see figure). The strength S of the beam is directly proportional to the width and the square of the height. What are the dimensions of the strongest beam that can be cut from a round log of diameter 20 inches? S = kh²w, where k
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(x) is a cubic polynomial such that ƒ'(x) is never zero, then any initial guess will force Newton's Method to converge to the zero of ƒ.
Two factories are located at the coordinates (-x, 0) and (x, 0), and their power supply is at (0, h) (see figure). Find y such that the total length of power line from the power supply to the factories is a minimum. (-x, 0) y (0, h) y (x, 0) X
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The roots of √ƒ(x) = 0 coincide with the roots of ƒ(x) = 0.
An offshore oil well is 2 kilometers off the coast. The refinery is 4 kilometers down the coast. Laying pipe in the ocean is twice as expensive as laying it on land. What path should the pipe follow in order to minimize the cost?
The graph of ƒ(x) = -sin x has infinitelymany tangent lines that pass through the origin. Use Newton'sMethod to approximate to three decimal places the slope of thetangent line having the greatest slope.
Find, with explanation, the maximum value of ƒ(x)= x³ -3x on the set of all real numbers x satisfyingx4 +36 ≤ 13x².
In Exercises consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4, 1), (5, 6), and (10, 3). A trunk line will run from the distribution center along the line y = mx, and
In Exercises consider a fuel distribution center located at the origin of the rectangular coordinate system (units in miles; see figures). The center supplies three factories with coordinates (4, 1), (5, 6), and (10, 3). A trunk line will run from the distribution center along the line y = mx, and
The graph shows the profit P (in thousands of dollars) of a company in terms of its advertising cost x (in thousands of dollars).(a) Estimate the interval on which the profit is increasing.(b) Estimate the interval on which the profit is decreasing.(c) Estimate the amount of money the company
Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 12%. Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)
A sector with central angle θ is cut from a circle of radius 12 inches (see figure), and the edges of the sector are brought together to form a cone. Find the magnitude of θ such that the volume of the cone is a maximum. 12 in. Ꮎ 12 in.
When light waves traveling in a transparent medium strike the surface of a second transparent medium, they change direction. This change of direction is called refraction and is defined by Snell’s Law of Refraction,where θ1 and θ2 are the magnitudes of the angles shown in the figure and v1 and
The conditions are the same as in Exercise 41 except that the man can row at v1 miles per hour and walk at v2 miles per hour. If θ1, and θ2 are the magnitudes of the angles, show that the man will reach point Q in the least time whenData from in exercise 41A man is in a boat 2 miles from the
A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located 3 miles down the coast and 1 mile inland (see figure). He can row at 2 miles per hour and walk at 4 miles per hour. Toward what point on the coast should he row in order to reach point Q in the least
The graph of ƒ(x) = cos x and a tangent line to ƒ through the origin are shown. Find the coordinates of the point of tangency to three decimal places. -1 f(x) = cos x 2π X
A light source is located over the center of a circular table of diameter 4 feet (see figure). Find the height h of the light source such that the illumination I at the perimeter of the table is maximum whenwhere s is the slant height, a is the angle at which the light strikes the table, and k is a
In Exercises match the graph of ƒ in the leftcolumn with that of its derivative in the right column.(a)(b)(c)(d) Graph of f' y 3 VALA 2 -2 -3+ X
In Exercises match the graph of ƒ in the left column with that of its derivative in the right column.(a)(b)(c)(d) Graph of f' 3 VAA 2 -2 -3+ X
In Exercises match the graph of ƒ in the left column with that of its derivative in the right column.(a)(b)(c)(d) Graph of f' 3 VAA 2 -2 -3+ X
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = X x² + 1
In Exercises match the graph of ƒ in the left column with that of its derivative in the right column.(a)(b)(c)(d) Graph of f' 3 VAA 2 -2 -3+ X
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y X2 x² + 3
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y 1 x-2 - 3
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = x2 + 1 x² - 4
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y 3x x² - 1
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f(x) = x - 3 X
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f(x) = x + 32 x²
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. У x² - 6x + 12 X x - 4
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f(x) x³ x² - 9
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y=x√√4-x
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y -x² - 4x - 7 x + 3
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = 3x2/3 - 2x
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. g(x)=x√√√9 - x²
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y=2x-x³
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = (x + 1)² - 3(x + 1)2/3
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = -(x3 - 3x + 2)
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = 3x4 + 4x³ -
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = - 2x4 + 3x²
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = r5 – 5r 5x
In Exercises analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = (x - 1)5
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x) 20x x² + 1 1 X
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x) = x + 4 x² + 1
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x) 4x x² + 15 -2
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x): = - 2x x² + 7
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x) = -x + 2 cos x, 0 ≤ x ≤ 2π
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f(x) = 2x - 4 sin x, 0≤ x ≤ 2π
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. y = cos x - cos 2x, 0≤ x ≤ 2π
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. y = 2x - tan x,
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. y = 2(csc x + sec x), 0 < x < ²
In Exercises use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. g(x) = x cotx, -2π < x < 2TT
In Exercises the graphs of ƒ, ƒ', and ƒ" are shown on the same set of coordinate axes. Which is which? Explain your reasoning. -4 -2 4 y 1-4+ 2 4 X
In Exercises the graphs of ƒ, ƒ', and ƒ" are shown on the same set of coordinate axes.Which is which? Explain your reasoning. -2 -2 y 2 X
Let ƒ'(t) < 0 for all t in the interval (2,8). Explain why ƒ(3) > ƒ(5).
Let ƒ(0) = 3 and 2 ≤ ƒ'(x) ≤ 4 for all x in the interval [-5, 5]. Determine the greatest and least possible values of ƒ(2).
In Exercises use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? f(x) = 4(x - 1)² x²
In Exercises use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? h(x) = sin 2x X
In Exercises use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? g(x) = 3x45x +
In Exercises use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? f(x): = cos 3x 4x
In Exercises use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. h(x) 6 - 2x 3-x
In Exercises use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur? f(x) = x² - 3x - 1 - x - 2

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