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mathematics
precalculus

Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions

a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur. -3-2-1 y 2 1 2 y = f(x) /1 2/3 X
Show that if ƒ″ > 0 throughout an interval [a, b], then ƒ″ has at most one zero in [a, b]. What if ƒ″ < 0 throughout [a, b] instead?
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 4x3 - x4 = x3(4 - x)
Use l’Hôpital’s rule to find the limit. 20 - T lim 0T/2 COS (2T -
A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.g(t) = -t2 - 3t + 3
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.i) y = x2 - 4ii) y = x2 + 8x + 15iii) y = x3 - 3x2 + 4 = (x + 1)(x - 2)2iv) y = x3 - 33x2 + 216x = x(x - 9)(x - 24)b. Use Rolle’s Theorem to prove that between every two zeros of xn + an-1.xn-1 + +
Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?
For what values of a, m, and b does the functionsatisfy the hypotheses of the Mean Value Theorem on the interval [0, 2]? f(x) = 3, -x² + 3x + a, mx + b. x = 0 0 < x < 1 1 ≤ x ≤ 2
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur. -3-2-1 y 2 1 2 y = f(x) 1 2 3 > X
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = -x4 + 6x2 - 4 = x2(6 - x2) - 4
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x4 - 2x2 = x2(x2 - 2)
Use l’Hôpital’s rule to find the limit. lim x-0 sin x - x3 x X
A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = -π to x = π. What are the dimensions of the rectangle with largest area, and what is the largest area?
A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.a. Write a formula V(x) for the volume of
Suppose that ƒ″ is continuous on [a, b] and that ƒ has three zeros in the interval. Show that ƒ″ has at least one zero in (a, b). Generalize this result.
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = (sin x 1)(2 cos x + 1), 0 ≤ x ≤ 2π
a. Find the open intervals on which the function is increasing and decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur. -3-2-1 y 2 1 2 y = f(x) 1 2 3 > X
Match the table with a graph. x a b C f'(x) does not exist does not exist -1.7
Use l’Hôpital’s rule to find the limit. 8.x lim x-0 COS X - 1
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 1 - (x + 1)3
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = (x - 2)3 + 1
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = (sin x + cos x)(sin x - cos x), 0≤ x
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.is zero at x = 0 and x = 1 and differentiable on (0, 1), but its derivative on (0, 1) is never zero. How can this be? Doesn’t Rolle’s Theorem say the
Use l’Hôpital’s rule to find the limit. sin 5t lim 10 2t
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. f(x) = (x² - x₂ -2≤ x ≤-−1 12x² 3x3, -1 < x≤0 -
Match the table with a graph. X a b C f'(x) does not exist 0 -2
You are designing a 1000 cm3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of aluminum used up by
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 1 - 9x - 6x2 - x3
What are the dimensions of the lightest opentop right circular cylindrical can that will hold a volume of 1000 cm3? Compare the result here with the result in Example 2.Example 2You have been asked to design a one-liter can shaped like a right circular cylinder. What dimensions will use the least
Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3. 3 X 3
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = -2x3 + 6x2 - 3
Use l’Hôpital’s rule to find the limit. lim 1-0 sin 1² t
Match the table with a graph. X a b C f'(x) 0 0 -5
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. f(x) = sin x 0, -T≤ x ≤ 0 x = 0
Match the table with a graph. x a b C f'(x) 0 0 5
Two sides of a triangle have lengths a and b, and the angle between them is θ. What value of θ will maximize the triangle’s area?
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = x¹/²(x - 3)
Use l’Hôpital’s rule to find the limit. x - 8x² lim xx12x² + 5x
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? ƒ'(x) = x¯¹/³(x + 2)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x(6 - 2x)2
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. f(x) = √x(1 − x), [0, 1] -
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x3 - 3x + 3
Use l’Hôpital’s rule to find the limit. 5x³ - 2x lim xxx 7x³ + 3
Use l’Hôpital’s rule to find the limit. 31³ + 3 43 14t³ - t + 3 lim
You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?
Find the absolute extreme values and where they occur. -3 y 2 -11 (1,2) 2 X
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? 0 X Vx 6 - ε = (x), f E
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 6 - 2x - x2
A 1125 ft3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.ƒ(x) = x4/5, [0, 1]
Use l’Hôpital’s rule to find the limit. 1³ - 4t + 15 12 lim 13 12² t
Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. y = 2 cos x - √2x, √2x, -# ≤x≤ ³7 y -TT 0 LX 3TT 2
Find the absolute extreme values and where they occur. 5- 0 2 X
Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. y sin |x|, -27 ≤ x ≤ 2T n 0 NOT TO SCALE
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval. f(b) f(a) b-a = f'(c)
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = 1 1 – 4 x2 ' x ≠ 0
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x2 - 4x + 3
Use l’Hôpital’s rule to find the limit. x². - 25 lim x 5 x + 5
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval.ƒ(x) = x3 - x2, [-1, 2] f(b) f(a) b-a = f'(c)
Find the absolute extreme values and where they occur. -2 y 2 0 2 X
Find the absolute extreme values and where they occur. -1 y 10- -10 1 X
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = (x − 2)(x + 4) (x + 1)(x − 3) x = -1,3
Your iron works has contracted to design and build a 500 ft3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base
Which of the functions satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.ƒ(x) = x2/3, [-1, 8]
Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. -
Determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with .Theorem 1 THEOREM 1-The Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains both an absolute maximum value M and an abso- lute
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = = x²(x − 1) x + 2 x = -2
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval.ƒ(x) = ln (x - 1), [2, 4] f(b) f(a) b-a = f'(c)
Use l’Hôpital’s rule to find the limit. x - 2 lim x2x²4
A 216 m2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? ƒ'(x) = (x − 7)(x + 1)(x + 5)
Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. y = x + sin 2x, -²5 ≤ x ≤ what 0 2πT LX 2T 3
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. 2x2 + 3x lim x→rs + x + 1 1.3
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval.ƒ(x) = sin-1 x, [-1, 1] f(b) f(a) b-a = f'(c)
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = (x − 1)e¯x -
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval. f(b) f(a) b-a = f'(c)
You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0, b). Show that the area of the triangle enclosed by the segment is largest when a = b.
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. lim x-0 1 - COS X X²
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval. f(b) f(a) b-a = f'(c)
Determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with .Theorem 1 THEOREM 1-The Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains both an absolute maximum value M and an abso- lute
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = (x − 1)²(x + 2)²
You are planning to make an open rectangular box from an 8-in.-by- 15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. x³ - 1 lim x14x³ - x - 3
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? ƒ'(x) = (x − 1)²(x + 2)
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. 5x 2 3x lim xxx 7x² + 1
The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.a. Express the y-coordinate of P in terms of x.b. Express the area of the rectangle in terms of x.c. What is the largest area the rectangle can have, and what are its dimensions? -1 0 B X P(x,
Find the value or values of c that satisfy the equationin the conclusion of the Mean Value Theorem for the functions and interval.ƒ(x) = x2/3, [0, 1] f(b) f(a) b-a = f'(c)
Determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with .Theorem 1 THEOREM 1-The Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains both an absolute maximum value M and an abso- lute
Answer the following questions about the functions whose derivatives are given.a. What are the critical points of ƒ?b. On what open intervals is ƒ increasing or decreasing?c. At what points, if any, does ƒ assume local maximum and minimum values? f'(x) = x(x − 1)
Show that among all rectangles with an 8-m perimeter, the one with largest area is a square.
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. x + 2 lim x2x² - 4
Assume that an ice cube retains its cubical shape as it melts. If we call its edge length s, its volume is V = s3 and its surface area is 6s2. We assume that V and s are differentiable functions of time t. We assume also that the cube’s volume decreases at a rate that is proportional to its
If x2 + y2 = 25 and dx/dt = -2, then what is dy/dt when x = 3 and y = -4?
Give examples of still other applications of derivatives.
Find dp/dq. P q sin q q² - 1
Water is flowing at the rate of 6 m3/min from a reservoir shaped like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius R is V = (π/3)y2(3R - y) when the water is y meters deep.a. At what
Find the derivatives of the function.y = (2x - 5)(4 - x)-1 S −1 15(15t-1)3
Find the derivatives of the function S −1 15(15t-1)3
What is logarithmic differentiation? Give an example.

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