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mathematics
precalculus
Questions and Answers of
Precalculus
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
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 -
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 =
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
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
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 =
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
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),
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 <
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
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
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
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
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
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
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
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, -#
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
Find the derivatives of the function.y = x2 cot 5x
Find the derivatives of the function S 4t t + 1 -2
Sand falls from a conveyor belt at the rate of 10 m3/min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) Height
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