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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = x√8 - x²
The height above ground of an object moving vertically is given by s = -16t2 + 96t + 112, with s in feet and t in seconds. Finda. The object’s velocity when t = 0;b. Its maximum height and when it occurs;c. Its velocity when s = 0.
Find all possible functions with the given derivative.a.b.c. y' 1 2√x
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. f(x) = 好 3x² + 1
Use l’Hôpital’s rule to find the limit. lim y-0 Vay + a² y a a> 0
Find all possible functions with the given derivative.a.b.c. y' || I x²
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x2/3(x - 5)
What values of a and b make ƒ(x) = x3 + ax2 + bx havea. A local maximum at x = -1 and a local minimum at x = 3?b. A local minimum at x = 4 and a point of inflection at x = 1?
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. f(x) x² - 3 x 2' x = 2
Use l’Hôpital’s rule to find the limit. lim y 0 V5y + 25 - 5 y
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = x2/3 5 NIU X
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(x)=x²√5 - x
Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3. h W r = 3
Use l’Hôpital’s rule to find the limit. lim x→0+ In (ex - 1) In x
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. zx - 8^x = (x)8 8.
What value of a makes ƒ(x) = x2 + (a/x) havea. A local minimum at x = 2?b. A point of inflection at x = 1?
Use l’Hôpital’s rule to find the limit. lim x-0+ 2 In (x² + 2x) In x
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. E + ₂x - x^+ = (x)8 x^t=
Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure. 5 h 3 W 4
Use l’Hôpital’s rule to find the limit. log₂ x lim x→∞log3 (x + 3) 83
Find all possible functions with the given derivative.a. y′ = 2x b. y′ = 2x - 1 c. y′ = 3x2 + 2x - 1
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 5x2/5 - 2x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = 1-x² 2 2x + 1
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = 2x - 3x2/3
Find all possible functions with the given derivative.a. y′ = x b. y′ = x2 c. y′ = x3
Use l’Hôpital’s rule to find the limit. In (x + 1) lim xxx log₂ x
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. f(x)= x 6√x - 1
What can be said about functions whose derivatives are constant? Give reasons for your answer.
Suppose that ƒ′(x) = 2x for all x. Find ƒ(2) ifa. ƒ(0) = 0 b. ƒ(1) = 0 c. ƒ(-2) = 3.
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y || X 2 √x² + 1
Use l’Hôpital’s rule to find the limit. 3x - 1 lim x-02-1
Suppose that ƒ(0) = 5 and that ƒ′(x) = 2 for all x. Must ƒ(x) = 2x + 5 for all x? Give reasons for your answer.
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.K(t) = 15t3 - t5
Use l’Hôpital’s rule to find the limit. x2x lim x-02-1
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 -214-16 H(t) =
Suppose that ƒ(-1) = 3 and that ƒ′(x) = 0 for all x. Must ƒ(x) = 3 for all x? Give reasons for your answer.
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x2/5
Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.
Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible.
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(x) = x4 - 4x3 + 4x2
Use l’Hôpital’s rule to find the limit. lim 0-0 (1/2)⁰ - 1 0
Show that the function have exactly one zero in the given interval.r(θ) = tan θ - cot θ - θ, (0, π/2)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x1/5
Find the point on the line x/a +y/b = 1 that is closest to the origin.
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = cos x + √3 sinx, 0≤ x ≤ 2πTT
Show that the function have exactly one zero in the given interval. r(0) = sec 0 1 03 + 5, (0, π/2)
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(x² - 1)²/3 y 0 X
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
Use l’Hôpital’s rule to find the limit. lim 0-0 3 sin 0 0 1
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.ƒ(x) = x4 - 8x2 + 16
A right triangle whose hypotenuse is √3 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way. h T V3
Show that the function have exactly one zero in the given interval. 20 - cos² 0 + √2, - cos² 0 + √2, (-∞, ∞) r(0) = 20
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = sin x cos x, 0 ≤ x ≤ π
Use l’Hôpital’s rule to find the limit. lim X→ (π/2)¯¯ ( 127 - x tan x
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.h(r) = (r + 7)3
Compare the answers to the following two construction problems.a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?b. The same sheet is to be revolved about one of
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y || = X tan X, F|C V V
Show that the function have exactly one zero in the given interval. r(0) = 0 + sin² Ө Ꮎ 3 - 8, (-∞, ∞)
Use l’Hôpital’s rule to find the limit. lim X→ (π/2)¯ X - TT 2 sec x
A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it
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.ƒ(r) = 3r3 + 16r
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = √3x - 2 cos x, 0 ≤ x ≤ 2T
Show that the function have exactly one zero in the given interval. g(t) = 1 1 + V1 + t - 3.1, (-1,1)
Use l’Hôpital’s rule to find the limit. lim 1-0 01 t sin t cos t
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.ƒ(θ) = 6θ - θ3
The trough in the figure is to be made to the dimensions shown. Only the angle θ can be varied. What value of θ will maximize the trough’s volume? a 20
Show that the function have exactly one zero in the given interval. g(t) = √t + V1 + 1 - 4, (0, ∞)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x - sin x, 0 ≤ x ≤ 2π
Use l’Hôpital’s rule to find the limit. t(1 cos t) lim 1-0 t - sin t
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.ƒ(u) = 3θ2 - 4θ3
A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and
Show that the function have exactly one zero in the given interval. f(x) = x³ + 4 +7, (-∞, 0)
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x + sin x, 0 ≤ x ≤ 2π
Use l’Hôpital’s rule to find the limit. In (csc x) lim xπ/2 (x- (π/2))² (X
A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit
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.h(x) = 2x3 - 18x
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = x( X 2 5 4
Use l’Hôpital’s rule to find the limit. -2 lim x-oln (sec x)
Show that the function have exactly one zero in the given interval.ƒ(x) = x4 + 3x + 1, [-2, -1]
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.h(x) = -x3 + 2x2
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. 뮤 0 2x² + 4 X
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)(x + 2)
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. 물-물- y 0 - 2x + 1/1/213 X
Use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method. lim x-0 sin 5x X
What is the smallest perimeter possible for a rectangle whose area is 16 in2, and what are its dimensions?
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 + 2x - 1, [0, 1] f(b) f(a) b-a = f'(c)
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. 0 X< A y (L = 2xx =
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
A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 12 - x2. What is the largest area the rectangle can have, and what are its dimensions?
a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are h by h by w and the girth is 2h + 2w. What dimensions will give the box its largest volume now?b. Graph the volume as a function of h and compare what you see with your answer in
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x5 - 5x4 = x4(x - 5)
a. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?b. Graph the volume of a 108-in. box (length plus girth equals 108 in.)
Use l’Hôpital’s rule to find the limit. lim x→iln x x - 1 sin TTX
Use l’Hôpital’s rule to find the limit. 1 sin 0 lim 07/21 cos 20 +
Show that a cubic polynomial can have at most three real zeros.
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) = -3t2 + 9t + 5
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. y = f(x) -3-2-1 y 2 1 -1 -2 1 2 3 > X
Use l’Hôpital’s rule to find the limit. 30 + T lim 0/3 sin (0+ (π/3))
Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.y = x4 + 2x3 = x3(x + 2)
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