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mathematics
precalculus

Questions and Answers of Precalculus

Evaluate the integral, if it exists. х 10 dx x2 – 4
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
Find the general indefinite integral. Vi? + 3t + 2) dt
Evaluate the indefinite integral. | sint /1 + cost dt
Use an integral to estimate the sum 10000 Σ ν. i-1
The graph of f consists of the three line segments shown. If g(x) = föf(t) dt, find g(4) and g'(4).
Use the guidelines of Section 4.5 to sketch the curve. y = x2/3(x – 3)
Use the guidelines of this section to sketch the curve. /1 — х2 y = х
Find f.
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. |4e¯* sin x = x² – x + 1
Use the guidelines of Section 4.5 to sketch the curve. у %3D х/2 + х y =
Sketch the graph of a function that satisfies all of the given conditions.Vertical asymptote x = 0, f' (x) > 0 if x < -2,f'(x) < 0 if x > -2 (x ± 0),f''(x) < 0 if x < 0, f' (x)
Use the guidelines of this section to sketch the curve. y = x/2 – x?
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1'Hospital’s Rule doesn’t apply, explain why. e"/10 lim .3 и"
Suppose that 3 < f '(x) < 5 for all values of x. Show that 18 < f (8) - f (2) < 30.
Find f.
A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere,
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. |cos(x? – x) = x*
Sketch the graph of a function that satisfies all of the given conditions.(a) f'(x) < 0 and f'' (x) < 0 for all x(b) f'(x) > 0 and f'' (x) > 0 for all x
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1'Hospital’s Rule doesn’t apply, explain why. VI+ 2х — 1 — 4х lim 1
Find f.
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. х --х .2 x² + 1
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.f (x) = |x|
Sketch the graph of a function that satisfies all of the given conditions.(a) f'(x) < 0 and f'' (x) < 0 for all x(b) f'(x) > 0 and f'' (x) > 0 for all x
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1'Hospital’s Rule doesn’t apply, explain why. 8' – 5' lim t
Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 4 – 3(1 + x²)¯!, F(1) = 0
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. - 3x* + x³ – x² – x + 6 = 0 х°
Use the guidelines of this section to sketch the curve. y = x/3 – 5r²/3
Use the guidelines of Section 4.5 to sketch the curve. y = sin-'(1/x)
Find f. f'(x) = 1 + 3/x, f(4) = 25
Sketch the graph of a function that satisfies all of the given conditions. 0 ifx < 2, | f"(x) < 0 if x > 2, ƒ has inflection point (2, 5), lim f(x) = 8, lim f(x) = 0 " style="" class="fr-fic
Use the guidelines of Section 4.5 to sketch the curve. y = e2x¬x3 2х-х
Find f. f'(x) = 5x* – 3x² + 4, f(-1) = 2 %3|
Use the guidelines of this section to sketch the curve. y = Vx3 + 1
Find f. f'(t) = t + 1/t³, t>0, ƒ(1) = 6
Find the critical numbers of the function.g(t) = |3t - 4|
Use the guidelines of Section 4.5 to sketch the curve. y = x + In(x² + 1)
Use the guidelines of Section 4.5 to sketch the curve. у 3 (х — 2)е-* —х
Find f. f'(t) = 4/(1 + t²), ƒ(1) = 0 %3D
Use the guidelines of Section 4.5 to sketch the curve. y = 3x* - 4x3 + 2
Suppose f 0 is continuous on (-∞, ∞).(a) If f'(2) = 0 and f'(2) = 25, what can you say about f ?(b) If f'(6) = 0 and f'(6) = 0, what can you say about f ?
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1'Hospital’s Rule doesn’t apply, explain why. t8 – 1 lim 15 →1
Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 5x* – 2x, F(0) = 4
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. -2x – 5x* + 9x³ + 5 = 0|
Use the guidelines of Section 4.5 to sketch the curve. х y : 1 — х2 .
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.f(t) = cos t, -3π/2 < t < 3π/2
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1'Hospital’s Rule doesn’t apply, explain why. In/x lim х х X-
Find the most general antiderivative of the function. 2x² + 5 f(x) x? + 1
Use Newton’s method to find all solutions of the equation correct to six decimal places. |sin x = x² – 2
Use the guidelines of Section 4.5 to sketch the curve. y x(х — 3)?
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.f(x) = sin x, -π/2 < x < π/2
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?f(x) = √x - 4√x
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1’Hospital’s Rule doesn’t apply, explain why. In x lim x-0+ x
Find the most general antiderivative of the function. 2x* + 4x3 — х х3 f(x) х
Use Newton’s method to find all solutions of the equation correct to six decimal places. tan'x .3 .3 х
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.f(x) = sin x, 0 < x < π/2
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which metho d do you prefer?f(x) = x2/x - 1
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1’Hospital’s Rule doesn’t apply, explain why. x + x² lim x-0 1- 2x?
Show that the equation has exactly one real root.x3 + ex = 0
Find the most general antiderivative of the function. 3 g(v) = 2 cos v 1 – v²
Use Newton’s method to find all solutions of the equation correct to six decimal places.√x + 1 = x2 - x
The most general antiderivative of f (x) = x-2 isF(x) = -1/x + C
Find the most general antiderivative of the function. h(0) = 2 sin e – sec20
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1’Hospital’s Rule doesn’t apply, explain why. 1 - sin 0 lim 0-T/2 1 +
Use Newton’s method to find all solutions of the equation correct to six decimal places.3 cos x = x + 1
If f is periodic, then f' is periodic. F(x)
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.f(x) = 2 - 1/3 x, x > -2
(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.f(x) = x2 ln x
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1’Hospital’s Rule doesn’t apply, explain why. .2 lim x-0 1 - cos x
Use Newton’s method to approximate the indicated root of the equation correct to six decimal places.The positive root of 3 sin x = x
Sketch the graph of a function that satisfies the given conditions. f(0) = 0, f is continuous and even, f'(x) = 2x if 0 3
If f is even, then f' is even.
(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.f(x) = e-x + e-x
Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.
Find, correct to two decimal places, the coordinates of the point on the curve y = sin x that is closest to the point (4, 2).
Find the point on the curve y = sx that is closest to the point (3, 0).
Find the point on the line y = 2x + 3 that is closest to the origin.
If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence? Data from in exercise 18A farmer wants to fence in a rectangular plot of
A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a
Do Exercise 16 assuming the container has a lid that is made from the same material as the sides.
A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width.Material for the base costs $10 per square meter. Material for the sides costs
If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points
Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.ex cos x ≈ 1 + x
Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points
Differentiate.y = x/2 - tan x
Find an equation of the tangent line to the curve at the given point. у %3Dх+ —, (2, 3) х
Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points
Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.4√1 + 2x ≈ 1 + 1/2 x
The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle
Differentiate the function.f (x) = e5
Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points
Differentiate.y = 2 sec x - csc x
Graph the function using as many viewing rectangles as you need to depict the true nature of the function. f(x) = e* + In|x – 4|
Calculate y'.y = x2 sin x
Find dy/dx by implicit differentiation.x2 - 4xy + y2 = 4
Use a computer algebra system to graph f and to find f' and f''. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and
Use a computer algebra system to graph f and to find f' and f''. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and
Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1.(1 + x)-3 ≈ 1 - 3x
Use a computer algebra system to graph f and to find f' and f''. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and

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