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mathematics
precalculus

Questions and Answers of Precalculus

If f (x) = x46 + x45 + 2/1 + x, calculate f (46)(3). Express your answer using factorial notation:n! − 1 · 2 · 3 · ∙ ∙ ∙ · (n - 1) · n.
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
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
Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the
Differentiate the function.y = x5/3 - x2/3
Differentiate.y = c cos t + t2 sin t
Calculate y'.y = ln(x ln x)
Use Newton’s method to approximate the indicated root of the equation correct to six decimal places.Use Newton’s method to approximate the indicated root of the equation correct to six decimal
Find the values of the constants a and b such that limx → 0 3√ ax + b - 2/x = 5/12
The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?
(a) Explain how we know that the given equation must have a root in the given interval.(b) Use Newton’s method to approximate the root correct to six decimal places. —2х5 + 9х4 — 7x3 — 11х
Differentiate the function. g(x) = x? – 3x + 12
Differentiate.g(θ) = eθ(tan θ - θ)
Calculate y'.y = x cos-1 x
Find dy/dx by implicit differentiation.2x2 + xy - y2 = 2
An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point (0, a) on the y-axis (see the figure). Find the value of a that minimizes the lengths of the
The radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 8 cm?
Differentiate.g(x) = 1 + 2x/3 - 4x
Calculate y'.y = t4 - 1/t4 + 1
Find dy/dx by implicit differentiation.x4 + x2y2 + y3 = 5
(a) Explain how we know that the given equation must have a root in the given interval.(b) Use Newton’s method to approximate the root correct to six decimal places. 3x* – 8x³ + 2 = 0, [2, 3]
Prove the identity.sinh(-x) = -sinh x (This shows that sinh is an odd function.)
Differentiate.G(x) = x2 - 2/2x + 1
Differentiate.f (t) = cot t/et
Calculate y'.xey4 = y sin x
Find the most general antiderivative of the function. f(x) = Jx² + x/x
For what values of c does the curve y = cx3 + ex have inflection points?
Use Newton’s method to approximate the given number correct to eight decimal places.8√500
Evaluate the limit. lim (x — п) csc x х>п
Find the most general antiderivative of the function. f(x) = 3/x – 2x
Use Newton’s method to approximate the given number correct to eight decimal places.4√75
Evaluate the limit. lim (x? – x³)e2x
Find the derivative of the function.y = cos(1 - e2x/1 + e2x)
The rate sin mg carbon/m3/hd at which photosynthesis takes place for a species of phytoplankton is modeled by the functionwhere I is the light intensity (measured in thousands of foot candles). For
Find dy/dx by implicit differentiation.√x + y = x4 + y4
Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly. 2
Evaluate the limit. e2x – e-2x lim In(x + 1)
Differentiate.f(z) = (1 - ez)(z + ez)
Prove the identity.sinh(x + y) = sinh x cosh y + cosh x sinh y
Find all values of c such that the parabolas y = 4x2 and x = c + 2y2 intersect each other at right angles.
Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and ecrease and intervals of concavity, and use calculus to find these intervals
Differentiate.y = cos x/1 - sin x
Evaluate the limit. — 2х e2x – e lim >0 In(x + 1)
Find the derivative of the function.g(θ) = cos(θ2)
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. х lim *—3 х? — 9 x-3
Calculate y'.y = (arcsin 2x)2
Find dy/dx by implicit differentiation.cos(xy) = 1 + sin y
Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f'' to estimate the intervals of increase and decrease, extreme values,
Find the most general antiderivative of the function. |f(x) = x³4 – 2rv2-1
Evaluate (x + 2)/* – x/x lim * (x + 3)/x - x/x
Prove the identity.cosh(x + y) = cosh x cosh y + sinh x sinh y
Differentiate.y = x2 + 1/x3 - 1
Differentiate.y = t sin t/1 + t
Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) + 4 = 0, x1 - 1
Evaluate the limit. tan 4.x lim 0 x + sin 2.x
Find the derivative of the function.y = x2e-3x
Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.f(x) = sin(x/2), [π/2,
Differentiate.y = sin t/1 + tan t
Find the derivative of the function.f (t) = t sin π t
Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f'' to estimate the intervals of increase and decrease, extreme values,
Find the most general antiderivative of the function. = 7x²/5 + 8x-4/5 f(x)
Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) x² + 1 = 0, X1 = 2
Calculate y'.y = ln sec x
Evaluate the limit. e* – 1 lim x→0 tan x
Prove the identity.tanh(x + y) = tanh x + tanh/1 + tanh x tanh y
Use the graph to state the absolute and local maximum and minimum values of the function. У y=g(x) 1 х 1
Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.f(x) − x3 - 2x2 - 4x +
Differentiate.f (t) = tet cot t
What is the minimum vertical distance between the parabolas |у%3D х* + 1 аnd yх — х*?
Calculate y'.y = √arctan x
Differentiate the function. у 3 x (2 + х)
Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f'' to estimate the intervals of increase and decrease, extreme values,
Find the most general antiderivative of the function. f(x) = (x – 5)?
Find the most general antiderivative of the function. 2 5 f(x)
Evaluate the limit. х lim In x х — 1
(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.
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. 6x? + 5x – 4 lim x-1/2
Verify that the function satisfies the hypotheses of the ean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.f(x) = x3 - 3x + 2,
If P(a, a2) is any point on the parabola y = x2, except for the origin, let Q be the point where the normal line at P intersects the parabola again (see the figure).(a) Show that the y-coordinate of
Sketch the graph of a function f that is continuous on [1, 5] and has the given properties.Absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or
(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) = 2x3 - 9x2 + 12x
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.Alternatively, we could
Let f(x) = tan x. Show that f (0) = f (π) but there is no number c in (0, π) such that f' (c) = 0. Why does this not contradict Rolle’s Theorem?
Find the most general antiderivative of the function. |f(x) = e
(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) = x3 - 3x2 - 9x + 4
A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) iswhere k is a positive constant. What nitrogen level gives the
Find the most general antiderivative of the function. f(x) = /2
The graph of the first derivative f' of a function f is shown.(a) On what intervals is f increasing? Explain.(b) At what values of x does f have a local maximum or minimum? Explain.(c) On what
Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.f(x) − x + 1/x, [12 , 2]
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.ln(1 + x) ≈ x
The graph of a function f and its tangent line at 0 are shown. What is the value of limx → 2 f(x)/ex - 1? y. у 3х y= f(x) х
Differentiate the function. 5 F(r) .3
Calculate y'.y = tan x/1 + cos x
Find the derivative of the function.F(x) = (5x6 + 2x3)4
If a, b, c, and d are constants such that find the value of the sum a + b + c + d. ax? + sin bx + sin cx + sin dx 3x? + 5x* + 7x lim 8
Use the graphs of f and t and their tangent lines at (2, 0) to find limx → 2 f(x) g(x). y y=1.5(x– 2) / х у%32—х y=2-
Use Newton’s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) 2x3 — 3х? + 2
Calculate y'.y = x2 - x + 2/√x
Find the local and absolute extreme values of the function on the given interval. f(x) = xe, [-1, 3]

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