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mathematics
precalculus

Questions and Answers of Precalculus

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. lim e* - 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. – cot x) lim (csc x l
Find the limit. Use l’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 In 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. lim cos x sec 5x x>(T/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. lim In x tan(7x/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. lim x/2 sin(1/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. lim x'e* 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. lim x In 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. lim sin 5x csc 3x
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. lim Vre2 x/2 -x/2 хе
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. lim x sin(7/x) TT
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. cos x In(x – a) lim In(e
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. 1.2 cos x - 1 + x lim 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* - e* – 2x lim х —
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. ха — 1 lim 1xв — Т
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* - 1 lim x0+ In x + 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. arctan(2.x) lim In 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. x sin(x – 1) lim 2x? —
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(1 + x) x0 cos x + er
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. cos mx - cos nx .2 lim 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. x3* lim x0 3* - 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. (In x)² lim 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. sinx lim х
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 - sin x lim x-0 x - tan 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. tanh x lim x 0 tan r
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. sinh x – x lim .3 х 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* – 1- x lim x2
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
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 + cos 0 lim cos e
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. tan 3x lim x-0 sin 2.x х
For what values of c is the function f(x) = cx + 1/x2 + 3 increasing on s?
Suppose f and t are both concave upward on (-∞, ∞). Under what condition on f will the composite function h(x) = f ( g(x)) be concave upward?
Assume that all of the functions are twice differentiable and the second derivatives are never 0.(a) If f and t are concave upward on I, show that f + t is concave upward on I.(b) If f is positive
Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.
(a) If the function f (x) = x3 + ax2 + bx has the local minimum value -2/9√ 3 at x = 1/√3 , what are the values of a and b?(b) Which of the tangent lines to the curve in part (a) has the smallest
Let f (t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case?(a) f' (3) = 2, f'' (3) = 4(b) f'
The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.
In an episode of The Simpsons television show, Homer reads from a newspaper and announces “Here’s good news! According to this eye-catching article, SAT scores are declining at a slower rate.”
Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f(x) = x - tan-1 x/1 + x3
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.(b) Use a graph of f 0 to give better estimates.f (x) = √(x - 1)2 (x + 1)3
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.(b) Use a graph of f 0 to give better estimates.f (x) = sin 2x + sin 4x, 0
Use the methods of this section to sketch the curve y = x3 - 3a2x + 2a3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each
(a)Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
(a)Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
Find the absolute maximum and absolute minimum values of f on the given interval.f (t) = t + cot (t/2), [π/4, 7π/4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (t) = 2cos t + sin 2t, [0, π/2]
Find the critical numbers of the function.f (θ) = 2 cosθ + sin2θ
Find the critical numbers of the function.g(θ) = 4θ - tanθ
Find the critical numbers of the function.f (x) = x3 + 6x2 - 15x
Find the critical numbers of the function.f (x) = 4 + 1/3 x - 1/2x2
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. 2x + 1 if 0
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
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) = 1/x, 1 < x < 3
If f(x) = limt → x sec t - sec x/t - x , find the value of f' (π/4).
Calculate y'.y = cot (3x2 + 5)
Calculate y'.y = e1/x/x2
Calculate y'.y = √x cos √x
Calculate y'.y = emx cos nx
Prove the identity.cosh x - sinh x = e2x
Prove the identity.cosh x + sinh x = ex
Prove the identity.cosh(-x) = cosh x (This shows that cosh is an even function.)
If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/dt in terms of dx/dt.
(a) If $1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a)–(c)
The graph of the derivative f' of a continuous function f is shown.(a) On what intervals is f increasing? Decreasing?(b) At what values of x does f have a local maximum? Local minimum?(c) On what
The graph of a function y = f (x) is shown. At which point(s) are the following true? d'y dy and dx? are both positive. (a) dx d²y dy and dx (b) are both negative. dx2 d'y is positive. dx? dy ·is
Suppose f(3) = 2, f '(3) = 1/2 , and f'(x) > 0 and f''(x), < 0 for all x.(a) Sketch a possible graph for f.(b) How many solutions does the equation f (x) = 0 have? Why?(c) Is it possible that
Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(4) = 0, f'(x) = 1 if x< -1, f'(x) > 0 if 0 < x < 2, f'(x) < 0 if –1 4
Sketch the graph of a function that satisfies all of the given conditions. 8, f"(x) > 0 fo" style="" class="fr-fic fr-dib"> f'(5) = 0, f'(x) < 0 when x < 5, f'(x) > 0 when x > 5, f"(2) = 0, f"(8) =
Sketch the graph of a function that satisfies all of the given conditions. 0 if x < 1 or x > 3, f"(x) < 0 if 1 < x< 3 " style="" class="fr-fic fr-dib"> f'(x) > 0 for all x + 1, vertical asymptote x =
(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) = x4e2x
(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 - x - ln
(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) = sin x + cos x, 0
A number a is called a fixed point of a function f if f (a) = a. Prove that if f'(x) ≠ 1 for all real numbers x, then f has at most one fixed point.
Show that sin x < x if 0 < x < 2π.
Let f (x) = 2 - |2x - 1|. Show that there is no value of c such that f (3) - f (0) = f' (c)(3 - 0). Why does this not contradict the Mean Value Theorem?
Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at (c, f (c)). Are the
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) = 1/x, [1, 3]
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) = 2x2 - 3x + 1,
The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function L(t) = 0.01441t3 - 0.4177t2 + 2.703t + 1060.1 where t is measured in
After an antibiotic tablet is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t) = 8 (e-0.4t - e-0.6t) where the time t is measured in hours and C is
After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline
(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimum values.f (x) = x2 - cos x, - 2 < x
(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimum values.f (x) = x√x - x2
(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimum values.f (x) = ex + e-2x, 0 < x
Use a graph to estimate the critical numbers of f (x) = |1 + 5x - x3| correct to one decimal place.
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = x - 2 tan-1 x, [0, 4]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = ln (x2 + x + 1), [-1, 1]
Find the absolute maximum and absolute minimum values of f on the given interval.f (x) = xex/2, [-3, 1]

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