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study help
mathematics
precalculus
Questions and Answers of
Precalculus
A model for the specific gravity of water S iswhere T is the water temperature in degrees Celsius.(a) Use the second derivative to determine the concavity of S.(b) Use a computer algebra system to
Sketch the graph of the arbitrary function f such that > 0, f'(x) undefined, 4
Can the graph of a function cross a horizontal asymptote? Explain.
(a) Use a computer algebra system to differentiate the function,(b) Sketch the graphs of f and f′ on the same set of coordinate axes over the given interval,(c) Find the critical numbers of f in
Consider(a) Use the definition of limits at infinity to find the value of N that corresponds to ε = 0.5.(b) Use the definition of limits at infinity to find the value of N that corresponds to ε =
Consider the function f(x) = 3√x.(a) Graph the function and identify the inflection point.(b) Does f″ exist at the inflection point? Explain.
Is the sum of two increasing functions always increasing? Explain.
Is the product of two increasing functions always increasing? Explain.
Use the graph of f′ to(a) Identify the critical numbers of f,(b) Identify the open intervals on which f is increasing or decreasing(c) Determine whether f has a relative maximum, a relative
The function f is differentiable on the indicated interval. The table shows f′(x) for selected values of x.(a) Sketch the graph of f,(b) Approximate the critical numbers, and(c) Identify the
A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is. The distance (in meters) the ball bearing rolls in t seconds is s(t) = 4.9(sin )t2.(a)
The function f is differentiable on the indicated interval. The table shows f′(x) for selected values of x.(a) Sketch the graph of f,(b) Approximate the critical numbers, and(c) Identify the
The end-of-year assets of the Medicare Hospital Insurance Trust Fund (in billions of dollars) for the years 2006 through 2014 are shown.(a) Use the regression capabilities of a graphing utility to
Consider the functions f(x) = x and g(x) = sin x on the interval (0, π).(a) Complete the table and make a conjecture about which is the greater function on the interval (0, π).(b) Use a graphing
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The graph of f(x) = 1/x is concave downward for x < 0 and concave upward for
The function s(t) describes the motion of a particle along a line.(a) Find the velocity function of the particle at any time t ≥ 0.(b) Identify the time interval(s) on which the particle is moving
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.There is no function with an infinite number of critical points.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The function f(x) = x has no extrema on any open interval.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Every nth-degree polynomial has (n − 1) critical numbers.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.An nth-degree polynomial has at most (n − 1) critical numbers.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.There is a relative extremum at each critical number.
Find the minimum value of ∣sin x + cos x + tan x + cot x + sec x + csc x∣ for real numbers x
Leta. Find limx→2+ ƒ(x) and limx→2- ƒ(x).b. Does limx→2 ƒ(x) exist? If so, what is it? If not, why not?c. Find limx→4- ƒ(x) and limx→4+ ƒ(x).d. Does limx→4 ƒ(x) exist? If so, what
Say whether the function graphed is continuous on [-1, 3]. If not, where does it fail to be continuous and why? y 2 1 0 y = h(x) 2 3 →X
For the function ƒ(t) graphed here, find the following limits or explain why they do not exist.a.b.c.d. lim f(t) t- 1--2
For the function g(x) graphed here, find the following limits or explain why they do not exist.a.b.c.d. lim g(x) x-1
Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?a. limx→2ƒ(x) does not exist.b. limx→2ƒ(x) = 2c. limx→1ƒ(x) does not
Leta. Does limx→0+ ƒ(x) exist? If so, what is it? If not, why not?b. Does limx→0- ƒ(x) exist? If so, what is it? If not, why not?c. Does limx→0 ƒ(x) exist? If so, what is it? If not, why
Say whether the function graphed is continuous on [-1, 3]. If not, where does it fail to be continuous and why? y 2 0 y = k(x) 2 3 X
Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?a.b.c.d.e.f.exists at every point c in (-1, 1).g.does not exist. lim f(x) exists. x-0
Say whether the function graphed is continuous on [-1, 3]. If not, where does it fail to be continuous and why? y 2 0 1 y = g(x) 2 3 X
The functiongraphed in the accompanying figure.a. Does ƒ(-1) exist?b. Does limx→-1+ ƒ(x) exist?c. Does limx→-1+ ƒ(x) = ƒ(-1)?d. Is ƒ continuous at x = -1? f(x) = x²-1, 2x, 1, -2x +
Leta. Find limx→2+ ƒ(x), limx→2- ƒ(x), and ƒ(2).b. Does limx→2 ƒ(x) exist? If so, what is it? If not, why not?c. Find limx→ -1- ƒ(x) and limx→ -1+ ƒ(x).d. Does limx→ -1
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = 1, b = 7, c = 5
Explain why the limits do not exist. X lim x-0 |x|
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = 1, b = 7, c = 2
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = -7>2, b = -1>2, c = -3
Leta. Does limx→0+ g(x) exist? If so, what is it? If not, why not?b. Does limx→0- g(x) exist? If so, what is it? If not, why not?c. Does limx→0 g(x) exist? If so, what is it? If not, why not?
The functiongraphed in the accompanying figure.a. Does ƒ(1) exist?b. Does limx→1 ƒ(x) exist?c. Does limx→1 ƒ(x) = ƒ(1)?d. Is ƒ continuous at x = 1? f(x) = x²-1, 2x, 1, -2x + 4, 0, -1≤ x <
The functiongraphed in the accompanying figure.At what values of x is ƒ continuous? f(x) = x²-1, 2x, 1, -2x + 4, 0, -1≤ x < 0 0 < x < 1 x = 1 1 < x < 2 2
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = -7/2, b = -1/2, c = -3/2
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
Explain why the limits do not exist. 1 lim x-1 X - 1
The functiongraphed in the accompanying figure.a. Is ƒ defined at x = 2? (Look at the definition of ƒ.)b. Is ƒ continuous at x = 2? f(x) = x²-1, 2x, 1, -2x + 4, 0, -1≤ x < 0 0 < x < 1 x = 1 1 <
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = 4/9, b = 4/7, c = 1/2
Sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of δ < 0 such that for all x, 0 < | x - c | < δ ⇒ a < x < b.a = 2.7591, b = 3.2391, c = 3
Suppose that a function ƒ(x) is defined for all real values of x except x = c. Can anything be said about the existence of limx→cƒ(x)? Give reasons for your answer.
Graph the function.a. What are the domain and range of ƒ?b. At what points c, if any, does limx→c ƒ(x) exist?c. At what points does only the left-hand limit exist?d. At what points does only the
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
Suppose that a function ƒ(x) is defined for all x in [-1, 1]. Can anything be said about the existence of limx→0ƒ(x)? Give reasons for your answer.
The functiongraphed in the accompanying figure.What value should be assigned to ƒ(2) to make the extended function continuous at x = 2? f(x) = x²-1, 2x, 1, -2x + 4, 0, -1≤ x < 0 0 < x < 1 x = 1 1
If limx→1ƒ(x) = 5, must ƒ be defined at x = 1? If it is, must ƒ(1) = 5? Can we conclude anything about the values of ƒ at x = 1? Explain.
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
The functiongraphed in the accompanying figure.To what new value should ƒ(1) be changed to remove the discontinuity? f(x) = x²-1, 2x, 1, -2x + 4, 0, -1≤ x < 0 0 < x < 1 x = 1 1 < x < 2 2
Graph the function.a. What are the domain and range of ƒ?b. At what points c, if any, does limx→c ƒ(x) exist?c. At what points does only the left-hand limit exist?d. At what points does only the
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
If ƒ(1) = 5, must limx→1ƒ(x) exist? If it does, then must limx→1ƒ(x) = 5? Can we conclude anything about limx→1ƒ(x)? Explain.
At what points are the function. y = 1 (x + 2)² +4
At what points are the function. У 1 x - 2 - 3x
Use the graphs to find a δ > 0 such that for all x 0 < |x − c < 8 = ⇒>> |f(x) − L| < €.
The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill.a. Estimate the slopes of secants PQ1, PQ2, PQ3, and PQ4, arranging them in order in a table
Use the Intermediate Value Theorem to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.x3 - 15x + 1 = 0 (three roots)
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y = x + 1 x² - 4x + 3
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. X x + 3 y x² - 3x - 10
At what points are the function. y = |x-1| + sin x
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. 1 x+1 2
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y COS X X
At what points are the function. y = x tan x 2 x² + 1
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y = V2x + 3
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y = √3x - 1
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y = (2 - x)1/5
Gives a function ƒ(x), a point c, and a positive number ϵ. Find L = limx→c ƒ(x). Then find a number δ > 0 such that for all x 0 < x − c < 8 - U |f(x) - L < €.
The accompanying graph shows the total distance s traveled by a bicyclist after t hours.a. Estimate the bicyclist’s average speed over the time intervals [0, 1], [1, 2.5] , and [2.5, 3.5].b.
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
At what points are the function. y = x + 2 COS X
Gives a function ƒ(x), a point c, and a positive number ϵ. Find L = limx→c ƒ(x). Then find a number δ > 0 such that for all x 0 < x − c < 8 - U |f(x) - L < €.
Prove the limit statement. lim (9-x) = 5 x→4
Prove the limit statement. lim (3x - 7) = 2 x-3
Prove the limit statement. lim Vx 5 = 2 x-9
Gives a function ƒ(x), a point c, and a positive number ϵ. Find L = limx→c ƒ(x). Then find a number δ > 0 such that for all x 0 < x − c < 8 - U |f(x) - L < €.
Gives a function ƒ(x), a point c, and a positive number ϵ. Find L = limx→c ƒ(x). Then find a number δ > 0 such that for all x 0 < x − c < 8 - U |f(x) - L < €.
The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after being driven for t days.a. Estimate the average rate of gasoline consumption over the time intervals
Prove the limit statement. - = 1 if f(x) = lim f(x) = 1 (x², 2, x = 1 1 X =
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
Gives a function ƒ(x) and numbers L, c, and ϵ > 0. In each case, find an open interval about c on which the inequality |ƒ(x) - L | < ϵ holds. Then give a value for δ > 0 such that for
Prove the limit statement. x² lim x→1 X - 1 1 = 2
For what values of a and b iscontinuous at every x? f(x) = -2, ax − b, −1
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