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mathematics
precalculus
Questions and Answers of
Precalculus
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
For what values of a and b iscontinuous at every x? g(x) = ax + 2b, x² + 3ab, 3x - 5, x≤0 0 < x≤ 2 x > 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
At what points are the function. y = (2x - 1)¹/3
Prove the limit statement. lim f(x) = 2 if f(x) x→1 4- 2x, 6x - 4, x < 1 x ≥ 1
Prove the limit statement. x² - 9 lim x -3x + 3 -6
For what value of b iscontinuous at every x? g(x): = x - b b + 1' x² + b, x > 0 x < 0
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 x² sin = 0 x->0 alk y 1 y = x² y = x² sin / alk y = -x²\ X<
At what points are the function. g(x) = x² - x - 6 x - 3 5, x = 3 x = 3
At what points are the function. f(x) = - 8 x² - 4' 3, 4, x = 2, x = -2 x = 2 x = -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
Prove the limit statement. lim_x sin = 0 == x-0 FALA 2π 2πT y = x sin X
Graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If
Graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If
Prove the limit statement. lim f(x) = 0 if f(x) x-0 [2.x, Lx/2₂ x < 0 x ≥ 0
Graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If
Make a table of values for the function F(x) = (x + 2)/(x - 2) at the points x = 1.2, x = 11/10, x = 101/100, x = 1001/1000, x = 10001/10000, and x = 1.a. Find the average rate of change of F(x) over
Suppose limx→c ƒ(x) = 5 and limx→c g(x) = -2. Finda.b.c.d. lim f(x)g(x) X-C
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_ f(x) = 4 if f(x) x→-2 [x², (1, x = -2 x = -2
Prove the limit statement. lim V4 - x = 2 x-0
Prove the limit statement. lim x→1 1 X = 1
Prove the limit statement. 1 lim x→√3 x² || 1 3
Suppose limx→4 ƒ(x) = 0 and limx→4 g(x) = -3. Finda.b.c.d. lim (g(x) + 3) x-4
Let g(x) = √x for x ≥ 0.a. Find the average rate of change of g(x) with respect to x over the intervals [1, 2], [1, 1.5] and [1, 1 + h].b. Make a table of values of the average rate of change of
Let ƒ(t) = 1/t for t ≠ 0.a. Find the average rate of change of ƒ with respect to t over the intervals (i) from t = 2 to t = 3, and (ii) from t = 2 to t = T.b. Make a table of values of the
At what points are the function.y = tan πx/2
At what points are the function.y = csc 2x
Graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If
Ohm’s law for electrical circuits like the one shown in the accompanying figure states that V = RI. In this equation, V is a constant voltage, I is the current in amperes, and R is the resistance
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
Define g(3) in a way that extends g(x) = (x2 - 9)/(x - 3) to be continuous at x = 3.
Suppose limx→b ƒ(x) = 7 and limx→b g(x) = -3. Finda.b.c.d. lim (f(x) + g(x)) x-b
Define g(4) in a way that extends g(x) = (x2 - 16)/ (x2 - 3x - 4) to be continuous at x = 4.
Once you know limx→a+ ƒ(x) and limx→a- ƒ(x) at an interior point of the domain of ƒ, do you then know limx→a ƒ(x)? Give reasons for your answer.
Define ƒ(1) in a way that extends ƒ(s) = (s3 - 1)/(s2 - 1) to be continuous at s = 1.
If you know that limx→c ƒ(x) exists, can you find its value by calculating limx→c+ ƒ(x)? Give reasons for your answer.
Define h(2) in a way that extends h(t) = (t2 + 3t - 10)/(t - 2) to be continuous at t = 2.
Suppose that limx→-2 p(x) = 4, limx→-2 r(x) = 0, and limx→-2 s(x) = -3. Finda.b.c. lim (p(x) +r(x) + s(x)) x→-2
Suppose that ƒ is an even function of x. Does knowing that limx→2- ƒ(x) = 7 tell you anything about either limx→-2- ƒ(x) or limx→-2+ ƒ(x)? Give reasons for your answer.
Suppose that ƒ is an odd function of x. Does knowing that limx→0+ ƒ(x) = 3 tell you anything about limx→0- ƒ(x)? Give reasons for your answer.
Prove that limx→c ƒ(x) = L if and only if limh→0 ƒ(h + c) = L.
Define what it means to say that limx→0g(x) = k.
A continuous function y = ƒ(x) is known to be negative at x = 0 and positive at x = 1. Why does the equation ƒ(x) = 0 have at least one solution between x = 0 and x = 1? Illustrate with a sketch.
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
Explain why the equation cos x = x has at least one solution.
Before contracting to grind engine cylinders to a cross-sectional area of 9 in2, you need to know how much deviation from the ideal cylinder diameter of c = 3.385 in. you can allow and still have the
Show that the equation x3 - 15x + 1 = 0 has three solutions in the interval [-4, 4].
Show that the function F(x) = (x - a)2 · (x - b)2 + x takes on the value (a + b)/2 for some value of x.
If ƒ(x) = x3 - 8x + 10, show that there are values c for which ƒ(c) equals (a) π; (b) -√3; (c) 5,000,000.
LetShow thata. limx→2 h(x) ≠ 4b. limx→2h(x) ≠ 3c. limx→2h(x) ≠ 2 h(x) = x², x < 2 x = 2 3, 2, x > 2.
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
Explain why the following five statements ask for the same information.a. Find the roots of ƒ(x) = x3 - 3x - 1.b. Find the x-coordinates of the points where the curve y = x3 crosses the line y = 3x
Give an example of a function ƒ(x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that ƒ is discontinuous at x = 2, and how you
For the function graphed here, explain whya. limx→3 f(x) ≠ 4b. limx→3f(x) ≠ 4.8c. limx→3f(x) ≠ 3 4.8 4 ننا 3 0 3 y = f(x) X
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
Give an example of a function g(x) that is continuous for all values of x except x = -1, where it has a nonremovable discontinuity. Explain how you know that g is discontinuous there and why the
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the functionis discontinuous at every point.b. Is ƒ right-continuous or
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the formoccur frequently in calculus. Evaluate this limit for the given value of x and function ƒ.ƒ(x) =
If functions ƒ(x) and g(x) are continuous for 0 ≤ x ≤ 1, could ƒ(x)/g(x) possibly be discontinuous at a point of [0, 1]? Give reasons for your answer.
If the product function h(x) = ƒ(x) · g(x) is continuous at x = 0, must ƒ(x) and g(x) be continuous at x = 0? Give reasons for your answer.
a. It can be shown that the inequalitieshold for all values of x close to zero. What, if anything, does this tell you aboutGive reasons for your answer.b. Graph y = 1 - (x2/6), y = (x sin x)/(2 - 2
If √5 - 2x2 ≤ ƒ(x) ≤ √5 - x2 for -1 ≤ x ≤ 1, find limx→0 ƒ(x).
a. Suppose that the inequalitieshold for values of x close to zero. What, if anything, does this tell you aboutGive reasons for your answer.b. Graph the equations y = (1/2) - (x2/24), y = (1 - cos
Prove that ƒ is continuous at c if and only if lim f(c + h) = f(c). h→0
Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.
If 2 - x2 ≤ g(x) ≤ 2 cos x for all x, find limx→0 g(x).
Give an example of functions ƒ and g, both continuous at x = 0, for which the composite f ° g is discontinuous at x = 0. Does this contradict Theorem 9? Give reasons for your answer.
Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer.
Let g(x) = (x2 - 2)/(x - √2).a. Make a table of the values of g at the points x = 1.4, 1.41, 1.414, and so on through successive decimal approximations of 22. Estimate limx→√2 g(x).b. Support
Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0 ≤ ƒ(x) ≤ 1 for every x in [0, 1]. Show that there must exist a number c in [0, 1] such that ƒ(c) = c (c is
Let ƒ be defined on an interval (a, b) and suppose that ƒ(c) ≠ 0 at some c where ƒ is continuous. Show that there is an interval (c - δ, c + δ) about c where ƒ has the same sign as ƒ(c).
Let G(x) = (x + 6)/(x2 + 4x - 12).a. Make a table of the values of G at x = -5.9, -5.99, -5.999, and so on. Then estimate limx→ -6 G(x). What estimate do you arrive at if you evaluate G at x =
Together with the identities sin (h + c) = sin h cos c + cos h sin c, cos (h + c) = cos h cos c - sin h sin c to prove that both ƒ(x) = sin x and g(x) = cos x are continuous at every point x = c.
Let h(x) = (x2 - 2x - 3) / (x2 - 4x + 3).a. Make a table of the values of h at x = 2.9, 2.99, 2.999, and so on. Then estimate limx→3 h(x). What estimate do you arrive at if you evaluate h at x =
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 - 3x - 1 = 0
Let ƒ(x) = (x2 - 1) / (|x| - 1).a. Make tables of the values of ƒ at values of x that approach c = -1 from above and below. Then estimate limx→ -1 ƒ(x).b. Support your conclusion in part (a) by
Let F(x) = (x2 + 3x + 2) / (2 - |x|).a. Make tables of values of F at values of x that approach c = -2 from above and below. Then estimate limx→ -2 F(x).b. Support your conclusion in part (a) by
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.2x3 - 2x2 - 2x + 1 = 0
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.x(x - 1)2 = 1 (one root)
Let g(θ) = (sin θ)/θ.a. Make a table of the values of g at values of θ that approach θ0 = 0 from above and below. Then estimate limθ→0 g(θ).b. Support your conclusion in part (a) by graphing
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.xx = 2
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. √x + √1 + x = 4
Let G(t) = (1 - cos t)/t2.a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate limt→0 G(t).b. Support your conclusion in part (a) by graphing G near
Let ƒ(x) = (3x - 1)/x.a. Make tables of values of ƒ at values of x that approach c = 0 from above and below. Does ƒ appear to have a limit as x→ 0? If so, what is it? If not, why not?b. Support
Let ƒ(x) = x1/(1-x).a. Make tables of values of ƒ at values of x that approach c = 1 from above and below. Does ƒ appear to have a limit as x→ 1? If so, what is it? If not, why not?b. Support
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.cos x = x (one root). Make sure you are using
Suppose that g(x) ≤ ƒ(x) ≤ h(x) for all x ≠ 2 and suppose thatCan we conclude anything about the values of ƒ, g, and h at x = 2? Could ƒ(2) = 0? Could limx→2 ƒ(x) = 0? Give reasons for
Use a CAS to perform the following steps:a. Plot the function near the point c being approached.b. From your plot guess the value of the limit. lim x→-1 x3 r3 – 2 – 5x – 3 (x + 1)²
a. Ifb. If f(x) lim x-2 X-2 5 = 3, find lim f(x). x-2
Use a CAS to perform the following steps:a. Plot the function near the point c being approached.b. From your plot guess the value of the limit. lim x-0 √1 + x - 1 X
If x4 ≤ ƒ(x) ≤ x2 for x in [-1, 1] and x2 ≤ ƒ(x) ≤ x4 for x < -1 and x > 1, at what points c do you automatically know limx→c ƒ(x)? What can you say about the value of the limit at
Use a CAS to perform the following steps:a. Plot the function near the point c being approached.b. From your plot guess the value of the limit. +4 lim x-2 X-2 16
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.2 sin x = x (three roots). Make sure you are
Use a CAS to perform the following steps:a. Plot the function near the point c being approached.b. From your plot guess the value of the limit. x² - 9 lim x 3 √√√x² + 7 - 4
a. Graph g(x) = x sin (1/x) to estimate limx→0 g(x), zooming in on the origin as necessary.b. Confirm your estimate in part (a) with a proof.
Use a CAS to perform the following steps:a. Plot the function near the point c being approached.b. From your plot guess the value of the limit. 1 - cos.x lim x0 xsinx
a. Graph h(x) = x2 cos (1/x3) to estimate limx→0 h(x), zooming in on the origin as necessary.b. Confirm your estimate in part (a) with a proof.
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