1.Evaluate the following limits (if it exists)Vx+6-xVx3-1+x2-x(a) limx-3(b) limズ→1ズー3xー12xVx2+1(d) lim2.(c) limX→00X1Xー12.(a) Let f(x) =(x+1)(x³+x² +1)(x? -x+1)find f'(-1)X+1Vx2+1 ex+2(b) Let f (x) = x* , find f'(2)(c) Let g(x) = 2* , find g(5) (x)3Use the linear approximation to estimate the following numbers(a) In(0.97)(b) V7.94(c) V100044.A deposit of $400 is made in an account that earns interest atan annual rate of 2.5%. How long will it take for the balance to doublewhen the interest is compounded (a) annually, (b) monthly, (c) daily,and (d) continuously?5.Let f (x) = xVx² + 3(a) Show that the function f (x) is one-to-one.(b) Find g'(f (-2)) for the function g(x) , that is the inverse function o- 師簽(r*)* -27 = 18 &6.Suppose that f (x) is twice differentiable, limズ→1ズ-1s'(1+t)-f'(1-4t)lim= 1. Findf (1), f'(1), &f"(1).rectangular package to be by a postal service can have a maximumcombined length and girth (perimeter of a cross section) of 276centimeters. Find the dimensions of the package with maximum volume.Assume that the package's dimensions are x by x by y (see figure).7.8Determine the points where the tangent line to the curve off(x) = x3 - 3x2 + 2 are parallel to the line y = 9x + 49.Given a normal probability density functionf(x) = e z) , find out the inflection points and the maximum1ov2n10.Sketch the graph of a rational function f (x) = 3ln(x² – 1) – 4xIn the graph, you should discuss the domain of f and the concavity of f,describe the rise and fall of f, and find asymptotes, local minimum andmaximum if they exist.

Answered: Image /qna-images/answer/03274490-f762-4774-89e3-c877144507a8.jpg

Answered: 1. Evaluate the following limits (if it…

1. Evaluate the following limits (if it exists) (a) lim X3 Vx+6-x (b) lim ズ→1 Vx-1+x2-x ズー3 xー1 2x (c) lim (* (d) lim ズ→1 Jx2+1 xー1

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.2: Polynomial Functions

Problem 96E: What is the purpose of the Intermediate Value Theorem?

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

1.
Evaluate the following limits (if it exists)
Vx+6-x
Vx3-1+x2-x
(a) lim
x-3
(b) lim
ズ→1
ズー3
xー1
2x
Vx2+1
(d) lim
2.
(c) lim
X→00
X1
Xー1
2.
(a) Let f(x) =
(x+1)(x³+x² +1)(x? -x+1)
find f'(-1)
X+1
Vx2+1 ex+2
(b) Let f (x) = x* , find f'(2)
(c) Let g(x) = 2* , find g(5) (x)
3
Use the linear approximation to estimate the following numbers
(a) In(0.97)
(b) V7.94
(c) V10004
4.
A deposit of $400 is made in an account that earns interest at
an annual rate of 2.5%. How long will it take for the balance to double
when the interest is compounded (a) annually, (b) monthly, (c) daily,
and (d) continuously?
5.
Let f (x) = xVx² + 3
(a) Show that the function f (x) is one-to-one.
(b) Find g'(f (-2)) for the function g(x) , that is the inverse function o-

師簽
(r*)* -27 = 18 &
6.
Suppose that f (x) is twice differentiable, lim
ズ→1
ズ-1
s'(1+t)-f'(1-4t)
lim
= 1. Findf (1), f'(1), &f"(1).
rectangular package to be by a postal service can have a maximum
combined length and girth (perimeter of a cross section) of 276
centimeters. Find the dimensions of the package with maximum volume.
Assume that the package's dimensions are x by x by y (see figure).
7.
8
Determine the points where the tangent line to the curve of
f(x) = x3 - 3x2 + 2 are parallel to the line y = 9x + 4
9.
Given a normal probability density function
f(x) = e z) , find out the inflection points and the maximum
1
ov2n
10.
Sketch the graph of a rational function f (x) = 3ln(x² – 1) – 4x
In the graph, you should discuss the domain of f and the concavity of f,
describe the rise and fall of f, and find asymptotes, local minimum and
maximum if they exist.

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