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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

17-28 Use the guidelines of Section 3.5 to sketch the curve.

y = 1 x + 1 + x

To determine

To Sketch:

The curve y=1-x+1+x

Explanation

1) Concept:

 i) A domain is the set of x values that satisfy the function.

ii) To find x-intercept, put y=0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so it has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) is a local maximum value of f  if fcf(x) when x is near c and is a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

The function y=1-x+1+x

3) Calculation:

The given function is y=1-x+1+x

A) Domain

First, we find the domain where the function defined

 1-x0and 1+x0 x1 and x-1

Therefore, the given function is defined on [-1, 1] therefore, the domain of the function is [-1, 1]

B) Intercept

For y intercept plug x=0  in the given function and solve it

y=1-0+1+0=1+1=2

The y-intercept is (0,2)

Now, there is no x-intercept because fx0  for all x

C) Symmetry

For symmetry, replace x by (-x), therefore,

f-x=1--x+1+-x=1-x+1+x

Since, f-x=fx so f is symmetric about y- axis

That is, f is an even function.

D) Asymptote

limx-y=limx-(1-x+1+x)=- , 

limx+y=limx+(1-x+1+x) = 

So there is no horizontal asymptote

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