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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

1-40 Use the guidelines of this section to sketch the curve.

y = 1 x 2 x

To determine

To sketch:

The curve of the given function.

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:

y=1-x2 x

3) Calculation:

Here, first find the domain of the given function and the x & y intercepts. Next, check the symmetry, asymptotes, intervals of increase and decrease, local maximum and minimum values, concavity, and points of inflection. Using these, sketch the curve.

A) Domain:

Since y=1-x2 x  is a rational expression, the value inside the radicals must be greater than zero, and at x=0, the denominator becomes 0, which makes the function undefined.

Therefore, its domain is -1,00, 1.

B) Intercepts:

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

y=1-02 0  y=

y  intercept: (0, )

For x intercept, plug y=0 in the original function, and solve it for x.

0=1-x2 x

x=±1

x  intercepts are 1, 0 and -1, 0.

C) Symmetry:

For symmetry, replace x by (-x).

f-x= 1-(-x)2 -x=-1-x2 x

f-x= -f(x)

The function is odd. Symmetry is about the origin.

D) Asymptote

Horizontal asymptotes:

limx-0  1-x2 x=-, limx0 1-x2 x=

No Horizontal asymptote.

Vertical asymptotes:

Since the function becomes undefined at x=0, the vertical asymptote is x=0.

Slant asymptotes:

No slant asymptote.

E) Intervals of increase or decrease.

To find the intervals of increase or decrease, find the derivative of the given function.

f'x= -1x21-x21/2

Equating this derivative with 0,

-1x21-x21/2=0

Solving this for x, it gives 1=0, that means no real solutions for x

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