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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 ( x 3 ) 2

To determine

To sketch:

The graph of the function

y=1xx-32

Explanation

1) Concept:

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

ii) To find x-intercept, put y=0,and to find y-intercept, put x=0in 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 fif 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=1xx-32

Calculation:

A) Domain

The given function is y=1xx-32

The function is defined for the all values of x except when the denominator is zero.

xx-32=0 gives x=0, x=3

Therefore, domain of function is (- ,0)(0, 3)(3 ,)

B) Intercepts

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

As the function is not defined for x=0, so there is no y-intercept

Now, for x-intercept set y = 0 then the equation becomes

1xx-32=0

As the numerator is constant, it has no x-intercepts

C) Symmetry

For symmetry replace x by (-x), therefore

f-x= 1-x -x-32

Since, f-x-fxand fxf(-x)

There is no symmetry

Therefore f is neither an even nor an odd function

D) Asymptote

limx±1xx-32=0

 So y=0 is horizontal asymptote

limx0-1xx-32= and limx0+1xx-32=

Therefore, x=0  is the vertical asymptote

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