   Chapter 3.R, Problem 23E

Chapter
Section
Textbook Problem

# 17-28 Use the guidelines of Section 3.5 to sketch the curve. y = ( x − 1 ) 3 x 2

To determine

To sketch:

The graph of the function y=x-13x2

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=x-13x2

3) Calculation:

The given function is y=x-13x2

A) Domain

The domain is a set of all values of x except when the denominator is zero.

Here x2=0  becomes x=0

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

B) Intercepts

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

As the function is not defined for x=0, it has no y-intercept.

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

x-13x2=0

x-13 =0

x=1

Therefore, x-intercept is x=1

C) Symmetry

For symmetry replace x by (-x)

f(-x)=-x-13(-x)2

Since, f-x-fxand fxf(-x)

That means there is no symmetry

Therefore f is neither an even nor an odd function

D) Asymptote

limx± x-13<

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