   Chapter 3.5, Problem 50E

Chapter
Section
Textbook Problem

# 49–54 Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. y = 1 + 5 x − 2 x 3 x − 2

To determine

To sketch:

The curve of y and use guidelines D to find an equation of the slant asymptote

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+5x-2x2x-2

3) Calculations:

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

Here, the denominator is zero when x=2. Therefore, Domain is (-,2)(2,)

B. Intercepts

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

y(0)=1+5(0)-2(0)2(0)-2 =-12

y  intercept is 0, -12

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

0=1+5x-2x2x-2

Solve x

x=-0.19, 2.69

x intercepts are -0.19, 0 and 2.69, 0

C. Symmetry

For symmetry, replace each x by -x therefore,

f-x= 1+5-x-2-x2-x-2

f-x= 1-5x-2x2-x-2

That means there is no symmetry

D. Asymptote:

Horizontal asymptotes: the degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote.

limn±1+5x-2x2x-2=±

So, there are no horizontal asymptotes.

Vertical asymptotes: the points where f is not defined.

Here, the function is not defined at 2, so vertical asymptote x=2

To find slant asymptote-

Use the long division method rule,

So we get,fx=-2x+1+3x-2

By guidelines for slant asymptote, slant asymptote is y=-2x+1

E

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