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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 = x 2 + 5 x 25 x 2

To determine

To sketch:

The curve of 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=x2+ 5x25-x2 

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=x2+ 5x25-x2  is a rational expression, its domain is -,-5 -5, 5(5, ).  Because at x=± 5, the denominator becomes 0, it makes the function undefined.

B. Intercepts

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

y=02+ 5025-02

y=0

y  intercept is (0, 0).

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

0=x2+ 5x25-x2

x=0

x  intercept is 0, 0.

C. Symmetry

For symmetry, replace x by (-x). Therefore,

f-x= (-x)2+ 5(-x)25-(-x)2=-xx+5

f-xfx -f(x)

The function is neither odd nor even; therefore, no symmetry.

D. Asymptote

a) Horizontal asymptotes:

limx-  x2+ 5x25-x2=-1, limx x2+ 5x25-x2=-1

Horizontal asymptote x=-1.

b) Vertical asymptotes:

Since, the function becomes undefined at x=±5, the vertical asymptote is x=5

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