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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 = sin x 1 + cos x

To determine

To sketch:

The curve of y

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=sinx1+cosx

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:

fx is not defined for (2k+1)π, where  kZ

So Domain is R-(2k+1)π

B. Intercepts:

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

y=sin01+cos0=0

The y-intercept is 0, 0

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

0= sinx1+cosx

x=2nπ, where nZ

C. Symmetry:

For f-x replace x by (-x)

f-x=sin(-x)1+cos(-x)

f-x=-sinx1+cosx

f-x=-f(x)

f(x) is an odd function

So f(x) has rotational symmetry about the origin

D. Asymptote:

a) Horizontal asymptotes:

There are no horizontal asymptotes

b) Vertical asymptotes:

limn-sinx1+cosx=

limn+(sinx1+cosx)=-

So there are vertical asymptotes at x=kπ

E

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