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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 + cos x

To determine

To sketch:

The curve of y

Explanation

1) Concept:

i) The domain is the set of x values for which the function is defined.

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, and its graph is symmetric about the y-axis. If f-x=-fx, then it is an odd function and its graph is symmetric about the origin. 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 f''x=0, gives the values of inflection points.

2) Given:

y= x+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

Domain is R because for every value in R, fx is defined.

B. Intercepts:

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

y=0+cos0=1

The y-intercept is 0, 1.

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

0= x+cosx

If x=-cosx then x-0.74

The x intercept is -0.74, 0.

C. Symmetry

For f-x, replace each x by -x.

 f-x= -x+cos(-x)

f-x= -x+cosx

So it has no symmetry.

D. Asymptote

a) Horizontal asymptotes

limx±x+cosx=±

So, there are no horizontal asymptotes.

b) Vertical asymptotes

There are no vertical asymptotes because there is no such value that the denominator becomes zero.

E. Intervals of increase or decrease

To find the intervals of increase or decrease, find the derivative of the given function.

f'(x)= 1-sinx

f'x0  x

Hence, f(x) is increasing in the entire domain

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