   Chapter 3.R, Problem 28E

Chapter
Section
Textbook Problem

# 17-28 Use the guidelines of Section 3.5 to sketch the curve. y = 4 x − tan x ,     − π / 2 < x < π / 2

To determine

To sketch:

The graph of the function y=4x-tanx

Explanation

1) Concept:

i) A domain is the set of xvalues that satisfy the function.

ii) To find x-intercept, put y=0,and to find y-intercept, put x=0in 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 fif 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=4x-tanx

3) Calculation:

The given function is y=4x-tanx

A) Domain

The given function is defined on -π2,π2  therefore, the domain of the function is -π2,π2

B) Intercepts

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

y=40-tan0=0

Therefore, y intercept is (0,0)

There is no x-intercept

C) Symmetry

f-x=4-x-tan-x= -f(x)

Therefore, f is an odd function, and hence, symmetric about origin

Also,f(x+p)f(x) hence f is not periodic on -π2,π</

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