   Chapter 3.5, Problem 35E

Chapter
Section
Textbook Problem

Use the guidelines of this section to sketch the curve.35. y = x tan x, −π/2 < x < π/2

To determine

To Sketch: The curve by plotting x and y coordinates using guidelines.

Explanation

Given:

The Cartesian equation is as below.

y=f(x)=xtanx (1)

Calculation:

a)

Calculate the domain.

The function is defined for the given interval π2<x<π2 . So the domain is (π2,π2)

b)

Calculate the intercepts.

Calculate the value of x -intercept.

Substitute 0 for y in the equation (1).

xtanx=0x=0

Hence, x -intercept is (0,0) .

Calculate the y -intercept.

Substitute 0 for x in the equation (1).

f(0)=0tan(0)=0

Therefore, the y -intercept is (0,0) .

c)

Calculate the symmetry.

Apply negative and positive values for x in the equation (1).

Substitute 1 for x in the equation (1).

f(1)=(1)tan(1)=1.557

Substitute +1 for x in the equation (1).

f(1)=(1)tan(1)=1.557

Here the condition f(x)=f(x) is true, hence it is an even function and the graph is symmetrical about y -axis.

d)

Calculate asymptotes.

Apply limit of x tends to π2 in the equation (1).

limxπ2y=π2tan(π2)=π2()=

Apply limit of x tends to π2 in the equation (1).

limxπ2y=π2tan(π2)=π2()=

Here, the value of limit gets infinity; this implies there is no horizontal asymptote.

For the given domain (π2,π2) , there will be vertical asymptotes at x=±π2

e)

Calculate the intervals.

Differentiate the equation (1) with respect to x .

Apply the product rule below.

(uv)'=uv'+vu'

Substitute x for f and tanx for v in the above equation.

f'(x)=xsec2x+tanx(1)

f'(x)=xsec2x+tanx (2)

Substitute 0 for f'(x) in the equation (2).

xsec2x+tanx=0tanx=xsec2x

Multiply cosx on both sides of the above equation.

cosxtanx=xsec2xcosx

Substitute (sinxcosx) for tanx and (1cos2x) for sec2x in the above equation.

cosx(sinxcosx)=x(1cos2x)cosxsinx=xcosxsinxcosx=x

Substitute sin2x2 for sinxcosx in the above equation.

sin2x2=xsin2x=2x

The only solution for the above equation is x=0

Take the interval (0,π2) .

Substitute π6 for x in the equation (2).

f'(π6)=π6sec2(π6)+tan(π6)=π6(1cos2(π6))+tan(π6)=π6(1cos2(π6))+13

f'(π6)=π6(1(32)2)+13=π6(43)+13=0.698+0.577=1.28

Here the condition f'(π6)>0 is true, hence the function f is increasing on (0,π2) .

Take the interval (π2,0) .

Substitute π6 for x in the equation (2)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find the missing power in the following calculation: 51/35=5.

Precalculus: Mathematics for Calculus (Standalone Book)

In Exercises 7-28, perform the indicated operations and simplify each expression. 14. xexx+1+ex

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Differentiate. g(x)=1+2x34x

Calculus (MindTap Course List)

Find the mean, median, and mode for the following scores; 8, 7, 5, 7, 0, 10, 2, 4, 11, 7, 8, 7

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Fill in each blank: 7040ft=mi

Elementary Technical Mathematics

Solve the differential equation. 7. 2yey2y=2x+3x

Single Variable Calculus: Early Transcendentals 