For 0<x<180 verify the monotony of the function xcosx+sin(-x).
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
We have the function f(x) = x*cos x + sin (-x)
f(x) = x*cos x - sin x
The first derivative of f(x) is f'(x)
=> f'(x)...
(The entire section contains 81 words.)
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.
Related Questions
- What are sin x, tan x, cot x, if 180<x<270 and cos x=-4/5?
- 1 Educator Answer
- Prove the following identity: (1+ sin x + cos x) / (1+ sin x - cos x) = cot x/2
- 1 Educator Answer
- finding a valuesin A=? cos A = -(3/5) 90 degrees < A < 180 degrees
- 1 Educator Answer
- Given 0=<x<2pi find the roots of the equation sinx=-1/2.
- 1 Educator Answer
The monotony of a function shows the behavior of the function: increasing or decreasing function.
A function is strictly increasing if it's first derivative is positive and it is decreasing if it's first derivative is negative.
We'll re-write the function, based on the fact that the sine function is odd:
f(x) = xcos x - sin x
We'll calculate the first derivative:
f'(x)= (xcos x – sin x)'
f'(x) = ( xcos x)'-( sin x)'
We notice that the first term is a product, so we'll apply the product rule:
f'(x) = x'*(cos x)+x*(cos x)' – cos x
f'(x)=1*cos x-x*sin x –cos x
We'll eliminate like terms:
f'(x)= -x*sin x
Since the sine function is positive over the range [0 ; 180], the values of x are positive and the product is negative, the first derivative is negative.
The function y = f(x) = xcosx+sin(-x) is decreasing over the range [0, pi].
Student Answers