if g(x)=x^2f'(x)

Question

if g(x)=x^2f'(x)

amy0799 · Answer

where $$f(x)=\frac{ x-cosx }{ x }$$ find the average rate of change of g on [pi, 2pi]

solomonzelman · Answer

differentiate the f(x).

solomonzelman · Answer

$\large\color{black}{ \displaystyle f(x) =\frac{x-\cos x}{x} }$

$\large\color{black}{ \displaystyle f(x) =1-\frac{\cos x}{x} }$

then derivative of 1 is 0, and derivative of the second peace will go by quotient rule (or you can use product or logarithmic)

solomonzelman · Answer

$\large\color{black}{ \displaystyle f'(x) =0-\frac{(\cos x)'x-\cos x(x)'}{x^2} }$

solomonzelman · Answer

continue please.

amy0799 · Answer

why did you take the 2nd derivative?

solomonzelman · Answer

I didn't take the second derivative.

amy0799 · Answer

oh wait I misread ur sentence, sorry

solomonzelman · Answer

My function is:

$\large\color{black}{ \displaystyle f(x) =\frac{x-\cos x}{x} }$

then I simplify it algebraically, to get:

$\large\color{black}{ \displaystyle f(x) =1-\frac{\cos x}{x} }$

And now, We have to differentiate it (once).

$\large\color{black}{ \displaystyle f'(x) =0-\frac{(\cos x)'x-\cos x(x)'}{x^2} }$

solomonzelman · Answer

if it is not making sense, then you are always welcome to express your concerns regarding the problem.

amy0799 · Answer

so then 
$$g(x)=x ^{2}(\frac{ xsinx+cosx}{ x ^{2} })$$

amy0799 · Answer

@SolomonZelman

solomonzelman · Answer

yes, very good, and you can tell that x² will can cancel

solomonzelman · Answer

Then the remaining part is an algebraic task of fining the slope (of the secant)
between $($ 2π,f(2π) $)$ and  $($ π,f(π) $)$

amy0799 · Answer

I put 2pi and pi into xsinx+cosx?

amy0799 · Answer

@SolomonZelman

solomonzelman · Answer

Yes, that is what you do:)

amy0799 · Answer

(2pi,1) (pi,-1)
is that correct?

solomonzelman · Answer

Yes

solomonzelman · Answer

then the slope formula...

amy0799 · Answer

pi/2 is the answer?

solomonzelman · Answer

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