Derivative of a trigonometric function

[Ocata]

Homework Statement

$\frac{d}{dx}7.5sin(\frac{pi}{10}x)$

The Attempt at a Solution

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$

Maximum: f'(x) = 0

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$ = 0

$7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x$

**$(\frac{pi}{10}\frac{10}{pi})7.5(90) = x$

$(1)(7.5)(90) = x = 675$

To me, this doesn't seem to be nearly the correct answer because it doesn't make sense given the graph of this function:

$\frac{pi}{10}x= pi$

x = 10

So, the first arch is at x=0 and x = 10,

so the the maximum of the curve can not be x = 675.

What am I doing incorrectly in the derivative of the trigonometric function?

Thank you

 

[lep11]

Homework Statement

$\frac{d}{dx}7.5sin(\frac{pi}{10}x)$

The Attempt at a Solution

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$

Maximum: f'(x) = 0

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$ = 0

$7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x$

**$(\frac{pi}{10}\frac{10}{pi})7.5(90) = x$

$(1)(7.5)(90) = x = 675$

To me, this doesn't seem to be nearly the correct answer because it doesn't make sense given the graph of this function:

$\frac{pi}{10}x= pi$

x = 10

So, the first arch is at x=0 and x = 10,

so the the maximum of the curve can not be x = 675.

What am I doing incorrectly in the derivative of the trigonometric function?

Thank you

Derivation part looks fine.

 

[HallsofIvy]

Homework Statement

$\frac{d}{dx}7.5sin(\frac{pi}{10}x)$

What, exactly, was the question? You only mention differentiating the function but then set that derivative to 0.

The Attempt at a Solution

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$

Yes, this is the correct derivative.

Maximum: f'(x) = 0

$7.5(\frac{pi}{10})cos(\frac{pi}{10}x)$ = 0

and so $cos(\frac{\pi}{10}x)= 0$

$7.5(\frac{pi}{10})cos^{-1}(0)= \frac{pi}{10}x$

No. To solve $Af(x)= B$, you take $f^{-1}(B/A)$, not $Af^{-1}(B)$.

**$(\frac{pi}{10}\frac{10}{pi})7.5(90) = x$/quote]

$(1)(7.5)(90) = x = 675$

To me, this doesn't seem to be nearly the correct answer because it doesn't make sense given the graph of this function:

$\frac{pi}{10}x= pi$

x = 10

So, the first arch is at x=0 and x = 10,

so the the maximum of the curve can not be x = 675.

What am I doing incorrectly in the derivative of the trigonometric function?

Thank you

 

[Ocata]

Derivation part looks fine.

Hi lep11,

Thank you. I had a feeling I was running into the issue at solving for x.
(1) $7.5\frac{π}{10}cos(\frac{π}{10}x)=0$

(2) $7.5\frac{π}{10}cos^{-1}(0)= \frac{π}{10}x$ Here is where I was getting stuck. The first thing I noticed, was that I let cos(0) = 90 instead of π/2.

(3) $7.5\frac{π}{10}\frac{10}{π}cos(0) = x$

(4) $7.5\frac{π}{2} = x = 11.78$ Which is closer to within the interval of the first arch of the function, but not quite where the maximum should be.

Playing around with the algebra, I finally arrived at an answer that makes sense:

If I divide the $7.5\frac{π}{10}$ out of the equation first at step (1):

$7.5\frac{π}{10}cos(\frac{π}{10}x)=0$

$cos(\frac{π}{10}x) = 0$

$cos(0) = \frac{π}{10}x$

$\frac{10}{π}\frac{π}{2} = 5$ That sounds more like it.However, what I don't understand, is how come I have to divide $7.5\frac{π}{10}$ first? In a regular equation, it doesn't matter when you decide to divide both sides by a number, why does it matter in this situation?

Thank you.

 

[Ocata]

What, exactly, was the question? You only mention differentiating the function but then set that derivative to 0. No. To solve $Af(x)= B$, you take $f^{-1}(B/A)$, not $Af^{-1}(B)$.

Thank you HallsofIvy,

$7.5\frac{π }{10}Cos(\frac{π}{10}x) = 0$

$Cos(\frac{0}{(7.5\frac{π }{10})}) = \frac{π}{10}x$

$Cos(0) = \frac{π}{10}x$

$\frac{10}{π}Cos(0) = x$

$\frac{10}{π}Cos(0) = x$

$\frac{10}{π}\frac{π}{2} = x = 5$!Thank you. Now that I know how to solve this and that there is a systematic way to approach, I will be looking into why the formula provided does in fact work. Thank you for your guidance :)

 

[SammyS]

Thank you HallsofIvy,

$7.5\frac{π }{10}Cos(\frac{π}{10}x) = 0$
Thank you. Now that I know how to solve this and that there is a systematic way to approach, I will be looking into why the formula provided does in fact work. Thank you for your guidance :)

The product of $\ 7.5\frac{π }{10} \$ and $\ \cos(\frac{π}{10}x) \$ is zero.The only way for a product to be zero is for one of the factors to be zero. The only one which can be zero is $\ \cos(\frac{π}{10}x) \$ .