How do you determine if your calculator gave you a legitimate answer when calculating the derivative of a function?

Question

I was instructed to use my calculator to find the derivatives of the following functions

a) find f'(pi/25) for f(x)=sin(200x)

b) find f'(pi/25) for f(x) = |sin(200x)|

c) find f'(pi/25) for f(x) = sin|200x|

d) find f'(pi/25) for f(x) = sin(200|x|)

My calculated answers are

a) 198.6693308

b) -3.9E-10

c) 198.6693308

d) 198.6693308

So, I know B is the one that is not legitimate, but I am not sure how to prove/word that.

Answer 1

First, 200π/25 = 8π, and therefore you can remove the absolute value signs in case c) and d), which makes them identical to case a).

For case a)

f'(x) = 200cos(200x), and therefore f'(π/25) = 200 for cases a), c), d)

For case b), f(x) reverses polarity at point x = π/25.

By that I mean that when close to π/25, for x <π/25 sin(200x) is negative, and for x >π/25 sin(200x) is positive.

Therefore |sin(200x)| would be differentiable at x = π/25, only if the derivative of sin(200x) there were 0.

But since the derivative there is 200, |sin(200x)| is not differentiable at x = π/25.

It looks like the calculator was trying to give you 0.

Answer 2

All answers are wrong.The answers to A, C and D must be 200, but the answer to B is really wrong since the derivative at pi/25 is undefined. You can find all the answers without a calculator using chain rule and Dx |x|=x/|x|. Or for B just graph sin(200x) and turn it into absolute value of sin(200x) by refelcting the negative parts across the x-axis. You will see that the function in B will have not derivtive at integer multiples of pi/200.