Question

Let f(x) = 7sin(x)
Part A) |f'(x)| <= __?__
Part B) By the Mean Value Theorem, |f(a) - f(b)| <= __?__ |a - b|

Answer 1

\(f'(x)=7\cos x \implies |f'(x)|\le7\). By the mean value theorem, we have \(f'(c)=\frac{f(a)-f(b)}{b-a}\). We can generalize this to any interval in the domain of f. We can then find that \(|f(a)-f(b)|\le 7|a-b|\), since \(|f'(c)|\le7\) for all c in the domain of f.

Answer 2

A = 7 cos x

Answer 3

I have a question. I know how to get \[f \prime (x) = 7\cos (x)\], but how does \[f \prime (x) = 7\cos x\] result in \[\left| f \prime (x) \right| \le 7\]?

Answer 4

I'm sure you know that \(-1\le \cos x \le 1\). This is the same as writing \(|\cos x|\le1\), then you just multiply both sides by \(7\).

Answer 5

Oh yeah, that's right. I forgot. Thanks.

Answer 6

You're welcome!