Science:Math Exam Resources/Courses/MATH100/December 2011/Question 01 (n)/Solution 1

The limit definition of the derivative at a point x=a is:

$f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}$

We notice that the term cos($\pi$/3 + h) in the statement of the question resembles the f(x+h) term in the limit definition of the derivative. If this is the case, f(x) = cos(x) and a = $\pi$/3.

We now test these guesses using the limit definition of the derivative. If f(x) = cos(x) and a = $\pi$/3, then

${\begin{aligned}f'\left({\frac {\pi }{3}}\right)&=\lim _{h\to 0}{\frac {\cos({\frac {\pi }{3}}+h)-\cos({\frac {\pi }{3}})}{h}}\\&=\lim _{h\to 0}{\frac {\cos({\frac {\pi }{3}}+h)-{\frac {1}{2}}}{h}}\end{aligned}}$

If we multiply both sides by 2 we get

$2f'\left({\frac {\pi }{3}}\right)=\lim _{h\to 0}{\frac {2\cos({\frac {\pi }{3}}+h)-1}{h}}$

This is the expression we were trying to find, so our guess of the function f(x) and the value a were correct: $f(x)=cos(x)$ and $a=\pi /3$.