Definition of the Derivative

Question

No calculator is allowed on this question.

The average rate of change of a function, f, on the interval [1, 1+Δx] is given by sin(Δx) + 3^-5*deltax.

Find f'(1).

Answer 1

Average rate of change of f on the given interval = [f(1+Δx) - f(1)] / Δx = sin(Δx) + 3^-5Δx

f'(1) = lim_Δx→0 [ (f(1+Δx) - f(1) ) / Δx ] = lim_Δx→0 [sin(Δx) + 3^{-5 (deltax)}] = sin0 + 3⁰ = 0 + 1 = 1

Answer 2

We can find f'(1) by taking the limit of the average rate of change in [1,1+Delta x], for Delta x going to 0.

f'(1) = lim_Δx→0sin(Δx) + 3^-5*deltax = 1