2.6: The Derivative

Maglev example (again)

Suppose a maglev's position at time $t$ is determined by the function \[f(t) = 4t^2 \qquad 0 \leq t \leq 10\]
  1. Find the average velocity over the time interval $[2, 4]$.
  2. Try to find the instantaneous velocity at $t = 2$ by doing the following:
    1. Set up the average velocity formula for $[2, t]$ where $t > 2$.
    2. Take the limit as $t \rightarrow 2$.

The above limit has been shown to be two things at once: the instantaneous rate of change at $t = 2$ and the slope of the tangent line touching the curve at $(2, f(2)) = (2, 16)$.

This phenomena is called the derivative.

The derivative of a function $f$ with respect to $x$ is the function $f'$ where \[f'(x) = \lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}\] The domain of $f'$ is the set of all $x$ for which the limit exists.