The derivative of a function F at 2 is the limit of the difference quotient for F at 2, written[br][br][math]F'\left(2\right)=\begin{matrix}lim\\h\longrightarrow0\end{matrix}\frac{F\left(2+h\right)-F\left(2\right)}{h}[/math],[br][br]It measures the slope of the graph of [math]x=F\left(t\right)[/math] at [math]\left(2,f\left(2\right)\right)[/math], the rate of change of [math]F[/math] at [math]2[/math].

If we divide the interval [math]\left[0,5\right][/math] into [math]n[/math] equal sub-intervals [math]\left[t_{i-1},t_i\right][/math] of size [math]\Delta t_i=\frac{5-0}{n}[/math] then [math]\sum_{i=1}^nf\left(t_i\right)\Delta t_i[/math] is called the (Right) Riemann sum for the function [math]f[/math] over [math]\left[0,5\right][/math].

The integral of [math]f[/math] over [math]\left[0,5\right][/math] is [math]\int_0^5f\left(t\right)dt=\begin{matrix}lim\\n\longrightarrow\infty\end{matrix}\sum_{i=1}^nf\left(t_i\right)\Delta t_i[/math], the limit of the (Right) Riemann sums. It measures the difference in areas under/over the graph of [math]v=f\left(t\right)[/math] over [math]\left[0,5\right][/math].

If [math]f[/math] is continuous on [math]\left[a,b\right][/math] then: [br][br]1) [math]\int_a^xf\left(t\right)dt[/math] is an anti-derivative of [math]f[/math] on [math]\left[a,b\right][/math], i.e., [math]\frac{d}{dx}\int_a^xf\left(t\right)dt=f\left(x\right)[/math] on [math]\left[0,5\right][/math][br][br]2) If [math]F[/math] is any anti-derivative of [math]f[/math] on [math]\left[a,b\right][/math] then [math]\int_a^bf\left(t\right)dt=F\left(b\right)-F\left(a\right)[/math]