Approximating tangent slope with a secant slope
Calculus consists of those problems in mathematics that can be solved with the
following basic approach:
Approximate and take a limit!
The simplest, standard, geometric example of this process is the approximation
of the slope of a tangent line with a secant line.
Here's an illustration of this basic idea.
Choose $f$:
$f(x)=x^2$
$f(x)=\frac{1}{3}e^x$
$f(x)=\sin(2x^2)+x/4$


Note that the slope of the secant line is computed via the difference quotient: $$ \frac{f(x+h)f(x)}{h} $$ and the slope of the tangent line is computed via the derivative: $$ f'(x) = \lim_{h\to0} \frac{f(x+h)f(x)}{h}. $$