Assume problem A is reducible to an easy problem B. If we can solve problem B, then we can solve problem A.
"Tangent" means touching. So tangent line to a curve is a line that touches the curve.
The Secant line \(\overline{PQ}\) is the line passing through point \(P\) and \(Q\).
How can you find the slope of the secant line \(\overline{PQ}\), \(m_{PQ}\)?
\[m_{PQ}=\frac{y_2-y_1}{x_2-x_1}\]
Find the slope of tangent line of \(f(x)=\frac{x^2}{4}\) at \(x=2\).
From above experiment, what do you expect?
Let \(y=f(x)\) be a position function of a car at time \(x\).
\[ average~velocity=\frac{\Delta distance}{\Delta time}=\frac{\Delta y}{\Delta x} \]
which is the slope of secant line.
How about an instantaneous velocity at time \(x\)?
\[ instantaneous~velocity =?=slope~of~tangent~line \]
Born: 25 December 1642