Instantaneous rate of change [2.1]

Notes (.ppt) and Handout (.pdf)

What is the slope of a tangent line to a curve at $x$?

We could calculate the slope of a line connecting two points, at $x=a$ and $x=b$... The slope of the line connecting two points is: $$m=\frac{f(b)-f(a)}{b-a}.$$ See what happens as $b\to a$.

Or we could take points on either side of some "point of interest" The slope is now: $$m=\frac{f(x+h)-f(x-h)}{(x+h)-(x-h)}=\frac{f(x+h)-f(x-h)}{2h}.$$ See what happens as $h\to 0$.

Graphical approximation

Either way, this suggests a graphical approach to estimating the slope at a point on a curve:

• Mark a point on a curve.
• Line up a ruler to be *tangent* to the curve at that point, and draw a line through the point.
• Pick two points on the line, and use them to calculate the slope.

Numerical approximation

Using a calculator and a formula for $f(x)$, evaluate the slope at $x=2$:

• Pick a "close" point on the curve, e.g. $(2.05, f(2.05))$,
• Calculate $f(2.05)$ and $f(2.0)$,
• Then calculate the slope between this point and $(2,f(2))$: $$m=\frac{f(2.05)-f(2.0)}{2.05-2.0}=\frac{f(2.05)-f(2.0)}{0.05}$$