2.4 – Rates of Change and Tangent Lines

Average Rates of Change

The average rate of change for some function $f(x)$ over an interval $[a,b]$ can be found using the slope formula.

$$\text{AROC}=\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$$

Applying the intermediate value theorem, we can infer that $f(x)$ takes on every value $y=f(c)$ for $f(a)<c<f(b)$ since motion is continuous.

Tangents

The AROC can be thought of as a secant line to $f(x)$. As $a\to b$, the slope of the secant approaches the slope of the tangent, which is the instantaneous rate of change at $f(b)$.

Slopes

The slope $m$ of a line at some point $(a,f(a))$ by applying the definition of the derivative

$$m=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$

To find the equation of the tangent line, we may use the point slope form

$$y-f(a)=m(x-a)$$

Normals

The normal is the line perpendicular to the tangent. The slope of the normal line at $a$ can be found by taking the negative reciprocal of the slope at $a$.