3.4 – Rates of Change

Velocity

$$\frac{\Delta \text{position}}{\Delta \text{time}}=\text{velocity}$$

Instantaneous Velocity

$$v(t)=\frac{ds}{dt}=\underset{t\to 0}{\lim }\frac{f(x+t)-f(x)}{t}$$

Where $s=f(t)$ is the position function at $t$.

Linear Approximations

The linear approximation of a function $y=f(x)$ in the neighbourhood of $a$ can be written as $f(x)\approx f(a)+(\frac{dy}{dx}{|}_{x=a})x$.

Why?

The local approximation follows the form $y=b+mx$

Where:

The derivative $\frac{dy}{dx}$ at point $a$ is the slope $m$;

The value of $f(a)$ is our $y$-intercept $b$;

The value of $x$ is the distance from $a$

Example

Given $A=\pi r^2$, find the rate of change in area with respect to radius.

$$\frac{dA}{dr}=\frac{d}{dr}\pi r^2$$ $$=\pi \frac{d}{dr}{r}^2$$ $$=2\pi r$$

Knowing that $\frac{dA}{dr}=2\pi r$, we can approximate the area of a circle near $r=3$ with the equation $A\approx 9\pi +6\pi (\Delta r)$.

At $r=3.1$:

$$A(3.1)\approx 9\pi +6\pi (0.1)$$ $$\approx 30.159289$$ $$A(3.1)=\pi (3.1{)}^2$$ $$=\mathrm{30.191...}$$

We know that our approximation must be an underestimate because the area formula, $\pi r^2$, is concave up.