# j-james/math All my math notes, now in Markdown.

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# 3-4 Rates of Change

## Learning Targets

You will be able to

• Use derivatives to describe rectilinear motion (straight line)
• Use derivatives to solve problems involving other rates of change like marginal cosr and marginal revenue

## Concepts / Definitions

### Velocity

The instantaneous velocity is the derivative of the position function $s = f(t)$ with respect to time. $$v(t) = \frac{ds}{dt}$$

### Speed

Speed is the absolute value of velocity.
Speed = $\lvert v(t)\rvert$

### Acceleration

Acceleration is the derivative of velocity with respect to time. If a body’s velocity at time $t$ is $v(t) = \frac{ds}{dt}$, then the body’s acceleration at time $t$ is $$a(t) = \frac{d}{dt}v(t) = \frac{d^2}{dt^2}$$

The jerk is the derivative of acceleration.

### Free-fall Constants (Earth)

$g = 32.2\frac{ft}{sec^2},\quad s = \frac 12 (32.2)t^2 = 16.2t^2$ where $s$ is in feet
$g = 9.8\frac{m}{sec^2},\;:\quad s = \frac 12 (9.8)t^2 = 4.9t^2$ where $s$ is in meters

### Rate of Change in Economics

In Economics, marginals are used instead of the derivative.
For example, marginal cost is the cost ot produce one more item when a certain amount of items are produced.

### Examples

A particle moves along a line so that its position at any time $t \geq 0$ is given by the function $s(t) = t^3 -6t^2 + 9t$, where $s$ is measured in meters and $t$ is measured in seconds.

1. Find the displacement of the particle during the first 2 seconds. Show work.
2. Find the average velocity of the particle suring the first 4 seconds and find the velocity after 4 seconds. Show work.