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3-4 Rates of Change

Learning Targets

You will be able to

Concepts / Definitions


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


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


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.


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.
  3. When is the particle moving forward? Explain your answer.
  4. Find the acceleration of the particle after 4 seconds.
  5. Find the distance travelled during the first 5 seconds.
  6. When is the particle speeding up?