Skip to main content
CalculusBy Educator Today

C13 · Unit C4 Contextual derivatives

Rates, motion and units

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 4.1–4.3
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

For position s(t), velocity is v equals s minutes and acceleration is a equals v minutes. Speed is |v|. A particle speeds up when v and a have the same sign and slows down when their signs differ, away from v equals 0. In other contexts, a derivative’s units are output units divided by input units; its sign indicates increase or decrease of the modeled quantity.

Worked example

If s equals t cubed minus 6t squared +9t, then v equals 3(t minus 1)(t minus 3) and a equals 6t minus 12. At t equals 2.5, v<0 and a>0: the particle moves in the negative direction while slowing down. Negative velocity alone does not establish slowing.

Explore the mathematics

Rates, motion and units

Bring two points together and compare secant and tangent slopes.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict how the secant changes as the two points get closer.xy-1123At x=1, h=0.125; secant slope=2.125

Predict before continuing

Predict how the secant changes as the two points get closer.

Read the complete explanation
  1. Predict how the secant changes as the two points get closer.
  2. The orange secant joins two points: its slope is Δy/Δx.
  3. The green tangent has slope 2. Reduce h to see the secant approach it.
  4. The derivative is a limit, not substitution of h=0 into a quotient with zero denominator. Here f′(1)=2.
See the idea: Rates, motion and units
See the idea: Rates, motion and unitsBoth velocity v=3(t−1)(t−3) and acceleration a=6t−12 are plotted. At t=2.5, v=−2.25 and a=3 have opposite signs, so speed is decreasing while motion is in the negative direction.xy0-4102438412v(2.5)a(2.5)

Both velocity v=3(t−1)(t−3) and acceleration a=6t−12 are plotted. At t=2.5, v=−2.25 and a=3 have opposite signs, so speed is decreasing while motion is in the negative direction.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Position, velocity and speed are different. Acceleration can be positive while speed decreases.

Investigate and explain

Draw aligned position, velocity and acceleration graphs for the same motion. Explain one interval in words.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.