M171 Calculus I · Unit 1: Understanding the derivative · on Exam 1
§1.1 How do we measure velocity?
Study notes for How do we measure velocity? in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- A position function $s(t)$ tells where an object is at time $t$. Average velocity over $[a,b]$ is the change in position divided by the change in time, which is the slope of the secant line through $(a,s(a))$ and $(b,s(b))$.
- Instantaneous velocity at $t=a$ is what average velocity approaches as the interval $[a,a+h]$ shrinks ($h\to0$). Numerically: compute $AV_{[a,a+h]}$ for smaller and smaller $h$ on both sides of $a$.
- Units of a rate are always $\dfrac{\text{units of output}}{\text{units of input}}$, e.g. feet per second, dollars per year, mg per hour.
- On a graph: average rate $=$ slope of a secant; instantaneous rate $=$ slope of the tangent.
Key formulas
Average velocity
$$AV_{[a,b]} = \frac{s(b)-s(a)}{b-a}$$
Interval of width h
$$AV_{[a,a+h]} = \frac{s(a+h)-s(a)}{h}$$
Worked example
A ball's height is $s(t) = 64 - 16(t-1)^2$ feet, $t$ in seconds. Find $AV_{[0.5,\,1]}$ and estimate the instantaneous velocity at $t=1$.
Show the solution
$s(1)=64$ and $s(0.5)=64-16(0.25)=60$, so $AV_{[0.5,1]} = \frac{64-60}{1-0.5} = 8$ ft/s. For $t=1$: $AV_{[1,1+h]} = \frac{64-16h^2-64}{h} = -16h \to 0$ as $h\to0$. The instantaneous velocity at $t=1$ is $0$ ft/s (the peak of the flight).
Common mistakes
- Forgetting units, or writing "per" backwards (seconds per foot).
- Reporting average speed (always ≥ 0) when the question asks for velocity (can be negative).
- Writing $AV_{[a,b]}$ as $s(b)-s(a)$ without dividing by $b-a$.
Exam tipAlways label your answer with proper notation, e.g. $AV_{[40,50]} = 0.3$ mi/min. The rubric rewards notation and explanation, not just the number.
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