M171 Calculus I · Unit 3: Implicit, limits & extrema · on Exam 3
§3.5 Related rates
Study notes for Related rates 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
- Several quantities change with time and are linked by an equation. Differentiate that equation with respect to $t$ (implicitly), then substitute the known values and rates to find the unknown rate.
- Steps: draw and label; list what is known (values and rates $\frac{d\square}{dt}$) and what is wanted; write the relating equation (geometry, Pythagoras, similar triangles, trig); differentiate with respect to $t$; substitute after differentiating; answer with units.
- Do not plug in a value that changes with time until after you differentiate.
Key formulas
Circle area
$$A=\pi r^2 \Rightarrow \frac{dA}{dt} = 2\pi r\frac{dr}{dt}$$
Sphere volume
$$V = \tfrac43\pi r^3 \Rightarrow \frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$$
Ladder / Pythagoras
$$x^2+y^2=L^2 \Rightarrow x\frac{dx}{dt} + y\frac{dy}{dt} = 0$$
Worked example
A 10 ft ladder leans on a wall. Its base slides away at 2 ft/s. How fast is the top sliding down when the base is 6 ft from the wall?
Show the solution
$x^2 + y^2 = 100$, so $2x x' + 2y y' = 0$. When $x=6$, $y=8$: $6(2) + 8y' = 0 \Rightarrow y' = -1.5$ ft/s. The top slides down at $1.5$ ft/s.
Common mistakes
- Substituting the current length before differentiating (kills the rate).
- Sign confusion: a decreasing quantity has a negative rate.
Exam tipWrite "want: dy/dt when x = 6" at the top of your work so the goal stays visible.
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