Two young mathematicians discuss a circle that is changing.

Check out this dialogue between two calculus students (based on a true story):

Devyn
Riley, I’ve been thinking about calculus.
Riley
YOLO.
Devyn
Consider a circle of some radius \(r\).
Riley
Ha! What else would we ever call the “radius?”
Devyn
Exactly. Now the formula for the perimeter of a circle is?
Riley
\(P=2\cdot \pi \cdot r\) baby.
Devyn
And its area?
Riley
You know it’s \(A=\pi \cdot r^2\).
Devyn
Right, but here’s what’s bugging me: If I know \(r'\), what is \(P'\)? What’s \(A'\)?
Riley
Oooh. Ouch. Hmmm. I wanna say it’s
\[ P' = 2\cdot \pi \cdot r' \qquad \text {and}\qquad A' = \pi (r')^2 \]
but I’m not sure that is right.
Devyn
Yeah…me too. But I’m not sure that’s right either. Are we forgetting something?
Do you think our young mathematicians above are correct?
Yes. \(P' = 2\cdot \pi \cdot r'\) and \(A' = \pi (r')^2\). No. While \(P' = 2\cdot \pi \cdot r'\), \(A' \ne \pi (r')^2\). No. While \(A' = \pi (r')^2\), \(P' \ne 2\cdot \pi \cdot r'\). No. \(P' \ne 2\cdot \pi \cdot r'\) and \(A' \ne \pi (r')^2\). There is no way to tell.
Set \(r(t)=3\cdot t\). What is \(r'(t)\) when \(r=15\)? \(r'(t)=\answer [given]{3}\)
Set \(r=3\cdot t\). Now \(P(t) = 2\cdot \pi \cdot 3\cdot t\). What is \(P'(t)\) when \(r=15\)? \(P'(t)=\answer [given]{2\cdot \pi \cdot 3}\)
Describe what \(P'\) means in this context.
It describes how fast the perimeter is changing at a particular instant in time.
Set \(r=3\cdot t\). Now \(A(t) = \pi \cdot (3\cdot t)^2\). What is \(A'(t)\) when \(r=15\)? \(A'(t)=\answer [given]{2\cdot \pi \cdot 15\cdot 3}\)
Describe what \(A'\) means in this context. Does it make sense that \(A'\) is positive?
It describes how fast the area is changing. It does make sense that \(A'\) is positive, since as the radius gets larger, the area should get larger, too.
What, if anything, did our two young mathematicians forget about above?
They forgot the chain rule.
2025-01-06 20:03:52