A changing circle

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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.