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Let $A(r)$ denote the area of the circle of the radius $r>0$, and let $C(r)$ denote the circumference of the circle, show $A'(r)=C(r)$ for all $r>0$.

I found a similar question which has many answers Calculus proof for the area of a circle. And I have a similar attempt as N.S.

Attempt: Let $C(r)=2\pi r$ is continuous $(0,a)$ where $a>0$ for all $r$. Let $\Delta r>0$, the differences of outer area and inner area is (If we draw the annulus and straightened it, the shape would look a trapezoid.)$$\Delta A = A(r+\Delta r)-A(r)=\left(\frac{C(r+\Delta r)+C(r)}{2}\right)\Delta r\qquad(\star)$$ Then $$\lim\limits_{\Delta r\rightarrow0^+}\frac{\Delta A}{\Delta r}=\lim\limits_{\Delta r\rightarrow0^+}\left(\frac{C(r+\Delta r)+C(r)}{2}\right)=2\pi r$$

For the equation $(\star)$, I don't know how to show it mathematically, can someone give me a suggestion or a hint to show it? Thanks.

EDIT For this question, I only can use the fundamental theorem of calculus, I can not do $\frac{d}{dr}\pi r^2=2\pi r$ or integration

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Simple
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  • Your title has it backwards... – symplectomorphic Jan 21 '16 at 06:09
  • @symplectomorphic thanks, edited – Simple Jan 21 '16 at 06:10
  • If your desire is simply to show that one is the derivative of the other, I would simply use integrals to solve for the area and circumference formulas and show equality. In fact, you can generalize this approach to $n$-space quite easily; for example, surface area of a sphere is the derivative of the volume of a sphere. This pattern holds for all $n$-spheres. Do you desire to show the equivalence without actually solving for either the area or circumference? – Brevan Ellefsen Jan 21 '16 at 06:15
  • @BrevanEllefsen I only can use the fundamental theorem of calculus – Simple Jan 21 '16 at 06:17
  • @Simple the fundamental theorem of calculus uses derivatives and integrals. What do you mean when you say you wish to use it but not integrals or derivatives, – Stella Biderman Jan 21 '16 at 06:26
  • @Simple hmm... That's a fairly odd theorem to use in a proof like this. Do you have any other examples you have worked that could provide enlightment as to what you know and don't know and what is expected? – Brevan Ellefsen Jan 21 '16 at 06:27
  • @StellaBiderman sorry for the confusing, use the idea in the proof of second fundamental theorem of calculus.http://www-math.mit.edu/~djk/18_01/chapter14/proof01.html – Simple Jan 21 '16 at 06:30
  • @Simple can you show us how to do what you want, but with a different problem? – Stella Biderman Jan 21 '16 at 06:31
  • Also, are you allowed to assume beforehand that the area of a circle is $\pi r^2$ (you can obviously assume the circumference formula from your question), or do you have to show first that circumference is the derivative of area (and thus area is the antiderivative, up to a constant $c$). The second way only shows this up to a constant though, which isn't a very great result... As such, I'm really unclear as to what you want here. – Brevan Ellefsen Jan 21 '16 at 06:33
  • @StellaBiderman I want to know how to show $(\star)$ mathematically, because in my attempt, I don't have any reason to show $(\star)$ – Simple Jan 21 '16 at 06:36
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    @Simple you've already explained why it's true informally, with your comment about the trapezoid. To show it formally you need integrals, because areas are defined via integrals – Stella Biderman Jan 21 '16 at 06:38

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Why don't you just say:

\begin{align*} A'(r)&=\frac{d}{dr}A(r)\\ &=\frac{d}{dr}\pi r^2\\ &=2\pi r\\ &=C(r) \end{align*}

Stella Biderman
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