Continuous Reflection

"If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries." - Carl Friedrich Gauss

A function defined by an integral

February 27, 2021

My daughter shared a lovely problem she had encountered in her calculus class:

Use the “function defined by an integral" part of the FTC(and explicitly not the “evaluation" part of the FTC)to find the values of a for which∫axcos⁡t dt=sin⁡x.Then find an antiderivative of cos⁡x which will not equal∫axcos⁡t dt for any value of a.\small\color{grey}\text{Use the ``function defined by an integral" part of the FTC}\\[6pt] \text{(and explicitly \textit{\textbf{not}} the ``evaluation" part of the FTC)}\\[3pt] \text{to find the values of } a \text{ for which} \int_{a}^{x}\cos t\,dt=\sin x.\\[14pt] \text{Then find an antiderivative of } \cos x \text{ which will not equal} \\[6pt] \int_{a}^{x}\cos t\,dt \hspace{}0.05in\text{ for any value of } a.

I was inspired to make a Desmos animation:

FTC (Evaluation part)

If f is continuous on [a,b], and F is the function defined byF(x)=∫abf(t) dt,a≤x≤bthen F is continuous on [a,b] and differentiable on (a,b),and F is an antiderivative of f.\small\color{grey}\\[6pt] \text{If } f \text{ is continuous on } [a,b] \text{, and } F \text{ is the function defined by} \\[6pt] F(x)=\int_{a}^{b}f(t) \, dt, \hspace{0.2in} a\leq x\leq b\\[6pt] \text{then } F \text{ is continuous on }[a,b] \text{ and differentiable on } (a,b), \\[6pt] \text{and } F \text{ is an antiderivative of } f. \\[6pt]
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About Me

I’m Laura Kinnel, a math teacher at George School. You can find my catalogued collection of interactive internet resources for high school math (some created by me, but most by others) at Interact with Math!

Speaking of math . . .

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.

— René Descartes, La Géométrie

Continuous Reflection

 

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