This file is a calculus worksheet Amanda Taylor Lipnicki created and used for teaching applications of calculus to crochet patterns. It is a TeX file, so you can edit it and use it in your own classes.

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%opening
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\title{Hooked on Calculus: Crochet Surfaces of Revolution Worksheet}
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\begin{document}
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\date{}
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\maketitle
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Choose a function $f(x)$  and horizontal bounds $a \leq x \leq b$ on which $f(x)$ is differentiable and positive valued.  Generate a crochet pattern for the corresponding surface of revolution.  You may use the calculator at https://www.desmos.com/calculator/8rfmnuciwg for this worksheet.  Enter your own function $f(x)$ and your own bounds $a$ and $b$ at the top.  
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\begin{enumerate} 
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\item For your chosen function and bounds, calculate the arc length using the formula $$\int_a^b \sqrt{1 + \left(f'(x)\right)^2} \, dx.$$ \item Compare the previous number with the number of rows in your pattern.  Are they similar?  Explain.
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\item Consider $$\int_a^{x_1} \sqrt{1 + \left(f'(x)\right)^2} \, dx.$$  Find an $x_1 > a$ such that the integral approximately equals 0.5.  
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\item Calculate $2\pi f(x_1)$ and compare this with the number of chain stitches in Row 1.  Explain.
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\item Consider $$\int_a^{x_2} \sqrt{1 + \left(f'(x)\right)^2} \, dx.$$  Find an $x_2 > x_1$ such that the integral approximately equals 1.5.  
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\item Calculate $2\pi f(x_2)$ and compare this with the number of stitches in Row 2.  Explain.  
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\item Using the above exploration as guidance, what are the main mathematical concepts behind our pattern creation for surfaces of revolution and why? 
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\item  For your chosen function, calculate the surface area of the surface of revolution using the formula $$\int_a^b 2\pi f(x) \sqrt{1 + \left(f'(x)\right)^2} \, dx.$$ 
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\item Compare the previous number with the number of stitches in your pattern.  Are they similar?  Explain.
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\item For your chosen function, calculate the volume inside the surface of revolution using the formula $$\int_a^b \pi (f(x))^2 \, dx.$$ 
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\item Include a printout of your program worksheet that shows your function, bounds, and the pattern generated.  
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\item Share your completed crochet project with Dr. Taylor, preferably in person, but pictures are also welcome!
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\end{enumerate}
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\end{document}
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