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Calculus Small Changes, Big Ideas

Bill Shillito

Section 5.4 Limits and integration

Introduction goes here.
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  • Zeno’s arrow paradox
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  • Limits at infinity
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  • Sums of powers:
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    \begin{align*} \sum_{k=1}^n c \amp = c n \\ \sum_{k=1}^n k \amp = \frac{n(n+1)}{2} \approx \frac{1}{2} n^2 \\ \sum_{k=1}^n k^2 \amp = \frac{n(n+1)(2n+1)}{6} \approx \frac{1}{3} n^3 \\ \sum_{k=1}^n k^3 \amp = \left(\frac{n(n+1)}{2}\right)^2 \approx \frac{1}{4} n^4 \end{align*}
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  • Riemann sum to compute \(\DS\int_{0}^{1} x\dd{x}\)
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  • Some functions aren’t integrable: great example is the Dirichlet function
    \begin{equation*} D(x)=\begin{cases}1 \amp \text{if $x$ is rational}\\ 0 \amp \text{if $x$ is irrational}\end{cases} \end{equation*}
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