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Useful to recall from calculus

1. Algebraic manipulations: we will rely on manipulation algebraic expressions and solving algebraic expressions throughout the semester.
   1. Power functions: expressions of the form $x^3+x^5+x^8$, etc.
   2. $e^x$, $a^x$, $\ln x$: manipulations of the form $a^t = e^{t \ln a}$, equations of the form $a^t = b$ then $t = \log_a b=\ln b/\ln a$.
   3. Rational functions.
2. Geometric series
   1. $1+x+x^2+\cdots+x^{n-1} =(x^n-1)/(x-1)$.
   2. $1+x+x^2+\cdots=1/(1-x)$ if $|x|\lt 1$.
3. Derivatives
   1. $(x^n)' = n x^{n-1}$.
   2. $(a^x)' = a^x \ln a$.
   3. Chain rule, log derivatives.
   4. L'Hopital: for example limits of the form $\lim_{x\to\infty}x(a^{1/x}-1)=\ln a$ and $\lim_{x\to\infty}(1+y/x)^x=e^y$.
   5. Approximations: $e^x\approx 1+x$ and $\ln(1+x)\approx x$ if $x$ is small, $(1+x)^y\approx 1+xy$ if $x,y$ are small.
4. Integrals
   1. $\int a^x dx$, $\int x^ndx$, etc.
   2. The Fundamental Theorem of Calculus
   3. Integration by parts
5. Partial derivatives
6. Implicit differentiation

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