Notes for the Coursera Single-Variable Calculus course
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... $$
Properties of the exponential
$$ \frac{d}{dx} e^x = e^x $$
$$ \int e^x dx = e^x + C $$
$ ln $ is the inverse of $ e^x $
$$ \frac d {dx} \ln(x) = \frac 1 x $$
$$ e^{ix} = \cos x + i \sin x $$
$$ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} $$
$$ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} $$
Taylor Series about $ x = 0 $ is
$$ f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + ... = \sum_{k=0}^{\infty} \frac{ f^{(k)}(0) }{ k! } x^k $$
where $ f^{(k)}(0) $ is the kth derivative of $ f $ evaluated at 0.
The Taylor series for $f$ about $ x=a $
$$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... = \sum_{k=0}^{\infty} \frac{ f^{(k)}(a) }{ k! } (x-a)^k $$
$$ \sinh(x) = \frac{e^x - e^{-x}}{2} $$
$$ \cosh(x) = \frac{e^x + e^{-x}}{2} $$
$$ \tanh(x) = \frac{ \sinh x }{ \cosh x} = \frac{e^x -e^{-x}}{e^x+ e^{-x}} $$
Note $ \cosh^2 - \sinh^2 = 1 $
fig, (ax1,ax2) = subplots(1,2, figsize=(16,5))
x=linspace(-2,2,100)
ax1.axhline(0, color="#c0c0c0")
ax1.axvline(0, color="#c0c0c0")
ax1.plot(x, cosh(x), label="cosh", lw=3)
ax1.plot(x, sinh(x), label="sinh", lw=3)
ax1.legend(loc="lower right")
ax2.axhline(0, color="#c0c0c0")
ax2.axvline(0, color="#c0c0c0")
ax2.plot(x, tanh(x), label="tanh", lw=3)
ax2.legend(loc="lower right")
Taylor Series for sinh and cosh
$$ \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + ... $$
$$ \cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ... $$
They are similar to $\sin x$ and $\cos x$ without the alternating signs.
$$ \frac{d}{dx}\sinh x = \cosh x $$
$$ \frac{d}{dx}\cosh x = \sinh x $$
$$ \frac 1 {1-x} = 1 + x + x^2 +x^3 + ... = \sum_{k=0}^{\infty}x^k \hspace{ 5 em } \text{for} |x| < 1 $$
This can be used to find the Taylor series for $ \ln(1+x) $. Note
$$ \frac d {dx} \ln(1+x) = \frac 1 {1+x} $$
Integrating as geometric series gets
\begin{align} \ln(1+x) &= x - \frac {x^2} 2 + \frac {x^3} 3 - ... \\ &= \sum_{k=1}^{\infty} (-1)^{k-1} \frac {x^k} { k } \hspace{ 5 em } (|x| < 1) \end{align}
$$ \frac d {dx} \arctan(x) = \frac 1 {1+x^2} $$
\begin{align} \arctan(x) &= x - \frac {x^3} 3 + \frac {x^5} 5 - ... \\ &= \sum_{k=0}^{\infty} (-1)^k \frac {x^{2k+1}} { 2k + 1 } \hspace{ 5 em } (|x| < 1) \end{align}
$$ (1 + x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k \hspace{ 5 em } \text{for} |x| < 1$$
\begin{align} \lim_{x \to a} f(x) = L \hspace{ 3 em } & \text{ iff for every} \epsilon > 0 \text{ there exits a } \delta > 0 \text{ such that } \\ |f(x) - L| < \epsilon \hspace{2 em} & \text{ whenever } 0 < | x-a | < \delta \end{align}
$$ \lim_{x \to a} (f+g)(x) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) $$
$$ \lim_{x \to a} (f \cdot g)(x) = ( \lim_{x \to a} f(x) ) ( \lim_{x \to a} g(x) ) $$
$$ \lim_{x \to a} (f / g)(x) = \frac {\lim_{x \to a} f(x)} { \lim_{x \to a} g(x) } \;\;\;\; \text{ if denominator } \neq 0 $$
$$ \lim_{x \to a} (f \circ g)(x) = f( \lim_{x \to a} g(x) ) \;\;\;\; \text{ if f is continuous } $$
If $ \lim_{x \to a} f(x) = 0 \; $ and $ \; \lim_{x \to a} g(x) = 0 \; $ then
$$ \lim_{x \to a} \frac {f(x)}{g(x)} = \lim_{x \to a} \frac {f'(x)}{g'(x)} $$
If $ \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = \infty $ then
$$ \lim_{x \to a} \frac {f(x)}{g(x)} = \lim_{x \to a} \frac {f'(x)}{g'(x)} $$
l'Hôpital's rule can be applied repeatly if the derivative is still $0/0$ or $\infty / \infty$.
Compare the asymptotic growth by taking limits of quotients
$$ \lim_{x \to \infty} \frac {f(x)} {g(x)} \; or \; \lim_{x \to 0} \frac {f(x)} {g(x)} $$
From greatest to smallest
$f(x)$ is in $O(g(x))$, as $ x \to 0 $ iff $ |f(x)| \leq C |g(x)| $
$f(x)$ is in $O(g(x))$, as $ x \to \infty $ iff $ |f(x)| \leq C |g(x)| $
$$ \tan \theta = \frac {\sin \theta} {\cos \theta} $$
Secant $ \; \sec \theta = \frac 1 {\cos \theta} $
Cosecant $ \; \csc \theta = \frac 1 {\sin \theta} $
Cotangent $ \; \cot \theta = \frac 1 {\tan \theta} $
arcsin, arccos, arctan, arcsec, arccsc, arccot
$$ \cos^2 \theta + \sin^2 \theta = 1 $$
\begin{align} \frac d {dx} \sin x &= \cos x \hspace{3em} & \frac d {dx} \arcsin x &= \frac 1 {\sqrt { 1-x^2}} \\ \frac d {dx} \cos x &= -\sin x \hspace{3em} & \frac d {dx} \arccos x &= \frac {-1} {\sqrt { 1-x^2}} \\ \frac d {dx} \tan x &= \sec^2 x \hspace{3em} & \frac d {dx} \arctan x &= \frac 1 {1 + x^2} \\ \end{align}
Trigonometric functions - Wikipedia, the free encyclopedia
List of trigonometric identities - Wikipedia, the free encyclopedia