# I. Functions

Notes for the Coursera Single-Variable Calculus course

• 2013-01-16 fixed arctan's Taylor series' summation form
• 2013-01-13 first draft

## The Exponential $e^x$

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$

Properties of the exponential

$$\frac{d}{dx} e^x = e^x$$

$$\int e^x dx = e^x + C$$

$ln$ is the inverse of $e^x$

$$\frac d {dx} \ln(x) = \frac 1 x$$

### Euler's formula

$$e^{ix} = \cos x + i \sin x$$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$

## Taylor Series

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 $x=a$

$$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... = \sum_{k=0}^{\infty} \frac{ f^{(k)}(a) }{ k! } (x-a)^k$$

## Hyperbolic Trigonometric Functions

$$\sinh(x) = \frac{e^x - e^{-x}}{2}$$

$$\cosh(x) = \frac{e^x + e^{-x}}{2}$$

$$\tanh(x) = \frac{ \sinh x }{ \cosh x} = \frac{e^x -e^{-x}}{e^x+ e^{-x}}$$

Note $\cosh^2 - \sinh^2 = 1$

In [29]:
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")

Out[29]:
<matplotlib.legend.Legend at 0x7a4a9f0>

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.

$$\frac{d}{dx}\sinh x = \cosh x$$

$$\frac{d}{dx}\cosh x = \sinh x$$

## The Geometric series

$$\frac 1 {1-x} = 1 + x + x^2 +x^3 + ... = \sum_{k=0}^{\infty}x^k \hspace{ 5 em } \text{for} |x| < 1$$

### ln

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}

### Arctan

$$\frac d {dx} \arctan(x) = \frac 1 {1+x^2}$$

\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}

### Binomial Series

$$(1 + x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k \hspace{ 5 em } \text{for} |x| < 1$$

## Limits

\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}

### Limit Rules

$$\lim_{x \to a} (f+g)(x) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$

$$\lim_{x \to a} (f \cdot g)(x) = ( \lim_{x \to a} f(x) ) ( \lim_{x \to a} g(x) )$$

$$\lim_{x \to a} (f / g)(x) = \frac {\lim_{x \to a} f(x)} { \lim_{x \to a} g(x) } \;\;\;\; \text{ if denominator } \neq 0$$

$$\lim_{x \to a} (f \circ g)(x) = f( \lim_{x \to a} g(x) ) \;\;\;\; \text{ if f is continuous }$$

## L'Hôpital's rule

### The 0/0 Case

If $\lim_{x \to a} f(x) = 0 \;$ and $\; \lim_{x \to a} g(x) = 0 \;$ then

$$\lim_{x \to a} \frac {f(x)}{g(x)} = \lim_{x \to a} \frac {f'(x)}{g'(x)}$$

### The $\infty / \infty$ Case

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 $\infty / \infty$.

## Orders Of Growth

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

• Factorial: $x!$
• Expontential: $c^x$ for any $c > 1$
• Polynomial: $x^k$ for any $k > 1$
• Logarithmic: $ln(x)$

### Big-O notation, $x \to 0$

$f(x)$ is in $O(g(x))$, as $x \to 0$ iff $|f(x)| \leq C |g(x)|$

### Big-O notation, $x \to \infty$

$f(x)$ is in $O(g(x))$, as $x \to \infty$ iff $|f(x)| \leq C |g(x)|$

## Trigonometric Functions Refresher

$$\tan \theta = \frac {\sin \theta} {\cos \theta}$$

Secant $\; \sec \theta = \frac 1 {\cos \theta}$

Cosecant $\; \csc \theta = \frac 1 {\sin \theta}$

Cotangent $\; \cot \theta = \frac 1 {\tan \theta}$

### Inverse functions

arcsin, arccos, arctan, arcsec, arccsc, arccot

### Pythagorean identity

$$\cos^2 \theta + \sin^2 \theta = 1$$

### Calculus

\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