Core Concepts

Limit Definition

The value a function approaches as the input approaches a number.

Continuity

A function is continuous at $x=a$ if $\lim[{x \to a} f(x) = f(a)$

Definition of Derivative

$$f'(x) = \lim[{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Power Rule

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Product Rule

$$(fg)' = f'g + fg'$$

Quotient Rule

$$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$$

Chain Rule

$$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$

d/dx [sinx]=

cos(x)

ddx[cos⁡x]$\frac{d}{dx}[\cos x] = -\sin x$

-sinx

ddx[tan⁡x]=sec⁡2x$\frac{d}{dx}[\tan x] = \sec^2 x$

string

ddx[ln⁡x]=1x$\frac{d}{dx}[\ln x] = \frac{1}{x}$

string

ddx[ex]=ex$\frac{d}{dx}[e^x] = e^x$

string

Critical Point

Where $f'(x) = 0$ or $f'(x)$ is undefined.

First Derivative Test

Determines local extrema using sign changes in $f'$.

Second Derivative Test

If f''(c) > 0, local min; if f''(c) < 0, local max.

Inflection Point

Where $f''(x) = 0$ and concavity changes.

Definition of Definite Integral

$$\int_a^b f(x)dx = \lim[{n \to \infty} \sum f(x_i^") \Delta x$$

Power Rule for Integrals

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

Substitution Rule

Use $u = g(x)$, then change all parts to $u$.

Integration by Parts

$$\int u \, dv = uv - \int v \, du$$

FTC Part 2

$$\int_a^b f(x)dx = F(b) - F(a)$$

Area Between Curves

$$\int_a^b [\text{top} - \text{bottom}] \, dx$$

Disk Method

$$\pi \int_a^b [R(x)]^2 dx$$

Washer Method

$$\pi \int_a^b [R(x)^2 - r(x)^2] dx$$

Shell Method

$$2\pi \int_a^b (radius)(height) dx$$

Convergent Series

Has a finite sum.

Divergent Series

Does not have a finite sum.

Nth-Term Test

If $\lim a_n \neq 0$, series diverges.

Geometric Series

\sum ar^n \text{ converges if } |r| < 1

P-Series

\sum \frac{1}{n^p} \text{ converges if } p > 1

Alternating Series Test

Converges if terms decrease and $\lim a_n = 0$.

Ratio Test

\lim \left|\frac{a{n+1}}{an}\right| < 1 \Rightarrow \text{converges}

Root Test

\lim \sqrt[n]{|a_n|} < 1 \Rightarrow \text{converges}

Maclaurin Series for $e^x$

$$\sum \frac{x^n}{n!}$$

Maclaurin Series for $\sin x$

$$\sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

Maclaurin Series for $\cos x$

$$\sum \frac{(-1)^n x^{2n}}{(2n)!}$$

Taylor Series Centered at $a$

$$\sum \frac{f^{(n)}(a)}{n!}(x - a)^n$$

Radius of Convergence

Use Ratio or Root Test.

IVT

If continuous on $[a,b]$, hits every value between $f(a)$ and $f(b)$

EVT

If continuous on $[a,b]$, has absolute max and min.

MVT

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Rolle’s Theorem

If $f(a) = f(b)$, then $f'(c) = 0$ somewhere.

FTC Part 1

$$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$