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)

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

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

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

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

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

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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)