Core Concepts

Limits and Continuity

• Limit Definition: The value a function approaches as the input approaches a number.

• Continuity: A function is continuous at x=ax = a if lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a).

• Removable Discontinuity: Hole in the graph where limit exists but function value is missing or different.

• Jump Discontinuity: Sudden jump; left-hand and right-hand limits are not equal.

• Infinite Discontinuity: Vertical asymptote; limit goes to ∞ or -∞.

Derivatives

• Definition of Derivative:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

• Power Rule: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}

• Product Rule: (fg)′=f′g+fg′(fg)' = f'g + fg'

• Quotient Rule: (fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

• Chain Rule: (f(g(x)))′=f′(g(x))⋅g′(x)(f(g(x)))' = f'(g(x)) \cdot g'(x)

Derivatives of Common Functions

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

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

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

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

• ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

Applications of Derivatives

• Critical Point: Where f′(x)=0f'(x) = 0 or f′(x)f'(x) is undefined.

• First Derivative Test: Determines local extrema using sign changes in f′f'.

• Second Derivative Test: If f′′(c)>0f''(c) > 0, local min; if f′′(c)<0f''(c) < 0, local max.

• Inflection Point: Where f′′(x)=0f''(x) = 0 and concavity changes.

• Related Rates: Differentiate with respect to time.

• Optimization: Use critical points to find max/min values in word problems.

Integrals

• Definition of Definite Integral:

∫abf(x)dx=lim⁡n→∞∑f(xi∗)Δx\int_a^b f(x)dx = \lim_{n \to \infty} \sum f(x_i^*) \Delta x

• Power Rule for Integrals:

∫xndx=xn+1n+1+C(n≠−1)\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

• Substitution Rule: Use u=g(x)u = g(x), then change all parts to uu.

• Integration by Parts:

∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Definite Integrals & Area

• FTC Part 2:

∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

• Area Between Curves:

∫ab[top−bottom] dx\int_a^b [\text{top} - \text{bottom}] \, dx

Volume

• Disk Method:

π∫ab[R(x)]2dx\pi \int_a^b [R(x)]^2 dx

• Washer Method:

π∫ab[R(x)2−r(x)2]dx\pi \int_a^b [R(x)^2 - r(x)^2] dx

• Shell Method:

2π∫ab(radius)(height)dx2\pi \int_a^b (radius)(height) dx

Series and Sequences

• Convergent Series: Has a finite sum.

• Divergent Series: Does not have a finite sum.

• Nth-Term Test: If lim⁡an≠0\lim a_n \neq 0, series diverges.

• Geometric Series:

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

• P-Series:

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

• Alternating Series Test: Converges if terms decrease and lim⁡an=0\lim a_n = 0.

• Ratio Test:

lim⁡∣an+1an∣<1⇒converges\lim \left|\frac{a_{n+1}}{a_n}\right| < 1 \Rightarrow \text{converges}

• Root Test:

lim⁡∣an∣n<1⇒converges\lim \sqrt[n]{|a_n|} < 1 \Rightarrow \text{converges}

Taylor and Maclaurin Series

• Maclaurin Series for exe^x:

∑xnn!\sum \frac{x^n}{n!}

• Maclaurin Series for sin⁡x\sin x:

∑(−1)nx2n+1(2n+1)!\sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}

• Maclaurin Series for cos⁡x\cos x:

∑(−1)nx2n(2n)!\sum \frac{(-1)^n x^{2n}}{(2n)!}

• Taylor Series Centered at aa:

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

• Radius of Convergence: Use Ratio or Root Test.

Core Theorems

• IVT: If continuous on [a,b][a, b], hits every value between f(a)f(a) and f(b)f(b).

• EVT: If continuous on [a,b][a, b], has absolute max and min.

• MVT:

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

• Rolle’s Theorem: If f(a)=f(b)f(a) = f(b), then f′(c)=0f'(c) = 0 somewhere.

• FTC Part 1:

ddx∫axf(t)dt=f(x)\frac{d}{dx} \int_a^x f(t) dt = f(x)