First derivative test: Change of sign of $f'(x)$ at critical point.

Second derivative test: If $f'(c)=0$, then if $f''(c)>0$ (local min), $f''(c)<0$ (local max).

Monotonicity: $f'(x) > 0$ (increasing), $f'(x) < 0$ (decreasing)

Rolle's Theorem: If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ such that $f'(c)=0$.

Mean Value Theorem (Lagrange): If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

4.5 Indefinite Integration

Standard Integrals:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$)
- $\int \frac{1}{x} dx = \ln|x| + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int e^x dx = e^x + C$
- $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(x/a) + C$
- $\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}(x/a) + C$
- $\int \frac{1}{x^2-a^2} dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$
- $\int \frac{1}{a^2-x^2} dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$
Integration by Parts: $\int u dv = uv - \int v du$ (LIATE rule for choosing $u$)

4.6 Definite Integration

Fundamental Theorem of Calculus: $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x)=f(x)$
Properties of Definite Integrals:
- $\int_a^b f(x) dx = \int_a^b f(t) dt$
- $\int_a^b f(x) dx = -\int_b^a f(x) dx$
- $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$
- $\int_0^a f(x) dx = \int_0^a f(a-x) dx$
- $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$
- $\int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a-x) dx$
  - If $f(2a-x) = f(x)$, then $2\int_0^a f(x) dx$
  - If $f(2a-x) = -f(x)$, then $0$
- $\int_{-a}^a f(x) dx$:
  - If $f(x)$ is even, $2\int_0^a f(x) dx$
  - If $f(x)$ is odd, $0$
Wallis Formula: $\int_0^{\pi/2} \sin^n x \cos^m x dx = \frac{[(n-1)!!][(m-1)!!]}{(n+m)!!} K$, where $K=\pi/2$ if $m,n$ both even, else $K=1$.

4.7 Area Under Curves

Area bounded by $y=f(x)$, x-axis, $x=a, x=b$: $\int_a^b |f(x)| dx$
Area between $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$: $\int_a^b |f(x)-g(x)| dx$

4.8 Differential Equations

Order: highest derivative present. Degree: power of highest derivative (after making it polynomial).
Variable Separable: $\frac{dy}{dx} = f(x)g(y) \implies \int \frac{dy}{g(y)} = \int f(x) dx$
Homogeneous: $\frac{dy}{dx} = f(x,y)$ where $f(x,y)$ is homogeneous of degree 0. Substitute $y=vx$.
Linear: $\frac{dy}{dx} + P(x)y = Q(x)$
- Integrating Factor (IF): $e^{\int P(x) dx}$
- Solution: $y \cdot (\text{IF}) = \int Q(x) \cdot (\text{IF}) dx + C$

5. Vectors

Dot Product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x+a_yb_y+a_zb_z$
- $\vec{a} \cdot \vec{a} = |\vec{a}|^2$
- If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$
Cross Product: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$
- $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$
- $|