Derivatives
Equation of a tangent line
$$y = f(a) + f'(a)(x - a)$$
Taylor Polynomials
$$T_{n,a}(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n$$
Taylor Series
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$$
Taylor Series for e^x at a = 0
$$e^x \text{ at } a = 0: \quad \sum_{n=0}^{\infty} \frac{x^n}{n!}$$
Taylor Series for sin x at a = 0
$$\sin x \text{ at } a = 0: \quad \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$
Integrals
Integration by Parts
$$\int u \, dv = uv - \int v \, du$$
Fundamental Theorem of Calculus (FTC I)
$$F(x) = \int_a^x f(t) \, dt$$
Fundamental Theorem of Calculus (FTC II)
$$\int_a^b f(x) \, dx = F(b) - F(a)$$
Leibniz Rule
$$\frac{d}{dx} \left[ \int_{a(x)}^{b(x)} f(t) \, dt \right] = f(b(x)) \cdot \frac{d}{dx} b(x) - f(a(x)) \cdot \frac{d}{dx} a(x)$$
P-Integrals (Convergence)
$$\text{For } a > 0, \int_a^{\infty} \frac{1}{x^p} \, dx \text{ converges if } p > 1 \text{ and diverges if } p \leq 1$$
P-Integrals (Divergence)
$$\text{For } a > 0, \int_0^a \frac{1}{x^p} \, dx \text{ converges if } p < 1 \text{ and diverges if } p \geq 1$$
Multivariable Functions
Magnitude of a vector
$$\|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$
Converting a vector to unit vector
$$\vec{u} = \left( \frac{v_1}{\|\vec{v}\|}, \frac{v_2}{\|\vec{v}\|}, \ldots, \frac{v_n}{\|\vec{v}\|} \right)$$
Directional Derivative
$$D_{\vec{u}} f = \nabla f \cdot \vec{u}$$
Directional derivative given angle θ
$$D_{\vec{u}} f = \|\nabla f(\mathbf{x}_0)\| \cos(\theta)$$
Unit vector given angle θ
$$\vec{u} = (\cos \theta, \sin \theta)$$
Chain Rule
$$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$$
Equation of a Tangent Plane
$$z - f(x_0, y_0) = \frac{\partial f}{\partial x}(x_0, y_0) \cdot (x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0) \cdot (y - y_0)$$
Linear Approximation
$$f(x, y) \approx f(x_0, y_0) + \frac{\partial f}{\partial x}(x_0, y_0) \cdot (x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0) \cdot (y - y_0)$$
Optimizations
Discriminant
$$D = \frac{\partial^2 f}{\partial x^2} \cdot \frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2$$
Lagrange Multipliers
$$F(x, y, z, \lambda) = f(x, y, z) - \lambda \left( g(x, y, z) - k \right)$$
Multivariable Integrals
Fubini's Theorem
$$\iint_R f(x, y) \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy$$
Polar Coordinates
$$x = r \cos \theta, \quad y = r \sin \theta$$
Double Integral in Polar Coordinates
$$\int_{\theta_1}^{\theta_2} \int_{r_1}^{r_2} r \, dr \, d\theta$$
Vector Fields
Tangent line
$$\text{The tangent line to } c \text{ at } t \text{ is } \{ c(t) + \lambda c'(t) : \lambda \in \mathbb{R} \}$$
Arc Length
$$\text{The arc length of a curve } c \text{ between } t_1 \text{ and } t_2 \text{ is } \ell = \int_{t_1}^{t_2} \|c'(t)\| \, dt.$$
Line integral of a scalar function
$$\int_C f \, ds = \int_a^b f(x(t), y(t), z(t)) \|c'(t)\| \, dt$$
Line integral of a vector function
$$\int_C \mathbf{F} \cdot d\mathbf{s} = \int_a^b \mathbf{F}(\mathbf{c}(t)) \cdot \mathbf{c}'(t) \, dt.$$
Fundamental Integration Theorem
$$\int_C \mathbf{F} \cdot d\mathbf{s} = \int_C \nabla f \cdot d\mathbf{s} = f(c(b)) - f(c(a))$$