Formulas

Useful Integrals

[\int \tan x , dx = \ln |\sec x|]

[\int \sec x , dx = \ln |\sec x + \tan x|]

[\int \frac{dx}{x^{2} + a^{2}} = \frac{1}{a} \arctan(\frac{x}{a})]

[\int \frac{dx}{\sqrt{a^{2} - x^{2}}} = \arcsin(\frac{x}{a})]

[\int \frac{dx}{x\sqrt{x^{2} - a^{2}}} = \frac{1}{a} \arcsec(\frac{x}{a})]

Integration Approximation

  • On interval ([a, b]) with midpoints and (\Delta x = \frac{b - a}{n})

  • Midpoint Rule:

    • [M_n = \Delta x \sum_{i=1}^{n} f(x_i)]

  • Trapezoidal Rule:

    • [T_n = \Delta x \left( \frac{1}{2} f(x_0) + \sum_{i=1}^{n-1} f(x_i) + \frac{1}{2} f(x_n) \right)]

  • Simpson’s Rule (n even only):

    • [S_n = \Delta x \left( \frac{1}{3} f(x_0) + 4 \sum_{i=1}^{n-1 \text{ odd}} f(x_i) + 2 \sum_{i=2}^{n-2 \text{ even}} f(x_i) + \frac{1}{3} f(x_n) \right)]

Error Bounds

  • For Midpoint:[\left|E_M\right| \leq \frac{K(b - a)^3}{24 n^2}]

  • For Trapezoidal:[\left|E_T\right| \leq \frac{K(b - a)^3}{12 n^2}]

  • For Simpson's Rule:[\left|E_S\right| \leq \frac{K(b - a)^5}{180 n^4}]

    • where (f'' \leq K) for the first two, and (f^{(4)} \leq K) for Simpson's on ([a,b])

Arc Length

  • For curve (y = g(x)) in (a \leq x \leq b):[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} , dx]

  • Alternatively for (x = g(y)):[L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^{2}} , dy]

Surface Area

  • Revolving around the x-axis:[S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} , dx]

  • Revolving around the y-axis:[S = 2\pi \int_{c}^{d} g(y) \sqrt{1 + \left(\frac{dx}{dy}\right)^{2}} , dy]

Center of Mass (Centroid)

  • For region bounded by (f(x) \geq g(x)):

  • [\bar{x} = \frac{1}{A} \int_{a}^{b} x[f(x) - g(x)] , dx]

  • [\bar{y} = \frac{1}{2A} \int_{a}^{b} [f(x)]^{2} - [g(x)]^{2} , dx]

Pappus Theorem

  • When a region revolves around a line it does not intersect, the volume is given by the area of the region times the distance traveled by the centroid.

Economic Surplus

  • Consumer Surplus:

  • [= \int_{0}^{A} p_D(x) - B , dx ]

  • Producer Surplus:

  • [= \int_{0}^{A} B - p_S(x) , dx ]

Probability

  • Probability Density Function (pdf) on ([a, b]):

  • (f(x) \geq 0) and (\int_{a}^{b} f(x) , dx = 1)

  • Probability:

  • [P(a \leq X \leq b) = \int_{a}^{b} f(x) , dx]

Expected Value and Variance

  • Discrete:

  • [\mu = \sum_{i} x_i p_i]

  • [Var(X) = \sum_{i} (x_i - \mu)^{2} p_i]

  • Continuous:

  • [\mu = E(X) = \int_{a}^{b} x f(x) , dx]

  • [Var(X) = \left(\int_{a}^{b} x^{2} f(x) , dx\right) - \mu^{2}]

  • Median: m satisfies

  • [\int_{a}^{m} f(x) , dx = \frac{1}{2}]

  • Standard Deviation:

  • [\sigma = \sqrt{Var(X)}]

Exponential Density Function

[f(t) = be^{-bt} \text{ on } [0,\infty)]

[\mu = \frac{1}{b}, \sigma = \frac{1}{b}]

Normal Density Function

[f(t) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}} \text{ on } (-\infty,\infty)]

Remainder Estimate for Integral

  • If (f(k) = ak) is continuous, positive, decreasing:

  • Remainder Test:

  • [R_n = S - S_n]

  • [\int_{n}^{\infty} f(x) , dx \leq R_n \leq \int_{n - 1}^{\infty} f(x) , dx]

Alternating Series Remainder Estimate

  • If (f(k) = b_k) is positive, decreasing with terms approaching 0,

  • Then (R_n \leq |S - S_n| \leq b_{n+1}]

General Taylor Series

  • Taylor series:[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^{n} ]

  • Maclaurin series:[a = 0]

Specific Maclaurin Series

  • [\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n} \text{ for } x \in (-1, 1)]

  • [e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} \text{ for } x \in \mathbb{R}]

  • [\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n + 1}}{(2n + 1)!} \text{ for } x \in \mathbb{R}]

  • [\cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \text{ for } x \in \mathbb{R}]

  • [\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n + 1}}{2n + 1} \text{ for } x \in [-1, 1]]

  • [\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{n}}{n} \text{ for } x \in (-1, 1]]

  • [(1 + x)^{k} = \sum_{n=0}^{\infty} \binom{k}{n} x^{n}\text{ for } x \in (-1, 1)]

Euler’s Method

  • For (y' = F(x, y)) with given point ((x_0, y_0))

  • For (n \geq 1, )

  • [y_n = y_{n-1} + hF(x_{n-1}, y_{n-1})]

Logistic Curve/Population with Carrying Capacity

  • Logistic Function:

  • [f(t) = \frac{M}{1 + b e^{-kt}}]

  • Rate of change of (y):

  • [y' = ky\left(1 - \frac{y}{M}\right)]

Extinction with Capacity

  • If the population has a carrying capacity (M) and extinction marker (m):

  • Rate:

  • [y' = ky\left(1 - \frac{y}{M}\right) ext{ at } \left(1 - \frac{m}{y}\right)]

Capacity with External Removal

  • If the capacity is (M) but we remove (c) units per time:

  • Rate:

  • [y' = ky\left(1 - \frac{y}{M}\right) - c]

Second Order Homogeneous Equations

  • For the equation (ay'' + by' + cy = 0):

  • Solve the characteristic equation:

  • [ar^{2} + br + c = 0]

Second Order Nonhomogeneous Equations

  • Use undetermined coefficients:

  • General solution:

  • [y_p + y_c]

  • Where (y_p) is a particular solution.

  • For cases like (G(x) = e^{kx} P(x) \ (P(x) \text{ is a polynomial})]:

  • Assume [y_p = e^{kx} Q(x) \cos(mx) + e^{kx} R(x) \sin(mx)]

  • Similar for (G(x)) having (cos) instead of (sin).

Variation of Parameters

  • For (ay'' + by' + cy = G(x)):

  • Try:

  • [y_p = u_1y_1 + u_2y_2]

  • Then solve:

  • [u'{1}y{1} + u'{2}y{2} = 0]

  • [u'{1}y'{1} + u'{2}y'{2} = G(x)]

Springs

  • Hooke’s Law:

  • [F = -kx]

  • For spring's extension beyond normal length (x).

  • Simple Harmonic Motion:

  • [m \frac{d^2x}{dt^{2}} + kx = 0]

  • With Damping:

  • [m \frac{d^2x}{dt^{2}} + c \frac{dx}{dt} + kx = 0]

    • where (c) is the damping constant.

  • Types of Damping:

  • Overdamping: (c^2 - 4mk > 0)

  • Critical Damping: (c^2 - 4mk = 0)

  • Underdamping: (c^2 - 4mk < 0)

  • Forced Vibrations:

  • [m \frac{d^2x}{dt^{2}} + c \frac{dx}{dt} + kx = F(t)]

    • Where (F(t)) is the external force.