Calculus Cheat Sheet

Cheatsheet Content

### Sequences - **Definition:** An ordered list of numbers $a_1, a_2, ..., a_n, ...$. - **Notation:** $a_n$ represents the $n$-th term. - **Explicit:** $a_n = f(n)$ (e.g., $a_n = n^2$). - **Recursive:** $a_n = f(a_{n-1})$ (e.g., $a_n = a_{n-1} + 2$, $a_1=1$). - **Convergent:** $\lim_{n \to \infty} a_n = L$ (a finite number). - **Divergent:** Limit does not exist or is $\pm\infty$. #### Important Limits - $\lim_{n \to \infty} \frac{1}{n^p} = 0$ for $p > 0$. - $\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$. - $\lim_{n \to \infty} \frac{\ln n}{n^p} = 0$ for $p > 0$. - $\lim_{n \to \infty} \frac{n^p}{e^{qn}} = 0$ for $p > 0, q > 0$. #### Theorems - **Monotone Convergence Theorem:** If a sequence is bounded and monotonic (always increasing or always decreasing), then it converges. - **Squeeze Theorem:** If $b_n \le a_n \le c_n$ for all $n$ and $\lim_{n \to \infty} b_n = \lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} a_n = L$. #### Techniques - **Dominant Term Analysis:** For rational functions of $n$, divide numerator and denominator by the highest power of $n$. - **Ratio Comparison:** $\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| ### Limits (Functions) - **Limit of a Function:** $\lim_{x \to c} f(x) = L$ if $f(x)$ approaches $L$ as $x$ approaches $c$. - **One-Sided Limits:** $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$. Limit exists iff both one-sided limits exist and are equal. - **Infinite Limits:** $\lim_{x \to c} f(x) = \pm\infty$ (vertical asymptote). - **Limits at Infinity:** $\lim_{x \to \pm\infty} f(x) = L$ (horizontal asymptote). - **Continuity:** $f(x)$ is continuous at $c$ if $\lim_{x \to c} f(x) = f(c)$. #### Limit Laws - **Sum Rule:** $\lim (f(x) \pm g(x)) = \lim f(x) \pm \lim g(x)$ - **Product Rule:** $\lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x)$ - **Quotient Rule:** $\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$, if $\lim g(x) \ne 0$. - **Composition Rule:** $\lim_{x \to c} f(g(x)) = f(\lim_{x \to c} g(x))$ if $f$ is continuous. #### Indeterminate Forms | Form | Example | Technique | |:------------|:--------------------------------------|:-----------------------------------------------| | $0/0$ | $\lim_{x \to 0} \frac{\sin x}{x}$ | L'Hôpital's Rule, Factoring, Rationalizing | | $\infty/\infty$ | $\lim_{x \to \infty} \frac{x^2}{e^x}$ | L'Hôpital's Rule, Dominant Term Analysis | | $0 \cdot \infty$ | $\lim_{x \to 0^+} x \ln x$ | Convert to $0/0$ or $\infty/\infty$ ($x/(1/\ln x)$) | | $\infty - \infty$ | $\lim_{x \to \infty} (\sqrt{x^2+x} - x)$ | Common Denominators, Rationalizing | | $1^\infty$ | $\lim_{x \to \infty} (1 + 1/x)^x$ | Logarithms, $e^{\lim g(x) \ln f(x)}$ | | $0^0$ | $\lim_{x \to 0^+} x^x$ | Logarithms, $e^{\lim g(x) \ln f(x)}$ | | $\infty^0$ | $\lim_{x \to \infty} x^{1/x}$ | Logarithms, $e^{\lim g(x) \ln f(x)}$ | #### Techniques - **Factoring:** Cancel common factors in rational expressions. - **Rationalizing:** Multiply by conjugate for expressions with radicals. - **Common Denominators:** Combine fractions. - **L’Hôpital’s Rule:** If $\lim \frac{f(x)}{g(x)}$ is $0/0$ or $\infty/\infty$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$. - **Series Approximation:** (Briefly) For limits as $x \to 0$, use Taylor series expansions (e.g., $\sin x \approx x$). #### Important Standard Limits - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ - $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ - $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$ ### Derivatives - **Definition:** $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ (Difference Quotient). - **Geometric Meaning:** $f'(x)$ is the slope of the tangent line to $f(x)$ at $x$. #### Derivative Table | Function $f(x)$ | Derivative $f'(x)$ | |:----------------|:-------------------------| | $c$ (constant) | $0$ | | $x^n$ | $nx^{n-1}$ | | $e^x$ | $e^x$ | | $a^x$ | $a^x \ln a$ | | $\ln|x|$ | $1/x$ | | $\log_a x$ | $\frac{1}{x \ln a}$ | | $\sin x$ | $\cos x$ | | $\cos x$ | $-\sin x$ | | $\tan x$ | $\sec^2 x$ | | $\sec x$ | $\sec x \tan x$ | | $\csc x$ | $-\csc x \cot x$ | | $\cot x$ | $-\csc^2 x$ | | $\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ | | $\arctan x$ | $\frac{1}{1+x^2}$ | #### Differentiation Rules - **Product Rule:** $(uv)' = u'v + uv'$ - **Quotient Rule:** $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ - **Chain Rule:** $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ #### Higher Derivatives - **Second Derivative ($f''(x)$):** Rate of change of slope. - $f''(x) > 0 \implies$ concave up. - $f''(x) 0 \implies$ local min. - If $f'(c)=0$ and $f''(c) ### Integrals #### Indefinite Integrals (Antiderivatives) - **Definition:** $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$. - **Constant of Integration:** $\int f(x) dx = F(x) + C$. #### Basic Rules - $\int c \cdot f(x) dx = c \int f(x) dx$ - $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$ #### Integral Table | Function $f(x)$ | Indefinite Integral $\int f(x) dx$ | |:----------------|:-----------------------------------| | $x^n$ ($n \ne -1$) | $\frac{x^{n+1}}{n+1} + C$ | | $1/x$ | $\ln|x| + C$ | | $e^x$ | $e^x + C$ | | $a^x$ | $\frac{a^x}{\ln a} + C$ | | $\sin x$ | $-\cos x + C$ | | $\cos x$ | $\sin x + C$ | | $\sec^2 x$ | $\tan x + C$ | | $\csc^2 x$ | $-\cot x + C$ | | $\sec x \tan x$ | $\sec x + C$ | | $\csc x \cot x$ | $-\csc x + C$ | | $\frac{1}{\sqrt{1-x^2}}$ | $\arcsin x + C$ | | $\frac{1}{1+x^2}$ | $\arctan x + C$ | #### Techniques - **Substitution Method ($u$-substitution):** 1. Let $u = g(x)$ (inner function). 2. Find $du = g'(x) dx$. 3. Substitute $u$ and $du$ into the integral. 4. Integrate with respect to $u$. 5. Substitute back $g(x)$ for $u$. - **Integration by Parts:** $\int u \, dv = uv - \int v \, du$. - **LIATE Rule:** Choose $u$ based on priority: **L**ogarithmic, **I**nverse Trig, **A**lgebraic, **T**rig, **E**xponential. - **Basic Trigonometric Integrals:** Use identities to simplify (e.g., $\sin^2 x = \frac{1-\cos(2x)}{2}$). #### Definite Integrals - **Fundamental Theorem of Calculus (Part 1):** $\frac{d}{dx} \int_a^x f(t) dt = f(x)$. - **Fundamental Theorem of Calculus (Part 2):** $\int_a^b f(x) dx = F(b) - F(a)$, where $F'(x) = f(x)$. - **Area Interpretation:** $\int_a^b f(x) dx$ is the signed area between $f(x)$ and the x-axis from $a$ to $b$. - **Average Value of a Function:** $f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$. #### Improper Integrals - **Infinite Limits:** $\int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$. - **Vertical Asymptotes:** $\int_a^b f(x) dx = \lim_{c \to b^-} \int_a^c f(x) dx$ if $f(x)$ has a VA at $b$. - Converges if limit is finite, diverges otherwise. ### Ordinary Differential Equations (ODE) #### First-Order ODE - **Separable Equations:** $dy/dx = g(x)h(y)$. - Method: $\int \frac{1}{h(y)} dy = \int g(x) dx$. - General Solution: Implicit or explicit function $y(x)$. - **Linear First-Order:** $y' + P(x)y = Q(x)$. - Integrating Factor: $\mu(x) = e^{\int P(x) dx}$. - General Solution: $y(x) = \frac{1}{\mu(x)} \int \mu(x)Q(x) dx$. #### Second-Order Linear ODE (Homogeneous) - Form: $ay'' + by' + cy = 0$. - **Characteristic Equation:** $ar^2 + br + c = 0$. - **Case 1: Two Distinct Real Roots $r_1, r_2$.** - $y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}$. - **Case 2: One Repeated Real Root $r$.** - $y(x) = C_1 e^{rx} + C_2 x e^{rx}$. - **Case 3: Complex Conjugate Roots $\alpha \pm i\beta$.** - $y(x) = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))$. #### Exponential Growth/Decay - **Equation:** $y' = ky$. - **General Solution:** $y(t) = y_0 e^{kt}$. - $k > 0$: Growth; $k ### Common Mistakes - **Forgetting +C:** Always include the constant of integration for indefinite integrals. - **Misusing Chain Rule:** Don't forget to multiply by the derivative of the inner function. - **Incorrect L’Hôpital Application:** Only use for $0/0$ or $\infty/\infty$; convert other indeterminate forms first. - **Confusing Definite vs. Indefinite Integral:** Definite integrals yield a number (area), indefinite integrals yield a family of functions. - **Sign Mistakes:** Be careful with signs, especially in derivatives of trigonometric functions (e.g., $\frac{d}{dx}\cos x = -\sin x$). - **Algebra Errors:** Simplify expressions carefully. - **Units:** Include units in final answers for application problems.