Calculus Formulas and Theorems

A comprehensive reference of key concepts in calculus

Limits

\[\begin{aligned} \lim_{x \to a} [f(x) \pm g(x)] &= L \pm M \\ \lim_{x \to a} [f(x) \cdot g(x)] &= L \cdot M \\ \lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] &= \frac{L}{M} (M \neq 0) \\ \lim_{x \to 0} \frac{\sin x}{x} &= 1 \end{aligned}\]

Squeeze Theorem: If \(g(x) \leq f(x) \leq h(x)\) near \(a\) and both approach \(L\), then \(\lim_{x \to a} f(x) = L\).

Differentiation

\[\begin{aligned} \frac{d}{dx}[x^n] &= nx^{n-1} \\ \frac{d}{dx}[e^x] &= e^x \\ \frac{d}{dx}[\ln x] &= \frac{1}{x} \\ \frac{d}{dx}[\sin x] &= \cos x \\ \frac{d}{dx}[\cos x] &= -\sin x \\ \frac{d}{dx}[\tan x] &= \sec^2 x \end{aligned}\]

\[\begin{aligned} \text{Product Rule: } &\frac{d}{dx}[f \cdot g] = f'g + fg' \\ \text{Quotient Rule: } &\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2} \\ \text{Chain Rule: } &\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \end{aligned}\]

Integration

\[\begin{aligned} \int x^n \, dx &= \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \\ \int \frac{1}{x} \, dx &= \ln|x| + C \\ \int e^x \, dx &= e^x + C \\ \int \sin x \, dx &= -\cos x + C \\ \int \cos x \, dx &= \sin x + C \\ \int \sec^2 x \, dx &= \tan x + C \end{aligned}\]

Techniques

\[\begin{aligned} \text{U-Substitution: } &\int f(g(x))g'(x) \, dx = \int f(u) \, du \\ \text{Integration by Parts: } &\int u \, dv = uv - \int v \, du \\ \text{Partial Fractions: } &\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} \end{aligned}\]

Fundamental Theorem of Calculus

Part 1: If \(f\) is continuous, then \(\frac{d}{dx}\left[\int_a^x f(t) \, dt\right] = f(x)\).

Part 2: If \(f\) is continuous on \([a,b]\), then \(\int_a^b f(x) \, dx = F(b) - F(a)\) where \(F' = f\).

Important Theorems

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

L'Hpital's Rule: For \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) forms, \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\) (if the limit exists).

Applications

\[\begin{aligned} \text{Area: } &A = \int_a^b [f(x) - g(x)] \, dx \\ \text{Volume (Disk): } &V = \pi \int_a^b [f(x)]^2 \, dx \\ \text{Volume (Shell): } &V = 2\pi \int_a^b x f(x) \, dx \\ \text{Arc Length: } &L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx \end{aligned}\]

\[\begin{aligned} \text{Second Derivative Test: } &f''(c) > 0 \Rightarrow \text{local minimum} \\ &f''(c) < 0 \Rightarrow \text{local maximum} \end{aligned}\]

Series

\[\begin{aligned} \text{Geometric: } &\sum_{n=0}^{\infty} ar^n \text{ converges to } \frac{a}{1-r} \text{ if } |r| < 1 \\ \text{p-Series: } &\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1 \\ \text{Ratio Test: } &\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L \\ &L < 1 \text{ (convergent)}, L > 1 \text{ (divergent)} \end{aligned}\]

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

Multivariable Calculus

\[\begin{aligned} \frac{\partial f}{\partial x} &= \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h} \\ \nabla f(x,y) &= \langle f_x(x,y), f_y(x,y) \rangle \\ \iint_R f(x,y) \, dA &= \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx \end{aligned}\]

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Reference Files For Calculus Formulas And Theorems

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