Formal Definition:
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Calculus Cheatsheet
A comprehensive calculus cheat sheet covering essential concepts, formulas, and techniques. This cheat sheet is designed to serve as a quick reference guide for students and professionals alike, providing a concise overview of calculus principles and methods.
Limits and Continuity
Limit Definitions
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For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. |
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Intuitive Definition: |
As x approaches a, f(x) approaches L. |
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One-Sided Limits: |
\lim_{x \to a^-} f(x) and \lim_{x \to a^+} f(x) |
Limit Laws
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\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) |
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\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x) |
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\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) |
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\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, if \lim_{x \to a} g(x) \neq 0 |
Continuity
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Definition: |
A function f(x) is continuous at x = a if \lim_{x \to a} f(x) = f(a). This means that f(a) exists, the limit exists, and they are equal. |
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Types of Discontinuities: |
Removable, Jump, Infinite |
Derivatives
Basic Differentiation Rules
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Power Rule: |
\frac{d}{dx}(x^n) = nx^{n-1} |
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Constant Rule: |
\frac{d}{dx}(c) = 0 |
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Constant Multiple Rule: |
\frac{d}{dx}(cf(x)) = c \frac{d}{dx}(f(x)) |
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Sum/Difference Rule: |
\frac{d}{dx}(f(x) \pm g(x)) = \frac{d}{dx}(f(x)) \pm \frac{d}{dx}(g(x)) |
Product and Quotient Rules
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Product Rule: |
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) |
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Quotient Rule: |
\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} |
Chain Rule
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Chain Rule: |
\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) |
Derivatives of Trig Functions
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\frac{d}{dx}(\sin x) |
\cos x |
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\frac{d}{dx}(\cos x) |
-\sin x |
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\frac{d}{dx}(\tan x) |
\sec^2 x |
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\frac{d}{dx}(\csc x) |
-\csc x \cot x |
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\frac{d}{dx}(\sec x) |
\sec x \tan x |
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\frac{d}{dx}(\cot x) |
-\csc^2 x |
Integrals
Basic Integration Rules
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Power Rule: |
\int x^n dx = \frac{x^{n+1}}{n+1} + C, for n \neq -1 |
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Constant Rule: |
\int c dx = cx + C |
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Constant Multiple Rule: |
\int cf(x) dx = c \int f(x) dx |
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Sum/Difference Rule: |
\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx |
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\int \frac{1}{x} dx |
ln|x| + C |
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\int e^x dx |
e^x + C |
Integration by Parts
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Formula: |
\int u dv = uv - \int v du |
Trigonometric Integrals
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\int \sin x dx |
-\cos x + C |
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\int \cos x dx |
\sin x + C |
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\int \sec^2 x dx |
\tan x + C |
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\int \csc^2 x dx |
-\cot x + C |
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\int \sec x \tan x dx |
\sec x + C |
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\int \csc x \cot x dx |
-\csc x + C |
Trigonometric Substitution
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Use when you have integrals involving \sqrt{a^2 - x^2}, \sqrt{a^2 + x^2}, or \sqrt{x^2 - a^2}. Substitute x = a\sin \theta, x = a\tan \theta, or x = a\sec \theta respectively. |
Applications of Derivatives
Related Rates
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Identify the variables, find the equation relating them, differentiate with respect to time, and solve for the desired rate. |
Optimization
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Find critical points by setting the first derivative to zero or undefined, then use the first or second derivative test to determine local maxima and minima. Check endpoints for absolute extrema. |
L'Hôpital's Rule
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When to Use: |
For limits of the form \frac{0}{0} or \frac{\infty}{\infty}. |
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Rule: |
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} |
Mean Value Theorem
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Theorem: |
If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = \frac{f(b) - f(a)}{b - a} |