Calculus I Cheatsheet

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Limits Definition: $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 One-Sided Limits: Left-hand limit: $\lim_{x \to a^-} f(x)$ Right-hand limit: $\lim_{x \to a^+} f(x)$ $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$. Limit Properties: (Assume $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist) Sum: $\lim (f(x) + g(x)) = \lim f(x) + \lim g(x)$ Difference: $\lim (f(x) - g(x)) = \lim f(x) - \lim g(x)$ Constant Multiple: $\lim (c \cdot f(x)) = c \cdot \lim f(x)$ Product: $\lim (f(x) \cdot g(x)) = \lim f(x) \cdot \lim g(x)$ Quotient: $\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$, provided $\lim g(x) \neq 0$ Power: $\lim (f(x))^n = (\lim f(x))^n$ L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists. Continuity A function $f$ is continuous at a number $a$ if $\lim_{x \to a} f(x) = f(a)$. This implies three conditions: $f(a)$ is defined. $\lim_{x \to a} f(x)$ exists. $\lim_{x \to a} f(x) = f(a)$. Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists a number $c$ in $(a, b)$ such that $f(c) = N$. Derivatives Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ Alternative Form: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ Interpretation: Slope of the tangent line to $y=f(x)$ at $x$. Instantaneous rate of change of $y$ with respect to $x$. Differentiation Rules Constant Rule: $\frac{d}{dx}(c) = 0$ Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Constant Multiple Rule: $\frac{d}{dx}(cf(x)) = c f'(x)$ Sum/Difference Rule: $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$ Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$ 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: $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ Derivatives of Common Functions $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(a^x) = a^x \ln a$ $\frac{d}{dx}(\ln x) = \frac{1}{x}$ $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$ $\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(\tan x) = \sec^2 x$ $\frac{d}{dx}(\cot x) = -\csc^2 x$ $\frac{d}{dx}(\sec x) = \sec x \tan x$ $\frac{d}{dx}(\csc x) = -\csc x \cot x$ $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$ Applications of Derivatives Related Rates: Use implicit differentiation to find the rate of change of one variable with respect to time when other variables and their rates are known. Optimization: Find maximum or minimum values of a function by setting $f'(x) = 0$ and checking critical points. Critical Numbers: Values $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ is undefined. Increasing/Decreasing Test: If $f'(x) > 0$ on an interval, $f$ is increasing. If $f'(x) First Derivative Test: If $f'$ changes from $+$ to $-$ at $c$, $f(c)$ is a local maximum. If $f'$ changes from $-$ to $+$ at $c$, $f(c)$ is a local minimum. Concavity Test: If $f''(x) > 0$ on an interval, $f$ is concave up. If $f''(x) Inflection Point: A point where the concavity changes. $f''(x)=0$ or $f''(x)$ is undefined. Second Derivative Test: If $f'(c) = 0$: If $f''(c) > 0$, $f(c)$ is a local minimum. If $f''(c) If $f''(c) = 0$, the test is inconclusive. Mean Value Theorem (MVT): If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a number $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. Integrals Antiderivative: A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$. Indefinite Integral: $\int f(x) dx = F(x) + C$, where $C$ is the constant of integration. Definite Integral: $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$ (Riemann Sum) Fundamental Theorem of Calculus (FTC) Part 1: If $f$ is continuous on $[a, b]$, then the function $g(x) = \int_a^x f(t) dt$ has derivative $g'(x) = f(x)$. Fundamental Theorem of Calculus (FTC) Part 2: If $f$ is continuous on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$. Integration Techniques Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ Log Rule: $\int \frac{1}{x} dx = \ln|x| + C$ Exponential Rule: $\int e^x dx = e^x + C$ Trigonometric Integrals: $\int \cos x dx = \sin x + C$ $\int \sin x dx = -\cos x + C$ $\int \sec^2 x dx = \tan x + C$ $\int \csc^2 x dx = -\cot x + C$ $\int \sec x \tan x dx = \sec x + C$ $\int \csc x \cot x dx = -\csc x + C$ Substitution Rule (u-substitution): $\int f(g(x))g'(x) dx = \int f(u) du$, where $u = g(x)$ and $du = g'(x) dx$. Applications of Integrals Area under a curve: If $f(x) \ge 0$ on $[a, b]$, Area $= \int_a^b f(x) dx$. Area between curves: Area $= \int_a^b [f(x) - g(x)] dx$, where $f(x) \ge g(x)$. Average Value of a Function: $f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$. Distance and Displacement: Displacement: $\int_{t_1}^{t_2} v(t) dt$ Total Distance: $\int_{t_1}^{t_2} |v(t)| dt$