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# Calculus ## 1. Limits ### 1.1 Definition The limit of a function $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$ if $f(x)$ gets closer and closer to $L$ as $x$ gets closer and closer to $c$. ### 1.2 Properties Suppose that $b$ and $c$ are real numbers and that $\lim_{x...

# Calculus ## 1. Limits ### 1.1 Definition The limit of a function $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$ if $f(x)$ gets closer and closer to $L$ as $x$ gets closer and closer to $c$. ### 1.2 Properties Suppose that $b$ and $c$ are real numbers and that $\lim_{x \to c} f(x)$ and $\lim_{x \to c} g(x)$ exist. Then, 1. $\lim_{x \to c} b = b$ 2. $\lim_{x \to c} x = c$ 3. $\lim_{x \to c} [bf(x)] = b \lim_{x \to c} f(x)$ 4. $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$ 5. $\lim_{x \to c} [f(x)g(x)] = [\lim_{x \to c} f(x)][\lim_{x \to c} g(x)]$ 6. $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, if $\lim_{x \to c} g(x) \ne 0$ 7. $\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$, where $n$ is a positive integer 8. $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$, where $n$ is a positive integer ### 1.3 Techniques for Evaluating Limits 1. **Direct Substitution:** - If $f(x)$ is a polynomial or a rational function, then $\lim_{x \to c} f(x) = f(c)$. 2. **Factoring:** - If direct substitution results in an indeterminate form like $\frac{0}{0}$, try factoring the numerator and/or the denominator. 3. **Rationalizing:** - If the function involves radicals, rationalizing the numerator or denominator might help. 4. **Common denominator** - Use a common denominator to simplify the equation ### 1.4 Squeeze Theorem If $l(x) \le f(x) \le u(x)$ for all $x$ in an open interval containing $c$, except possibly at $c$ itself, and if $\lim_{x \to c} l(x) = L = \lim_{x \to c} u(x)$ then $\lim_{x \to c} f(x)$ exists and is equal to $L$. ### 1.5 Limits Involving Infinity 1. $\lim_{x \to \infty} \frac{1}{x^p} = 0$ for $p > 0$ 2. $\lim_{x \to -\infty} \frac{1}{x^p} = 0$ for $p > 0$ ### 1.6 Indeterminate Forms and L'Hôpital's Rule If $\lim_{x \to c} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$ provided that the limit on the right exists. ## 2. Continuity ### 2.1 Definition A function $f(x)$ is continuous at $x = c$ if the following three conditions are met: 1. $f(c)$ is defined 2. $\lim_{x \to c} f(x)$ exists 3. $\lim_{x \to c} f(x) = f(c)$ ### 2.2 Types of Discontinuities 1. **Removable Discontinuity:** The limit exists, but it does not equal the function's value at that point. 2. **Jump Discontinuity:** The limit from the left and the limit from the right exist, but they are not equal. 3. **Infinite Discontinuity:** The function approaches infinity at that point. ### 2.3 Properties of Continuous Functions If $f$ and $g$ are continuous at $c$, then the following functions are also continuous at $c$: 1. $f \pm g$ 2. $fg$ 3. $\frac{f}{g}$, provided $g(c) \ne 0$ ### 2.4 Intermediate Value Theorem If $f$ is a continuous function on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists a number $c$ in $[a, b]$ such that $f(c) = k$. ## 3. Derivatives ### 3.1 Definition The derivative of $f$ at $x$ is defined as $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ ### 3.2 Basic Differentiation Rules 1. **Constant Rule:** $\frac{d}{dx}(c) = 0$ 2. **Power Rule:** $\frac{d}{dx}(x^n) = nx^{n-1}$ 3. **Constant Multiple Rule:** $\frac{d}{dx}[cf(x)] = cf'(x)$ 4. **Sum/Difference Rule:** $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ 5. **Product Rule:** $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ 6. **Quotient Rule:** $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$ 7. **Chain Rule:** $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ ### 3.3 Derivatives of Trigonometric Functions 1. $\frac{d}{dx}(\sin x) = \cos x$ 2. $\frac{d}{dx}(\cos x) = -\sin x$ 3. $\frac{d}{dx}(\tan x) = \sec^2 x$ 4. $\frac{d}{dx}(\csc x) = -\csc x \cot x$ 5. $\frac{d}{dx}(\sec x) = \sec x \tan x$ 6. $\frac{d}{dx}(\cot x) = -\csc^2 x$ ### 3.4 Derivatives of Exponential and Logarithmic Functions 1. $\frac{d}{dx}(e^x) = e^x$ 2. $\frac{d}{dx}(a^x) = a^x \ln a$ 3. $\frac{d}{dx}(\ln x) = \frac{1}{x}$ 4. $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$ ### 3.5 Implicit Differentiation 1. Differentiate both sides of the equation with respect to $x$. 2. Use the chain rule when differentiating terms involving $y$. 3. Solve for $\frac{dy}{dx}$. ### 3.6 Higher-Order Derivatives The second derivative is the derivative of the first derivative: $f''(x) = \frac{d^2}{dx^2}f(x)$. The $n$th derivative is: $f^{(n)}(x) = \frac{d^n}{dx^n}f(x)$. ## 4. Applications of Derivatives ### 4.1 Related Rates 1. Identify all given quantities and quantities to be determined. 2. Write an equation relating the variables. 3. Differentiate both sides with respect to time $t$. 4. Substitute all known values and solve for the required rate. ### 4.2 Critical Points and Extrema 1. Find the critical points by setting $f'(x) = 0$ or finding where $f'(x)$ is undefined. 2. Use the first derivative test or the second derivative test to determine if each critical point is a local maximum, a local minimum, or neither. ### 4.3 First Derivative Test If $f'(x)$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$. If $f'(x)$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$. ### 4.4 Second Derivative Test If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$. If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$. ### 4.5 Optimization Problems 1. Identify the quantity to be maximized or minimized. 2. Write the objective function. 3. Find the critical points and test them to find the absolute maximum or minimum. ### 4.6 Mean Value Theorem If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$. ## 5. Integrals ### 5.1 Definition The indefinite integral of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$. We write $\int f(x) \, dx = F(x) + C$ where $C$ is the constant of integration. ### 5.2 Basic Integration Rules 1. **Power Rule:** $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, for $n \ne -1$ 2. **Constant Multiple Rule:** $\int cf(x) \, dx = c \int f(x) \, dx$ 3. **Sum/Difference Rule:** $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$ ### 5.3 Integrals of Trigonometric Functions 1. $\int \sin x \, dx = -\cos x + C$ 2. $\int \cos x \, dx = \sin x + C$ 3. $\int \sec^2 x \, dx = \tan x + C$ 4. $\int \csc^2 x \, dx = -\cot x + C$ 5. $\int \sec x \tan x \, dx = \sec x + C$ 6. $\int \csc x \cot x \, dx = -\csc x + C$ ### 5.4 Integrals of Exponential and Logarithmic Functions 1. $\int e^x \, dx = e^x + C$ 2. $\int a^x \, dx = \frac{a^x}{\ln a} + C$ 3. $\int \frac{1}{x} \, dx = \ln |x| + C$ ### 5.5 Substitution Rule $\int f(g(x))g'(x) \, dx = \int f(u) \, du$ where $u = g(x)$ and $du = g'(x) \, dx$ ### 5.6 Integration by Parts $\int u \, dv = uv - \int v \, du$ ### 5.7 Definite Integrals $\int_a^b f(x) \, dx = F(b) - F(a)$ where $F'(x) = f(x)$. ### 5.8 Fundamental Theorem of Calculus 1. If $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$. 2. $\int_a^b f(x) \, dx = F(b) - F(a)$ where $F'(x) = f(x)$. ## 6. Applications of Integrals ### 6.1 Area Between Curves The area of the region bounded by $y = f(x)$, $y = g(x)$, $x = a$, and $x = b$ is $A = \int_a^b |f(x) - g(x)| \, dx$ ### 6.2 Volume 1. **Disk Method:** - $V = \pi \int_a^b [f(x)]^2 \, dx$ (rotation about the x-axis) 2. **Washer Method:** - $V = \pi \int_a^b ([f(x)]^2 - [g(x)]^2) \, dx$ (rotation about the x-axis) 3. **Cylindrical Shells Method:** - $V = 2\pi \int_a^b x f(x) \, dx$ (rotation about the y-axis) ### 6.3 Average Value of a Function The average value of $f$ on $[a, b]$ is $f_{avg} = \frac{1}{b - a} \int_a^b f(x) \, dx$