Calculus I: Key Definitions, Theorems, and Reference Formulas

Definitions and Extrema

Local and Absolute Extrema

In calculus, understanding the behavior of functions at their maximum and minimum values is fundamental. These points are called extrema and are classified as local (relative) or absolute (global) maxima and minima.

• Local Maximum at c: f has a local maximum at c if there is an open interval (a, b) containing c such that $f(c) \geq f(x)$ for all $x$ in $(a, b)$.

• Local Minimum at c: f has a local minimum at c if there is an open interval (a, b) containing c such that $f(c) \leq f(x)$ for all $x$ in $(a, b)$.

• Absolute Maximum: f has an absolute maximum if there exists a point c in the domain such that $f(c) \geq f(x)$ for all $x$ in the domain.

• Absolute Minimum: f has an absolute minimum if there exists a point c in the domain such that $f(c) \leq f(x)$ for all $x$ in the domain.

The absolute maximum and minimum, if they exist, are called the extreme values of the function.

Critical Values

• A critical value of f is a value c in its domain such that either $f'(c) = 0$ or $f'(c)$ does not exist.

Increasing and Decreasing Functions

• Increasing: f is increasing if $a < b \implies f(a) < f(b)$ for all $a$ and $b$ in the domain.

• Decreasing: f is decreasing if $a < b \implies f(a) > f(b)$ for all $a$ and $b$ in the domain.

Concavity and Inflection Points

Concavity

• Concave Up: f is concave up on an interval if $f''(x) > 0$ for all $x$ in the interval.

• Concave Down: f is concave down on an interval if $f''(x) < 0$ for all $x$ in the interval.

Inflection Points

• A point c is an inflection point if the concavity of f changes at c.

• If $f''(c) = 0$ and the sign of $f''$ changes at c, then c is an inflection point.

• If $f''(c) = 0$ and the concavity changes, but the tangent is horizontal, it is called a saddle point.

Theorems and Tests

First Derivative Test

• If $f'(x)$ changes from positive to negative at $x = c$, then $f$ has a local maximum at $c$.

• If $f'(x)$ changes from negative to positive at $x = c$, then $f$ has a local minimum at $c$.

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 $c$.

• If $f''(c) = 0$, the test is inconclusive.

Extreme Value Theorem

• If $f$ is continuous on $[a, b]$, then $f$ attains both a maximum and minimum value at either a critical point or at the endpoints.

Fundamental Theorem of Calculus (FTC)

• Suppose $f$ is continuous on $[a, b]$.

• Part 1: $F(x) = \int_a^x f(t)dt \implies F'(x) = f(x)$.

• Part 2: $\int_a^b f(x)dx = F(b) - F(a)$, where $F$ is any anti-derivative of $f$.

Reference Formulas

Derivative Rules

• $\frac{d}{dx}c = 0$

• $\frac{d}{dx}x^n = nx^{n-1}$

• $\frac{d}{dx}(f+g) = f' + g'$

• $\frac{d}{dx}(cf) = c f'$

• $\frac{d}{dx}(fg) = f'g + fg'$

• $\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}$

• $\frac{d}{dx}e^x = e^x$

• $\frac{d}{dx}a^x = a^x \ln a$

• $\frac{d}{dx}\ln x = \frac{1}{x}$

• $\frac{d}{dx}\log_a x = \frac{1}{x \ln a}$

Trigonometric Derivatives

• $\frac{d}{dx}\sin x = \cos x$

• $\frac{d}{dx}\cos x = -\sin x$

• $\frac{d}{dx}\tan x = \sec^2 x$

• $\frac{d}{dx}\cot x = -\csc^2 x$

• $\frac{d}{dx}\sec x = \sec x \tan x$

• $\frac{d}{dx}\csc x = -\csc x \cot x$

Inverse Trigonometric Derivatives

• $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$

Logarithmic Differentiation

• $\frac{d}{dx}f(x) = f(x) \frac{d}{dx}\ln f(x)$

Logarithm Rules

• $\ln(e^x) = x$

• $\ln(1) = 0$

• $\ln(ab) = \ln a + \ln b$

• $\ln\left(\frac{a}{b}\right) = \ln a - \ln b$

• $\ln(a^r) = r \ln a$

Exponent Rules

• $a^0 = 1$

• $a^m a^n = a^{m+n}$

• $\frac{a^m}{a^n} = a^{m-n}$

• $(a^m)^n = a^{mn}$

• $(ab)^n = a^n b^n$

Algebraic Identities

• $(a + b)^2 = a^2 + 2ab + b^2$

• $(a - b)^2 = a^2 - 2ab + b^2$

Anti-derivative Rules

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)

• $\int \frac{1}{x} dx = \ln|x| + C$

• $\int e^x dx = e^x + C$

• $\int a^x dx = \frac{a^x}{\ln a} + C$

• $\int \sin x dx = -\cos x + C$

• $\int \cos x dx = \sin x + C$

• $\int \sec^2 x dx = \tan x + C$

• $\int \csc^2 x dx = -\cot x + C$

• $\int \sec x \tan x dx = \sec x + C$

• $\int \csc x \cot x dx = -\csc x + C$

• $\int \frac{1}{1+x^2} dx = \arctan x + C$

• $\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C$

Properties of Integrals

• Linearity: $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$

• Definite Integral as Area: $\int_a^b f(x) dx$ gives the net area between $f(x)$ and the $x$-axis from $x=a$ to $x=b$.

• Sum of Areas: $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

Example: Area Under |x|

• $|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

• $\int_{-1}^2 |x| dx = \int_{-1}^0 -x dx + \int_0^2 x dx = \frac{1}{2} + 2 = 2.5$

Trigonometric Reference

Graphs and Properties

• sin(x): Domain: $(-\infty, \infty)$, Range: $[-1, 1]$, Periodicity: $2\pi$

• cos(x): Domain: $(-\infty, \infty)$, Range: $[-1, 1]$, Periodicity: $2\pi$

• tan(x): Domain: $x \neq \frac{\pi}{2} + k\pi$, Range: $(-\infty, \infty)$, Periodicity: $\pi$

• sec(x): Domain: $x \neq \frac{\pi}{2} + k\pi$, Range: $(-\infty, -1] \cup [1, \infty)$, Periodicity: $2\pi$

Trigonometric Identities

• $\sin^2(x) + \cos^2(x) = 1$

• $\tan(x) = \frac{\sin(x)}{\cos(x)}$

• $\sec(x) = \frac{1}{\cos(x)}$

• $\csc(x) = \frac{1}{\sin(x)}$

Unit Circle and Special Angles

• Values of $\sin(\theta)$ and $\cos(\theta)$ for $\theta = 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$, etc.

• Periodicity: $\sin(\theta + 2\pi n) = \sin(\theta)$, $\cos(\theta + 2\pi n) = \cos(\theta)$ for $n \in \mathbb{Z}$

Exponential and Logarithmic Functions

Properties of $e^x$ and $\ln x$

• Exponential: $e^x$ is defined for all real $x$, $e^0 = 1$, $\lim_{x \to -\infty} e^x = 0$, $\lim_{x \to \infty} e^x = \infty$

• Logarithm: $\ln x$ is defined for $x > 0$, $\ln(1) = 0$, $\lim_{x \to 0^+} \ln x = -\infty$, $\lim_{x \to \infty} \ln x = \infty$

Vertical Asymptotes of Rational Functions

• For $f(x) = \frac{1}{x-a}$, $\lim_{x \to a^-} f(x) = -\infty$, $\lim_{x \to a^+} f(x) = \infty$

• For $f(x) = \frac{1}{(x-a)^2}$, $\lim_{x \to a^-} f(x) = \infty$, $\lim_{x \to a^+} f(x) = \infty$

Additional Algebraic and Trigonometric Facts

• Square root properties: $\sqrt{a^2} = |a|$

• Examples of evaluating roots, exponents, and trigonometric values at special points.

Summary Table: Trigonometric Functions

Function	Domain	Range	Periodicity	Key Properties
$\sin x$	$(-\infty, \infty)$	$[-1, 1]$	$2\pi$	Odd, $\sin(0) = 0$
$\cos x$	$(-\infty, \infty)$	$[-1, 1]$	$2\pi$	Even, $\cos(0) = 1$
$\tan x$	$x \neq \frac{\pi}{2} + k\pi$	$(-\infty, \infty)$	$\pi$	Odd, vertical asymptotes at $x = \frac{\pi}{2} + k\pi$
$\sec x$	$x \neq \frac{\pi}{2} + k\pi$	$(-\infty, -1] \cup [1, \infty)$	$2\pi$	Even, vertical asymptotes at $x = \frac{\pi}{2} + k\pi$

Additional info: Some explanations and context have been expanded for clarity and completeness, including the summary table and explicit statements of theorems and properties.