Calculus I-II Review - Functions, Derivatives, Integrals, and Trigonometric Identities

Calculus I/II Review

Functions and Their Properties

Understanding different types of functions is fundamental in calculus. Here are some key types:

• Linear Function: A function of the form $y = mx + b$, where m is the slope and b is the y-intercept.

• Quadratic Function: A function of the form $y = ax^2 + bx + c$, where a, b, and c are constants.

• Exponential Function: A function of the form $y = e^x$, where $e$ is Euler's number (approximately 2.718).

• Natural Logarithm Function: A function of the form $y = \ln x$, the inverse of the exponential function.

Example: The natural exponential function $y = e^x$ crosses the y-axis at (0,1) and has a slope of 1 at that point.

Limits and Continuity

Limits describe the behavior of a function as the input approaches a certain value.

• Notation: $\lim_{x \to a} f(x)$ denotes the limit of $f(x)$ as $x$ approaches $a$.

• Continuity: A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

Example: For $f(x) = 2x^2 + 4x + 2$, find $\lim_{x \to 3} f(x)$.

Differentiation

Differentiation is the process of finding the rate at which a function changes.

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Exponential Rule: $\frac{d}{dx} e^x = e^x$

• Logarithmic Rule: $\frac{d}{dx} \ln x = \frac{1}{x}$

• 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: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Example: $\frac{d}{dx} (x^2 + 4x + 2) = 2x + 4$

Derivatives of Trigonometric Functions

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} \tan x = \sec^2 x$

Derivatives of Inverse Trigonometric Functions

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

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

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

Integration

Integration is the reverse process of differentiation and is used to find areas, volumes, and accumulated quantities.

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

• Constant Multiple: $\int c f(x) dx = c \int f(x) dx$

• Definite Integral: $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

• Substitution: $\int f(g(x))g'(x) dx = \int f(u) du$ where $u = g(x)$.

Common Integral Formulas

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

• $\int \frac{1}{x} dx = \ln|x| + C$

• $\int e^x dx = e^x + C$

• $\int \sin x dx = -\cos x + C$

• $\int \cos x dx = \sin x + C$

• $\int \sec^2 x dx = \tan x + C$

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

Integration by Substitution

Used when an integral contains a function and its derivative. Set $u = g(x)$, then $du = g'(x) dx$.

Integration Using Trigonometric Identities

Trigonometric identities can simplify integrals involving trigonometric functions.

• Double Angle Formula: $\sin(2x) = 2\sin x \cos x$

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

Trigonometric Functions and Identities

Angle (θ)	0	π/6	π/4	π/3	π/2
sin(θ)	0	1/2	√2/2	√3/2	1
cos(θ)	1	√3/2	√2/2	1/2	0
tan(θ)	0	1/√3	1	√3	undefined

Key Trigonometric Identities

• $\sin^2 x + \cos^2 x = 1$

• $\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

• $\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

• $\sin(2x) = 2\sin x \cos x$

• $\cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$

Solving Trigonometric Equations

Trigonometric equations can often be solved using identities and algebraic manipulation.

• Example: Solve $3\cos x + 5 = 8$ for $x$.

• Example: Solve $\sin x + 1 = 3\sin(2x)$ for $x$.

Applications: Derivatives and Integrals of Trigonometric Functions

Examples

• Find $\frac{d}{dt} (2\sin t \cos t)$

• Evaluate $\int_0^{\pi/3} \sin(x/2) dx$ using substitution.

• Evaluate $\int_0^{\pi/3} 7\sin(x)\cos(x) dx$ using substitution.

• Evaluate $\int \cos^2 x dx$ using double angle formula.

• Evaluate $\int 4\sin^2(x)\cos^2(x) dx$ using double angle formula.

• Evaluate $\int \sqrt{2 + 2\cos x} dx$ using double angle formula.

Summary Table: Basic Derivative and Integral Rules

Function	Derivative	Integral
$x^n$	$n x^{n-1}$	$\frac{x^{n+1}}{n+1} + C$
$e^x$	$e^x$	$e^x + C$
$\ln x$	$\frac{1}{x}$	$x \ln x - x + C$
$\sin x$	$\cos x$	$-\cos x + C$
$\cos x$	$-\sin x$	$\sin x + C$
$\tan x$	$\sec^2 x$	$-\ln|\cos x| + C$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and applications are based on standard Calculus I/II/III curricula.