Calculus Study Notes: Trigonometry, Limits, Continuity, and Derivatives

Trigonometry

Trigonometry of Unit Circle

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. It is fundamental in defining trigonometric functions for all real numbers.

• Sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) functions are defined using the coordinates of points on the unit circle.

• For an angle $\theta$, the coordinates $(x, y)$ on the unit circle are $(\cos \theta, \sin \theta)$.

• Example: $\sin(\pi/4) = \frac{\sqrt{2}}{2}$, $\cos(5\pi/6) = -\frac{\sqrt{3}}{2}$, $\tan(7\pi/2)$ is undefined because tangent is not defined where cosine is zero.

Trigonometry of Right Triangles

Trigonometric functions can also be defined using right triangles, relating the angles to ratios of side lengths.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Quadrants: Sine is negative and cosine is positive in the fourth quadrant.

Limits

Limits: Interpretation & Application

A limit describes the value that a function approaches as the input approaches a certain point. Limits are foundational for calculus, especially in defining continuity and derivatives.

• Notation: $\lim_{x \to a} f(x)$

• Interpretation: The value $f(x)$ gets closer to as $x$ approaches $a$.

• Application: Used to analyze behavior near points of interest, such as discontinuities or boundaries.

Limits: Evaluation Techniques

There are several techniques for evaluating limits:

• Direct substitution: Plug in the value of $x$.

• Factoring: Factor expressions to cancel terms causing indeterminate forms.

• Rationalization: Multiply by conjugate to simplify expressions with roots.

• Special limits: Recognize standard limits, such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

• Indeterminate forms: $\frac{0}{0}$ or $\frac{\infty}{\infty}$ require further analysis.

• Example: $\lim_{x \to 2} \frac{(x^2 - 4)}{(x - 2)}$ is indeterminate ($\frac{0}{0}$), but factoring numerator gives $\lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$.

Continuity

A function is continuous at a point $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

• Discontinuity: Occurs when any of the above conditions fail.

• Example: Piecewise functions may be discontinuous at points where the definition changes.

Functions: Manipulation and Properties

Identifying & Solving Equations

Solving equations often involves manipulating algebraic expressions and applying properties of functions.

• Example: Solve $2\cos x = 1$ for $x$.

• Properties: Functions may be even, odd, periodic, or have specific domains and ranges.

Discontinuity in Functions

Discontinuities can be classified as:

• Jump discontinuity: The function "jumps" from one value to another.

• Infinite discontinuity: The function approaches infinity at a point.

• Removable discontinuity: A hole in the graph where the function is not defined, but the limit exists.

• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, $x=2$ is a removable discontinuity.

Logarithms: Manipulation

Properties of Logarithms

Logarithms are the inverses of exponential functions and have several useful properties:

• Product rule: $\log_b(xy) = \log_b x + \log_b y$

• Quotient rule: $\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$

• Power rule: $\log_b(x^k) = k \log_b x$

• Example: $\log_{81} = \frac{1}{4}$ because $81 = 3^4$ and $\log_3 81 = 4$.

Derivatives: Computation and Application

Definition of Derivative

The derivative of a function at a point measures the rate at which the function value changes as its input changes.

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Interpretation: The slope of the tangent line to the graph at $x$.

• Example: For $f(x) = x^2$, $f'(x) = 2x$.

Derivative Computation

To compute derivatives, use rules such as:

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

• Sum rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$

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

• Example: For $f(x) = x^2 - 3x$, $f'(x) = 2x - 3$.

Table: Assessment Topics Overview

The following table summarizes the main topics covered in the assessment:

Trigonometry	Limits	Functions	Logarithms	Derivatives
Unit circle	Interpretation & application	Meaning & properties	Manipulation	Computation
Right triangles	Evaluation techniques	Manipulation
Identifying & equations	Continuity

Examples and Applications

• Distance problems: Use trigonometry and the Pythagorean theorem to solve navigation problems.

• Discontinuity points: For piecewise functions, check where the definition changes or where denominators are zero.

• Continuous functions: Physical phenomena like the height of a bouncing ball or population growth are typically modeled by continuous functions.

• Derivative existence: If $f(x)$ is discontinuous at $x=a$, then $f'(a)$ does not exist.

Graphical Analysis

Limits and Continuity from Graphs

Analyzing graphs helps determine where functions are continuous or discontinuous, and to evaluate limits visually.

• Left-hand limit: $\lim_{x \to a^-} f(x)$

• Right-hand limit: $\lim_{x \to a^+} f(x)$

• Function value: $f(a)$

• Discontinuity: Points where the graph has jumps, holes, or vertical asymptotes.

Summary Table: Types of Discontinuity

Type	Description	Example
Removable	Hole in the graph; limit exists but function not defined	$f(x) = \frac{x^2 - 4}{x - 2}$ at $x=2$
Jump	Function jumps from one value to another	Piecewise function with different values at $x=a$
Infinite	Function approaches infinity	$f(x) = \frac{1}{x}$ at $x=0$

Additional info:

• Some context and examples were inferred to provide a complete study guide.

• Graphical analysis and table content were expanded for clarity.