Calculus Chapter: Functions and Limits

Definition: A function is a relation that assigns each input exactly one output.
Types of Functions:
- Linear Functions: f(x) = mx + b
- Quadratic Functions: f(x) = ax² + bx + c
- Polynomial Functions: f(x) = a_nx^n + ... + a_1x + a_0
- Rational Functions: Ratio of two polynomials.
- Exponential Functions: f(x) = a*b^x
- Trigonometric Functions: Sine, cosine, tangent, etc.
Domain and Range: The set of allowable inputs (domain) and the resulting outputs (range).
Composite Functions: (f◦g)(x) = f(g(x)).

Limit: The value that a function approaches as the input approaches a point.
Notation: lim (x→c) f(x) = L means as x approaches c, f(x) approaches L.
Types of Limits:
- One-sided limits: left-hand limit (LHL) and right-hand limit (RHL).
- Infinite limits and limits at infinity.
Continuity:
- A function is continuous at a point c if:
  1. f(c) is defined.
  2. lim (x→c) f(x) exists.
  3. lim (x→c) f(x) = f(c).
- Types of discontinuities: removable, jump, infinite.

Descriptive Statistics:
- Measures of Central Tendency: Mean, median, mode.
- Measures of Dispersion: Range, variance, standard deviation.
Inferential Statistics: Drawing conclusions from data samples about a population.
Probability: The likelihood of an event occurring, ranging from 0 to 1.
Distributions:
- Normal distribution: Bell-shaped curve, defined by mean and standard deviation.
- Binomial distribution: Discrete probability distribution of successes in a series of trials.
Hypothesis Testing: Method to test claims or hypotheses about a parameter in a population.

Definition: An integral represents the area under a curve or the accumulation of quantities.
Types of Integrals:
- Definite Integral: ∫[a,b] f(x) dx (calculates area between curve and x-axis from a to b).
- Indefinite Integral: ∫ f(x) dx (represents a family of functions and includes a constant of integration C).
Fundamental Theorem of Calculus: Connects differentiation and integration.
- If F is the antiderivative of f, then ∫[a,b] f(x) dx = F(b) - F(a).
Techniques of Integration: Substitution, integration by parts, partial fractions.

Definition: The process of finding the derivative of a function, representing the rate of change.
Notation: f'(x), df/dx, Df.
Rules of Differentiation:
- Power Rule: d/dx(x^n) = nx^(n-1).
- Product Rule: d/dx(fg) = f'g + fg'.
- Quotient Rule: d/dx(f/g) = (f'g - fg')/g².
- Chain Rule: d/dx(f(g(x))) = f'(g(x)) * g'(x).
Applications:
- Finding slopes of tangent lines.
- Determining local maxima and minima (critical points).
- Analyzing concavity and inflection points.

Functions relate each input to exactly one output.
Linear function form: f(x) = mx + b, where m is the slope and b is the y-intercept.
Quadratic function formula: f(x) = ax² + bx + c, representing a parabolic shape.
Polynomial functions take the form: f(x) = a_nx^n + ... + a_1x + a_0, including terms with different powers of x.
Rational functions are expressed as the ratio of two polynomials.
Exponential function formula: f(x) = a*b^x, characterized by a constant base raised to variable exponent.
Trigonometric functions include sine, cosine, and tangent, essential for analyzing periodic phenomena.
Domain represents all allowable inputs, while range indicates all potential outputs of a function.
Composite functions combine two functions, denoted as (f◦g)(x) = f(g(x)).

A limit describes the value a function approaches as the input nears a specific point.
Notation: lim (x→c) f(x) = L signifies that as x approaches c, f(x) approaches L.
Types of limits include one-sided limits (left-hand and right-hand) and infinite limits.
A function is continuous at point c if:
- f(c) is defined,
- lim (x→c) f(x) exists,
- lim (x→c) f(x) equals f(c).
Discontinuities are classified as removable, jump, or infinite.

Descriptive statistics summarize data through:
- Measures of central tendency: mean (average), median (middle value), mode (most frequent value).
- Measures of dispersion: range (difference between max and min), variance (average of squared deviations), and standard deviation (measure of variability).
Inferential statistics involve making predictions or generalizations about a population based on sample data.
Probability quantifies the likelihood of events, with values ranging between 0 (impossible) and 1 (certain).
Normal distribution is depicted as a bell-shaped curve, defined by its mean and standard deviation.
Binomial distribution assesses the number of successes in a fixed number of independent trials.
Hypothesis testing evaluates claims or hypotheses regarding population parameters.

An integral quantifies the area under a curve or accumulates quantities over an interval.
Definite integral form: ∫[a,b] f(x) dx calculates the area between the curve and the x-axis from point a to b.
Indefinite integrals, represented as ∫ f(x) dx, yield a family of functions and include a constant of integration, C.
Fundamental Theorem of Calculus establishes a connection between differentiation and integration, stating that if F is the antiderivative of f, then ∫[a,b] f(x) dx = F(b) - F(a).
Techniques for integration include substitution, integration by parts, and partial fractions.

Differentiation is used to find a function's derivative, indicating its rate of change.
Common notations for derivatives are f'(x), df/dx, and Df.
Differentiation rules include:
- Power Rule: d/dx(x^n) = nx^(n-1).
- Product Rule: d/dx(fg) = f'g + fg'.
- Quotient Rule: d/dx(f/g) = (f'g - fg')/g².
- Chain Rule: d/dx(f(g(x))) = f'(g(x)) * g'(x).
Applications of differentiation include calculating slopes of tangent lines, identifying local maxima and minima (critical points), and analyzing concavity and inflection points.