Calculus

Definition of a Function

Definition of a function. Evaluation of a function. Representing a function, e.g., verbally, numerical, algebraically, graphically, piecewise, recursive, parametric, integral, and series representation. The Vertical Line Test. Read more... 2806 words,🕔14 minutes read, May 14, 2022.

Graphing functions

A comprehensive guide to graphing mathematical functions covering systematic procedures from domain analysis to plotting key features. Detailed coverage of linear, quadratic, rational, exponential, logarithmic, trigonometric, and piecewise functions with emphasis on intercepts, asymptotes, symmetry, and continuity. Read more... 1862 words,🕔9 minutes read, May 14, 2022.

Intercepts of Graphs

A detailed guide to finding and interpreting x-intercepts and y-intercepts. Covers linear, quadratic, exponential, and trigonometric functions, as well as circles. Explains how to find intercepts from equations and graphs, and how to use them for graphing lines. Read more... 2033 words,🕔10 minutes read, May 14, 2022.

Monotonic Functions and Local Extrema

A comprehensive guide to monotonic functions (increasing, decreasing, constant) and their relationship with derivatives. Explores strict monotonicity conditions, critical points, and local extrema with practical methods for finding maxima and minima. Includes detailed examples like e^x, sqrt(x), absolute value, and piecewise functions. Read more... 2315 words,🕔11 minutes read, May 14, 2022.

Local Extrema. First and Second Derivative Tests

A detailed guide to finding local maxima and minima of functions. Covers the First Derivative Test, including its proof using the Mean Value Theorem, and the Second Derivative Test for classifying critical points. Includes worked examples with polynomial and trigonometric functions. Read more... 2101 words,🕔10 minutes read, May 14, 2022.

Asymptotes and End Behavior

A detailed guide to understanding asymptotes and the end behavior of functions. Covers vertical, horizontal, and oblique asymptotes, with a focus on rational, exponential, and logarithmic functions. Includes solved examples and graphical analysis. Read more... 3264 words,🕔16 minutes read, Jul 14, 2022.

Analyzing and Graphing Rational Functions

Provides detailed, step-by-step examples for analyzing and graphing rational functions. Covers finding domain, range, intercepts, asymptotes (including oblique), and using first and second derivative tests to determine monotonicity and concavity for accurate sketching Read more... 2568 words,🕔13 minutes read, Jul 14, 2022.

Increasing and Decreasing Functions

A comprehensive guide to understanding monotonic functions. Covers the intuitive and formal definitions of increasing and decreasing functions, explains the derivative test for monotonicity, and details how to find increasing and decreasing intervals by analyzing critical points and the sign of the derivative. Includes solved examples with linear, quadratic, cubic, exponential, and rational functions. Read more... 3854 words,🕔19 minutes read, Jul 14, 2022.

Algebra of functions

Comprehensive guide to algebra of functions covering arithmetic operations (sum, difference, product, quotient), composition, differentiation rules, limits, horizontal and vertical asymptotes, and the test-point method for sign analysis with detailed examples. Read more... 2988 words,🕔15 minutes read, May 14, 2022.

Analyzing Concavity with the Second Derivative Test

A comprehensive guide to understanding concavity and inflection points. Explains the geometric and analytic definitions, connects them to the second derivative, and provides detailed examples for quadratic, cubic, rational, trigonometric, and logarithmic functions to illustrate how to determine intervals of concavity and locate inflection points. Read more... 2862 words,🕔14 minutes read, May 14, 2022.

Exponential Functions. Growth, Decay, and Properties

Comprehensive exploration of exponential functions including definition, properties, derivatives, Taylor series, and applications. Covers exponential growth vs decay, the natural exponential e^x, and techniques for solving exponential equations. Read more... 2880 words,🕔14 minutes read, May 14, 2022.

Limits in Calculus

Comprehensive introduction to limits covering intuitive and formal epsilon-delta definitions, one-sided limits, evaluating limits from graphs, and techniques for solving limits including factoring and rationalizing. Read more... 3046 words,🕔15 minutes read, May 14, 2022.

Epsilon-Delta Proofs for Limit

Comprehensive collection of solved limit problems using rigorous epsilon-delta proofs. Detailed step-by-step solutions for polynomial, rational, radical, and trigonometric limits with careful epsilon and delta selection strategies. Read more... 1107 words,🕔6 minutes read, May 14, 2022.

Inverse functions

Definition, graphical interpretation, and the horizontal line test. Learn how to find inverses of linear, rational, exponential, and trigonometric functions. Calculate and understand the derivative of inverse functions with solved examples. Read more... 3027 words,🕔15 minutes read, May 14, 2022.

Logarithm functions

Explores the logarithm function as the inverse of exponentiation. Covers logarithmic properties (product, quotient, power), domain, range, vertical asymptotes, and monotonicity. Includes derivatives, integrals, Taylor series expansion, and solved examples of logarithmic equations. Read more... 3330 words,🕔16 minutes read, May 14, 2022.

Symmetry of Graphs

Comprehensive guide to symmetry of graphs covering even and odd functions, symmetry about axes and origin, vertical line symmetry, point symmetry, and algebraic tests. Includes even-odd decomposition formula and solved examples for various function types. Read more... 3471 words,🕔17 minutes read, May 14, 2022.

Types of functions

Comprehensive classification of mathematical functions covering linear, quadratic, polynomial, exponential, logarithmic, trigonometric, rational, piecewise, hyperbolic, and special functions. Discusses classification criteria including mapping, symmetry, smoothness, growth rates, monotonicity, and explores implicit, parametric, and multivariable functions. Read more... 4545 words,🕔22 minutes read, May 14, 2022.

Absolute Extrema

Definitions, the Extreme Value Theorem with proof, and solved examples for finding absolute extrema of functions. Read more... 2888 words,🕔14 minutes read, Apr 14, 2022.

Continuity and discontinuity

Definitions of continuity and discontinuity, including ε–δ formulation, types of discontinuities (jump, removable, infinite, mixed), examples with various functions, and proof that differentiability implies continuity. Read more... 2374 words,🕔12 minutes read, May 14, 2022.

Infinite Limits

Intuitive and formal definitions of infinite limits using epsilon-delta, vertical asymptotes, one-sided infinite limits, and solved examples with proofs. Read more... 1493 words,🕔8 minutes read, May 14, 2022.

Limits at infinity

Intuitive and formal definitions of limits at infinity with epsilon-delta proofs, horizontal asymptotes, growth rate hierarchies, rational functions using dominant term method, limits with radicals, L'Hôpital's Rule, and extensive solved examples. Read more... 1708 words,🕔9 minutes read, May 14, 2022.

Limits of Rational Functions and at Infinity

Definition of rational functions, techniques for computing limits including direct substitution, factoring, rationalization, and limits at infinity using the dominant term method with extensive solved examples. Read more... 2022 words,🕔10 minutes read, May 14, 2022.

One-Sided Limits

Intuitive and formal definitions of one-sided limits, relationship with two-sided limits, extensive solved examples with piecewise functions, absolute value, floor function, and epsilon-delta proofs. Read more... 2529 words,🕔12 minutes read, May 14, 2022.

Strategy in finding limits

Comprehensive strategies for calculating limits including L'Hôpital's Rule for various indeterminate forms, the Squeeze Theorem with geometric proofs, and techniques for trigonometric limits using identities and fundamental limits. Read more... 2178 words,🕔11 minutes read, May 14, 2022.

The Limit Laws

The Limit Laws. Definition and solved examples. Proof of some basic limit rules. Read more... 1884 words,🕔9 minutes read, May 14, 2022.