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... 3078 words,🕔15 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

Exponential. Definition and solved examples. Exponential Function derivate and series. Properties. Exponent rules. Read more... 2624 words,🕔13 minutes read, May 14, 2022.