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Calculus

Function Fundamentals

 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... 1216 words,🕔6 minutes read, May 14, 2022.

Intercepts of Graphs

 Intercepts of Graphs. Definition, graph, properties, and solved examples. Graphing Lines Using Intercepts. Finding Intercepts Given the Graph. Read more... 1513 words,🕔8 minutes read, May 14, 2022.

Local Extrema

 Local Extrema. Definition, graph, and solved examples. First Derivative Test. Second Derivative Test. Read more... 1217 words,🕔6 minutes read, May 14, 2022.

Asymptotes and End Behavior

 Horizontal, vertical, and oblique asymptotes. Definitions and solved examples. End Behaviour of rational, exponential, and logarithm functions. Read more... 1854 words,🕔9 minutes read, Jul 14, 2022.

Increasing and Decreasing Functions

 Monotonic functions. Increasing and Decreasing Functions. Intuitive and formal definition. How to find increasing and decreasing intervals. Solved examples. Plotting lineas and quadratic functions. Read more... 1754 words,🕔9 minutes read, Jul 14, 2022.

Algebra of functions

 Algebra of functions. Definition and solved examples. The test point method. Read more... 1215 words,🕔6 minutes read, May 14, 2022.

Concavity and inflection points

 Concave functions. Inflection points. Definition, graph, properties, and solved examples. Read more... 1613 words,🕔8 minutes read, May 14, 2022.

Exponential functions

 Exponential. Definition and solved examples. Exponential Function derivate and series. Properties. Exponent rules. Read more... 1446 words,🕔7 minutes read, May 14, 2022.

Limits

 Intuitive and formal definition. How To Evaluate Limits From a Graph. Solved easy exercises. Solved examples with epsilon-delta proofs. Read more... 1666 words,🕔8 minutes read, May 14, 2022.

Inverse functions

 Inverse functions. Definition and solved examples. Derivate of inverse functions. The horizontal line test. Read more... 1335 words,🕔7 minutes read, May 14, 2022.

Logarithm functions

 Logarithm function. Definition and solved examples. Logarithmic Properties. Derivate of the Logarithmic Function. Change of Base of Logarithm. Read more... 1468 words,🕔7 minutes read, May 14, 2022.

Symmetry of Graphs.

 Symmetry of Graphs. Definition and solved examples. Even and odd functions. Symmetry with respecto to the x-axis. Test for symmetries. Read more... 1290 words,🕔7 minutes read, May 14, 2022.

Types of functions.

 Types of functions and criteria, such as lineal, quadratic, cubic, exponential, logarithmic, trigonometric, piecewise, rational, step, radical, Dirichlet, implicit functions, etc. Read more... 1494 words,🕔8 minutes read, May 14, 2022.

Absolute Extrema

 Absolute Extrema. Definition and solved examples. Finding absolute extrema. Read more... 1098 words,🕔6 minutes read, Apr 14, 2022.

Continuity and discontinuity

 Definition and examples. What is a Discontinuous Function? Types of Discontinuity. If a function f is differentiable at x = a, then f is continuous at x = a. Read more... 734 words,🕔4 minutes read, May 14, 2022.

Infinite Limits

 Infinite Limits. Intuitive and formal definition. Solved examples. Vertical asymptotes. Read more... 1036 words,🕔5 minutes read, May 14, 2022.

Limits at infinity

 Limits at infinity. Horizontal asymptotes. Intuitive and formal definition. Solved exercises. Solved examples with epsilon-delta proofs. Read more... 872 words,🕔5 minutes read, May 14, 2022.

Limits of Rational Functions

 Limits of Rational Functions. Definition of Rational Functions. Solved examples. Read more... 1092 words,🕔6 minutes read, May 14, 2022.

One-Sided Limits

 One-Sided Limits. Intuitive and formal definition. Solved examples. Read more... 1824 words,🕔9 minutes read, May 14, 2022.

Strategy in finding limits

 Direct substitution. Factoring. Combining rational fractions. The conjugate method (Rationalization). The Squeeze Theorem. Trigonometric identities. L'Hôpital's rule. Read more... 1897 words,🕔9 minutes read, May 14, 2022.

The Limit Laws

 The Limit Laws. Definition and solved examples. Proof of some basic limit rules. Read more... 1179 words,🕔6 minutes read, May 14, 2022.

The Squeeze Theorem

 The Squeeze Theorem. Intuitive Explanation. If a function f lies between two functions g and h, and the limits of each of them at a point are equal to L, then the limit of f at that point is L. Solved examples. Read more... 1237 words,🕔6 minutes read, May 14, 2022.

Derivate

 Definition and solved exercises, 1/x, √x, |x|, xⁿ. Find the equation of the tangent line. Read more... 1707 words,🕔9 minutes read, May 14, 2022.

Derivate 2: Exponential and Logarithms

 Derivate of Exponential and Logarithms functions. Definition, proofs, and solved examples. Read more... 1000 words,🕔5 minutes read, May 14, 2022.

Derivate 3. Trigonometric Functions.

 Basic trigonometry. Angles measurements in degrees and radians. Common trigonometry formulas. Derivatives of Trigonometric Functions. Read more... 1716 words,🕔9 minutes read, May 14, 2022.

Derivate 4. Inverse functions.

 Derivate of inverse functions. Inverse function theorem. Definition, proofs, and solved exercises. Derivatives of Inverse Trigonometric Functions. Read more... 1237 words,🕔6 minutes read, May 14, 2022.

Derivative Rules

 Derivate General rules. Constant, Constant Multiple, Power, Sum, Product, quotient, and chain Rule. Read more... 1015 words,🕔5 minutes read, May 14, 2022.

Higher derivatives

 Higher Order Derivatives. Definition, interpretation, notations, applications (Taylor series), and solved exercises. Read more... 1371 words,🕔7 minutes read, May 14, 2022.

Implicit Differentiation

 Implicit Differentiation. Definition, interpretation, and solved exercises. Read more... 817 words,🕔4 minutes read, May 14, 2022.

Linear and Quadratic Approximation

 Linear and quadratic approximation. Definition, geometrical interpretation, and solved exercises. How to Perform Linear and Quadratic Approximation. Read more... 1260 words,🕔6 minutes read, May 14, 2022.

Advanced Calculus

Sketching the Graph of a function

 Sketching the Graph of a function. Solved examples. Read more... 1467 words,🕔7 minutes read, May 14, 2022.

Sketching the Graph of a function II

 General strategy to plot functions. Solved examples. Basic Transformations. Plotting inverse and piecewise functions. Read more... 1919 words,🕔10 minutes read, May 14, 2022.

Bolzano–Weierstrass theorem

 Bolzano–Weierstrass theorem. Every subsequence of a convergent sequence converges and to the same limit. Read more... 1197 words,🕔6 minutes read, May 14, 2022.

Boundedness theorem

 Boundedness theorem. Solved homework examples. Read more... 939 words,🕔5 minutes read, May 14, 2022.

Derivatives as Rates of Change

 Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1536 words,🕔8 minutes read, May 14, 2022.

Derivatives as Rates of Change II

 Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1519 words,🕔8 minutes read, May 14, 2022.

Derivatives as Rates of Change III

 Applications of Derivatives. Related rates. Solved homework exercises. How to solve rates of change problems. Read more... 1544 words,🕔8 minutes read, May 14, 2022.

Fermat's Theorem

 Fermat's Theorem. How to find Absolute Extrema given a function f on a close interval [a, b]. Solved homework exercises. Read more... 1573 words,🕔8 minutes read, May 14, 2022.

Newton's Method

 Applications of Derivatives. Newton's Method. Basic principle. Solved homework exercises. Read more... 1189 words,🕔6 minutes read, May 14, 2022.

Optimization Problems

 Applications of Derivatives. Optimization Problems. Solved homework exercises. Read more... 1532 words,🕔8 minutes read, May 14, 2022.

Optimization Problems II

 Applications of Derivatives. Optimization Problems. Solved homework exercises. Steps to solve an optimization problem. Read more... 2070 words,🕔10 minutes read, May 14, 2022.

Rolle's and the Mean Value Theorems

 Rolle's Theorem. The Mean Value Theorem. Increasing and decreasing functions. Solved homework exercises. Read more... 1734 words,🕔9 minutes read, May 14, 2022.

Th. Extreme Value. Nested Interval Property.

 Extreme Value Theorem. Proof. Solved homework exercises. How to find Absolute Extrema given a function f on a close interval [a, b]. Axiom of Completeness/the least-upper-bound property. Nested Interval Property. Read more... 1590 words,🕔8 minutes read, May 14, 2022.

Antiderivates

 Antiderivates or indefinite integrals. Definition and Examples. Uniqueness of Antiderivatives. Read more... 818 words,🕔4 minutes read, May 14, 2022.

Differential equations

 Differential equations. Separation of variables. Integral Factors. Solved exercises. Read more... 1220 words,🕔6 minutes read, May 14, 2022.

Definite integrals. Fundamental Theorem of Calculus

 Definite integrals. Definition and Examples. Fundamental Theorem of Calculus. Intuitive interpretation of Fundamental Theorem. Read more... 1498 words,🕔8 minutes read, May 14, 2022.

Properties of integrals

 Properties of integrals. Integration by substitution or change of variables. Linearity of Integrals. Read more... 1242 words,🕔6 minutes read, May 14, 2022.

Fundamental Theorems of Calculus. MVT for Integrals

 Alternative version of the Fundamental Theorem of Calculus. Proof of the Fundamental Theorem of Calculus. The Second Fundamental Theorem of Calculus. The Mean Value Theorem for Integrals Read more... 1351 words,🕔7 minutes read, May 14, 2022.

The Logarithm Defined as an Integral. The error function

 Alternative definition of the natural logarithm as a definite integral. The error function. Read more... 999 words,🕔5 minutes read, May 14, 2022.

Areas between curves

 Determine the area of a region between two curves by integrating with respect to the independent variable. Read more... 1222 words,🕔6 minutes read, May 14, 2022.

Average Value Theorem

 Average Function Value. Average Value Theorem. Find the Average Value with the Mean Value Theorem for Integrals. Solved exercises. Read more... 1167 words,🕔6 minutes read, May 14, 2022.

Determining volumes

 Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1317 words,🕔7 minutes read, May 14, 2022.

Determining volumes II

 Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1195 words,🕔6 minutes read, May 14, 2022.

Determining volumes III

 Determining Volumes by Slicing. Volumes of solid of revolution. The shell method. Washer method. Read more... 1240 words,🕔6 minutes read, May 14, 2022.

Numerical integration.

 Riemann Sums. Trapezoid Sums. Simpson's Rule. Read more... 956 words,🕔5 minutes read, May 14, 2022.

Weighted Average

 Weighted Average. Solved exercises. Average temperature. Calculus Probability Modeling. Read more... 1436 words,🕔7 minutes read, May 14, 2022.

Weighted Average II

 Weighted Average. Solved exercises. Average temperature. Calculus Probability Modeling. Read more... 1445 words,🕔7 minutes read, May 14, 2022.

Integration of Trigonometric Functions

 Integration of Trigonometric Functions. Formulas and solved examples. Trigonometry substitution for integrals. Completing the square. Read more... 1209 words,🕔6 minutes read, May 14, 2022.

Integration of Trigonometric Functions II

 Integration of Trigonometric Functions. Formulas and solved examples. Trigonometry substitution for integrals. Completing the square. Read more... 814 words,🕔4 minutes read, May 14, 2022.

Arc lengths

 Arc lengths. Surface Area. Solved exercises. Read more... 1197 words,🕔6 minutes read, May 14, 2022.

Integration by parts

 Integration by parts. Solved homework exercises. Geometrical interpretation. LIATE mnemonic. Read more... 1107 words,🕔6 minutes read, May 14, 2022.

Integration by parts II

 Integration by parts. Solved homework exercises. Geometrical interpretation. LIATE mnemonic. Read more... 821 words,🕔4 minutes read, May 14, 2022.

Integration of rational functions.

 Integration of rational functions. Integration of improper rational fraction. Partial fraction decomposition. Distinct linear factors. Repeated linear factors. Read more... 1130 words,🕔6 minutes read, May 14, 2022.

Parametric curves. Polar Coordinates.

 Parametric curves. Polar Coordinates. The area of the sector of a curve in polar coordinates. Read more... 1311 words,🕔7 minutes read, May 14, 2022.

L'Hôpital's Rule

 L'Hôpital's Rule. Motivation. General form. Solved examples. Evaluating Limits of Indeterminate Forms. Read more... 1020 words,🕔5 minutes read, May 14, 2022.

L'Hôpital's Rule II

 L'Hôpital's Rule. Motivation. General form. Solved examples. Evaluating Limits of Indeterminate Forms. Rates of growth and decay. Asymptotic Complexity. Read more... 1022 words,🕔5 minutes read, May 14, 2022.

Comparison Test For Improper Integrals

 Comparison Test For Improper Integrals. Solved examples. Read more... 1124 words,🕔6 minutes read, May 14, 2022.

Improper integrals type 2

 Improper integrals of second type. Read more... 1209 words,🕔6 minutes read, May 14, 2022.

Improper integration.

 Improper integration. Limit Comparison. Improper integrals of second type. Read more... 1064 words,🕔5 minutes read, May 14, 2022.

Alternating Series

 Definition. Alternating Series Test. Solved exercises. Read more... 1398 words,🕔7 minutes read, May 14, 2022.

Convergence/Divergence of series

 Necessary condition for the convergence of a series. Divergence test. Integral Test For Convergence and Divergence of Series. Read more... 1547 words,🕔8 minutes read, May 14, 2022.

Direct & Limit Comparison test

 Direct Comparison test. Limit Comparison Test. Solved homework exercises. Theorem p-series. Integral Comparison. Read more... 1588 words,🕔8 minutes read, May 14, 2022.

Infinite Series

 Infinite Series. Arithmetic and Geometric series. Convergent and divergent series. Solved examples. Algebraic Properties of Convergent Series. Read more... 1485 words,🕔7 minutes read, May 14, 2022.

Power Series. Convergence, derivatives, and Integrals

 Power Series. Power Series Convergence. Derivatives and Integrals of Power Series. Read more... 1684 words,🕔8 minutes read, May 14, 2022.

Root and Ratio Test

 Root Test. Ratio Test. Solved examples. Read more... 1308 words,🕔7 minutes read, May 14, 2022.

Taylor’s Formula.

 Taylor’s Formula. Taylor's theorem. Lagrange form of the remainder. Read more... 1614 words,🕔8 minutes read, May 14, 2022.

Multivariable Calculus

Vectors

 Vectors. Definition, components, representation, examples, solved exercises, and Properties Vector Arithmetic. Dot product. Read more... 1545 words,🕔8 minutes read, May 14, 2022.

Cross Products

 Vectors II. Determinant in space. Cross products. Properties. Solved exercises. Read more... 2701 words,🕔13 minutes read, May 14, 2022.

Vectors II. The Dot Product.

 Vectors. The Dot Product. Solved exercises. Read more... 2144 words,🕔11 minutes read, May 14, 2022.

Vectors III. Planes and Areas

 Calculating the Area of a Triangle Using Vectors Read more... 1543 words,🕔8 minutes read, May 14, 2022.

Equations of Planes

 Find the Equation of a Plane Given Three Points. Vectors and the Geometry of Space. Solved exercises. Read more... 2219 words,🕔11 minutes read, May 14, 2022.

Matrices: definitions, types, examples, and properties.

 Rotation Matrices. Inverse matrix. Properties. Trace of a matrix. Diagonal matrix. Lower and upper triangular matrix. Symmetric matrix. Read more... 3281 words,🕔16 minutes read, May 14, 2022.

Systems of Linear Equations.

 Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3041 words,🕔15 minutes read, May 14, 2022.

Parametric equations for lines and curves II

 Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. A cycloid. Read more... 1815 words,🕔9 minutes read, May 14, 2022.

Parametric equations for lines and curves.

 Parametric equations for lines and curves. Graphing a Parametrically Defined Curve. Eliminating the Parameter. Read more... 1978 words,🕔10 minutes read, May 14, 2022.

Systems of Linear Equations II

 Equations of planes. Solving Systems of Linear Equations. Exercises of vectors and planes. Read more... 3627 words,🕔18 minutes read, May 14, 2022.

Functions of two variables II

 Functions of two variables. Sketching graphs. Read more... 1510 words,🕔8 minutes read, May 14, 2022.

Functions of two variables.

 Functions of two variables. Definition, examples, domain, graphs, contour plots. Read more... 1795 words,🕔9 minutes read, May 14, 2022.

Least Squares Interpolation

 Least Squares Interpolation. Fitting a linear and a quadratic model. Moore's Law. Read more... 1812 words,🕔9 minutes read, May 14, 2022.

Partial derivatives

 Definition. Formal Definition. Geometric interpretation. Solved examples. Local extrema and critical points in multivariable functions. Read more... 2105 words,🕔10 minutes read, May 14, 2022.

Tangent Planes And Linear Approximations

 Tangent Planes And Linear Approximations Read more... 1969 words,🕔10 minutes read, May 14, 2022.

Velocity and Acceleration.

 Definition and Properties of Vectors in Motion. Kepler's Laws and Planetary Motion. Newton's explanation using Vector Calculus. Solved exercises. Read more... 3593 words,🕔17 minutes read, May 14, 2022.

Second derivative test

 Quadratic Functions and Critical Points. Steps to Perform the Second Derivative Test. Non-rigorous proof. Solved examples. Read more... 3505 words,🕔17 minutes read, May 14, 2022.

Total differential

 Total differential. Using Different Notation. The Chain Rule for multivariable functions. Proof of the Chain Rule. Solved exercises. Read more... 1378 words,🕔7 minutes read, May 14, 2022.

Directional derivatives

 Step to Compute the Directional derivative. Formal and alternative definition. Solved exercises. Read more... 3364 words,🕔16 minutes read, May 14, 2022.

Gradient vector

 The Gradient Vector is Perpendicular to Level Surfaces. Solved exercises. Read more... 3481 words,🕔17 minutes read, May 14, 2022.

Lagrange Multipliers

 Lagrange Multipliers. Solved exercises. Read more... 3917 words,🕔19 minutes read, May 14, 2022.

Double integrals

 Double integrals. Formal and informal definition, examples, and properties. Calculating a double integral. Fubini's Theorem. Read more... 2765 words,🕔13 minutes read, May 14, 2022.

Double integrals II

 Double integrals II. Solved examples. Exchanging order of integration. Read more... 2292 words,🕔11 minutes read, May 14, 2022.

Implicit partial differentiation

 Calculating ∂f/∂x, ∂f/∂y for a given implicit function Read more... 1624 words,🕔8 minutes read, May 14, 2022.

Partial Derivative with Constrained Variables

 Non-independent variables. Solving a Constraint. Avoiding confusion. Calculate the area of a triangle. Read more... 3756 words,🕔18 minutes read, May 14, 2022.

Partial differential equations

 Classification of differential equations. Ordinary Differential Equations. The heat and the harmonic oscillator equations. First-order differential equations. Read more... 3003 words,🕔15 minutes read, May 14, 2022.

Applications of Double Integrals

 Area Region. Volume under Surface. Average Value Function. Total mass of a flat object (lamina) over a region. Read more... 4029 words,🕔19 minutes read, May 14, 2022.

Applications of Double Integrals II

 Weighted Average of a Function with a Density Function. Center of Mass. Moment of Inertia. Kinetic Energy of a Rotating Object. Read more... 4441 words,🕔21 minutes read, May 14, 2022.

Double integrals in Polar Coordinates

 Double integrals in Polar Coordinates Read more... 3372 words,🕔16 minutes read, May 14, 2022.

Change of variables in double integrals

 Understanding the Jacobian in Change of Variables. Applying the Change of Variables Formula. Transformations with polar coordinates. Read more... 4070 words,🕔20 minutes read, May 14, 2022.

Vector fields

 Vector fields Read more... 2960 words,🕔14 minutes read, May 14, 2022.

Conservative vector fields

 Open, connected, and simply connected regions. The Fundamental theorem of Calculus for Line Integral. Equivalent Properties of Conservative Vector Fields. Read more... 2931 words,🕔14 minutes read, May 17, 2022.

Conservative vector fields II

 Path Independence and Conservative Vector Fields. Criterion for a Conservative Vector Field. Curl and Torque Read more... 2620 words,🕔13 minutes read, May 17, 2022.

Find potential functions for conservative fields

 Finding a potential function for conservative vector fields. Solved exercises. Read more... 3298 words,🕔16 minutes read, May 17, 2022.

Green's theorem

 Green's theorem. Fully Explained. Step by Step examples. Read more... 3678 words,🕔18 minutes read, May 17, 2022.

Surface Integrals of Vector Fields. Flux.

 Surface Integrals of Vector Fields. Flux Form of Green's Theorem. Explanation of divergence. Limitations of Green's Theorem. Read more... 3960 words,🕔19 minutes read, May 18, 2022.

Surface Integrals of Vector Fields. Flux. II

 Example with a Non-Simply Connected Region. Definition and Importance of Simply Connected Regions in Green's Theorem. Why Simply Connected Regions Matter. Criteria for Conservative Fields. Solved mixed exercises. Read more... 3257 words,🕔16 minutes read, May 18, 2022.

Triple Integrals

 Triple Integrals. Solved Exercises. Cylindrical Coordinates. Read more... 3410 words,🕔17 minutes read, May 17, 2022.

Triple Integrals 2. Applications.

 Triple Integrals. Solved Exercises. Applications. Read more... 4916 words,🕔24 minutes read, May 17, 2022.

Triple Integrals 3. Spherical coordinates

 Spherical coordinates. Solved Exercises. Applications. Calculation of Gravitational Force Exerted by an object. Read more... 4701 words,🕔23 minutes read, May 17, 2022.

Vector fields & flux 3D

 Vector fields in 3D. Flux in 3D. Solved Exercises. Read more... 4767 words,🕔23 minutes read, May 17, 2022.

Vector fields & Flux in 3D II

 Vector fields & flux in 3D. Solved Exercises. Explanation of Surface Area Elements Using Normal Vectors. Read more... 5023 words,🕔24 minutes read, May 17, 2022.

The Divergence Theorem

 The Divergence Theorem. Examples, Physical Interpretation, and Proof. Step by Step Solved Exercises. Read more... 4926 words,🕔24 minutes read, May 17, 2022.

The Divergence Theorem II

 The Divergence Theorem. Solved Exercises. The diffusion equation Read more... 4220 words,🕔20 minutes read, May 17, 2022.

Curl in 3D

 Curl in 3d, definition and solved exercises. Conditions for a Vector Field to be Conservative. Read more... 2751 words,🕔13 minutes read, May 17, 2022.

Line integrals in space

 Line integrals in space. Solved Exercises. Test for conservative fields. Find potential functions. Read more... 4028 words,🕔19 minutes read, May 17, 2022.

Stoke's Theorem

 Formal Statement, Proof of Stoke's Theorem. Orientation and the Right-Hand Rule. Intuitive explanation. Solved Exercises. Comparing Stokes’ Theorem with Green’s Theorem. Read more... 3433 words,🕔17 minutes read, May 17, 2022.

Stoke's Theorem II

 Stoke's Theorem II. Solved Exercises. Read more... 2699 words,🕔13 minutes read, May 17, 2022.

Stoke and Surface Independence. Curl.

 Solved examples. Stoke Theorem and Surface Independence. Curl and Topological considerations. Physical interpretation of Curl. Conservative Fields and Rotation. Irrotational and Conservative Fields. Read more... 4637 words,🕔22 minutes read, May 17, 2022.

Faraday's law. Maxwell's equations

 Faraday's law. Maxwell's equations. Read more... 1630 words,🕔8 minutes read, May 17, 2022.

Differential equations

Geometrical View.

 Differential equations. Geometrical View. Read more... 2405 words,🕔12 minutes read, May 17, 2022.

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