# Mathematics

# Exponential Models

This

**video**tutorial explains how to solve**exponential growth and decay****word problems**.

This

**video**provides an example of the**radioactive decay**of a substance, as well as**exponential growth**of a fast-food restaurant chain.

This

**video**explores how to**model populations**using**differential equations**using**separation of variables**. Assuming a quantity grows proportionally to its size results in the general equation**dy/dx=ky.**Solving it with**separation of variables**results in the**general exponential function****y=Ceᵏ**.

This

**video**shows the importance of using**initial conditions**, such as the population at time 0 and at time 50, to find the solution to an**exponential modeling problem**, given the general solution**P=Ceᵏᵗ**.

# Functions

This

**video**explains how to use the**first derivative**and a**sign chart**to determine the intervals where a**function is increasing and decreasing**and how to express the answer using**interval notation**with the help of a**number line**.

This

**video**describes how to find the**average rate of change**of a function**over an interval.**

This

**video**explains the**average rate of change**and its relation to the**slope of a secant line**.

This

**video**shows how to find the**interval in function's graph**where the function has an arbitrary value as its**rate of change**(in this case, 4).

This

**video**illustrates how to find the**average rate of change**of a function over a**specific interval**, given a**table of values**of the function.

# Exponential & Logarithmic Functions

By plotting points on a graph, this

**video**illustrates that the**inverse**of**exponential function**is a**logarithmic function**and the inverse of a**logarithmic function**is an**exponential function**.

By graphing the

**exponential function y=2ˣ**and the**logarithmic function y=log₂(x)**on the same coordinate plane, this**video**shows how the graphs relate as**inverse functions**.

# Arithmetic & Algebra

This

**video**highlights the correct order of operations to solve mathematics expressions using**PEMDAS**.**PEMDAS**is an acronym for the words**P**arenthesis,**E**xponents,**M**ultiplication,**D**ivision,**A**ddition,**S**ubtraction. For any expression, all expressions in**parenthesis**should be performed**first**, followed by**exponents**, followed by**multiplication and division**(from**left to right**) and, finally,**addition and subtraction**(from**left to right**).

This

**video**shows how**PEMDAS**is applied to arrive at the correct solution of a numerical example.

This

**video**uses**numerical examples**to illustrate how to convert between**percents**,**decimals**, and**fractions**.

## 🔢 Proportions 📺

This

**video**explains how to solve**proportions**by using**equivalent ratios**, or**equivalent fractions**.

This

**video**explains how the**binomial theorem**is used to**expand expressions**of the form**(a+b)ⁿ.**

## 🔢 Reading Matrices 📺

This

**video**teaches us how to read**matrices**: rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.**Matrices**are covered in our**mathematics**and**computer science**courses.

# Motion Problems

This

**video**defines the difference between**displacement**and**distance**. The**definite integral of a velocity function**gives us the**displacement**. To find the**actual distance traveled**, we need to use the**speed function**, which is the**absolute value of the velocity**.

This

**video**explains the relationship between**position**,**velocity**, and**acceleration**. If**position**is given by a function**p(t)**, then the**velocity, v(t),**is the**first derivative**of that function, and the**acceleration, a(t)**is the**second derivative**. This concept is covered in our**mathematics**and**physics**courses.

This

**study guide**defines:**distance**,**speed**and**acceleration**,**force and pressure**,**work**,**energy**,**conservation of energy**(**potential**and**kinetic**energy) and the**equivalence of work and energy**.

# Calculus, Linear Algebra & Differential Equations

This link provides comprehensive

**video**explanations for such concepts in**Calculus I (Differential Calculus)**as: Limits, Average Rate of Change, Definition of the Derivative, Differentiation of Basic Functions and Using the Power Rule, Product Rule, Quotient Rule, and Chain Rule, Differentiation of Exponential Functions, Differentiation of Hyperbolic Functions, Logarithmic Differentiation, Differentiation of Inverse Trigonometric Functions, Higher Order Differentiation, Applications of Differentiation and Relative Extrema [Concavity, Maximum/Minimum/Optimization Problems], Absolute Extrema, Differentials, Rolle’s Theorem and the Mean Value Theorem, Implicit Differentiation, Related Rates, Newton’s Method and L’Hôpital’s Rule, and Proofs.

This link provides comprehensive

**video**explanations for such topics in**Calculus II**(Integral Calculus) as: Approximating Area Under a Curve, The Antiderivative, Indefinite Integration, Definite Integral and The Fundamental Theorem of Calculus [FTC], The Second Fundamental Theorem of Calculus, Applications of Definite Integration, Area Bounded by Two Functions, Integration by Substitution, Integration by Parts, Integration Involving Inverse Trigonometric Function and Integration Tables, Numerical Integration, Improper Integrals, Introduction to Differential Equations, Business Applications of Integration, Volume of Revolution, Arc Length, Surface Area, Work, Force, Center of Mass, Integration Involving Powers of Trigonometric Functions, Integration Using Partial Fractions, Integration Using Trigonometric Substitution, Infinite Series, Power Series, Parametric Equations, Polar Coordinates and Equations, Graphing Polar Equations, Derivatives and Integrals with Polar Equations, Vectors in 2D, Applications of Vectors, Vectors in Space, Equations of Planes and Lines in Space, Quadric Surfaces, Cylindrical Coordinates and Spherical Coordinates, and Vector Valued Functions.

This link provides comprehensive

**video**explanations for such topics in**Calculus III (Multivariable Calculus)**as: Introduction to Functions of Several Variables, Limits and Partial Derivatives of Functions of Two Variables, The Chain Rule and Directional Derivatives and the Gradient Functions of Two Variables, Normal Vectors and Tangent Planes to Functions of Two Variables, Relative Extrema and Applications to Functions of Two Variables, Double Integrals, Double Integrals in Polar Coordinates, Applications of Double Integrals (Mass, Center of Mass, Jacobian), Triple Integrals, Triple Integrals in Cylindrical and Spherical Coordinates, Vector Fields, Line Integrals, and Surface Integrals.

## 🔢 Linear Algebra 📺

This link provides comprehensive

**video**explanations for such topics in**Linear Algebra**as:**Solving Systems of Equations by Graphing**,**Solving Systems of Equations Using Substitution**,**Solving Systems of Equations Using Elimination**,**Applications of Systems of Equations**,**Systems of Equations with Three Unknowns**,**Matrices**,**Augmented Matrices,****Matrix Operations**,**Inverse Matrices**,**Matrix Equations**,**Determinants**,**Vectors**,**Applications of Vectors**,**Vectors in Space**,**Eigenvalues and Eigenvectors**.

This link provides comprehensive

**video**explanations for such topics in**Differential Equations**as:**Introduction to Differential Equations**,**First Order Differential Equations**,**Separation of Variables**,**Exact First Order Differential Equations**,**Linear First Order Differential Equations,****Bernoulli Differential Equations**,**First Order Homogeneous Differential Equations**,**Interval of Validity (Existence and Uniqueness)**,**Applications of First Order Differential Equations**,**Equilibrium Solutions**,**Euler’s Method**,**Second Order Differential Equations**,**Linear Second Order Homogeneous Differential Equations**,**Cauchy-Euler**,**Reduction of Order**,**Method of Undetermined Coefficients**,**Variation of Parameters**,**Modeling with Higher Order Differential Equations**, and**Laplace Transforms**.

## 🔵 Matemáticas, Mathematics Khan Academy Videos en Español 📺

This link provides comprehensive

**video**tutorials from the**Khan Academy en Español**for such topics in**Mathematics**as: Elementary Mathematics, Arithmetic, Pre-Algebra, Algebra Fundamentals, Algebra I, Basic Geometry, Geometry, Algebra II, Trigonometry, Statistics and Probability, Statistics, Differential Calculus, Integral Calculus, Multivariable Calculation, Differential Equations, Linear Algebra, and Precalculus.

## 🔵 Aritmética, Arithmetic Khan Academy Videos en Español 📺

This link provides comprehensive

**video**tutorials from the**Khan Academy en Español**for such topics in**Arithmetic**as: Addition and Subtraction, Multiplication and Division, Negative Numbers, Fractions, and Decimals.

## 🔵 Cálculo Diferencial, Differential Calculus Khan Academy Videos en Español 📺

This link provides

**video**resources from the**Khan Academy en Español**for such topics in**Calculus I (Differential Calculus)**as: Limits and Continuity, Definition and Rules of Derivatives, Derivatives (Chain Rule, Logarithmic and Implicit Differentiation), Applications of the Derivative (Local Linear Approximation and L'Hôpital's Rule), Analysis of Functions, and Parameterized Equations, Polar Coordinates and Functions with Vector Values.

## 🔵 Cálculo Integral, Integral Calculus Khan Academy Videos en Español 📺

This link provides

**video**resources from the**Khan Academy en Españo**l for such topics in**Integral Calculus (Calculus II)**as: Integrals, Differential Equations, Applications of Integrals, Parameterized Equations, Polar Coordinates and Functions with Vector Values, and Series.

## 🔵 Cálculo Multivariable, Multivariable Calculus Khan Academy Videos en Español 📺

This link provides

**video**resources from the**Khan Academy en Español**for such topics in**Multivariable Calculus (Calculus III)**as: Multivariable Functions, Derivatives of Multivariable Functions, Applications of Multivariable Derivatives, Integration of Multivariable Functions, and Green's, Stokes' and Divergence Theorems.

## 🔵 Álgebra Lineal, Linear Algebra Khan Academy Videos en Español 📺

This link provides

**video**resources from the**Khan Academy en Español**for such topics in**Linear Algebra**as: Spaces and Vectors, Matrix Transformations, and Alternative Coordinate Base Systems (i.e. Orthogonal Complements, Orthonormal bases and the Gram-Schmidt Method, and Eigenvectors).

## 🔵 Ecuaciones Diferenciales, Differential Equations Khan Academy Videos en Español 📺

This link provides

**video**resources from the**Khan Academy en Español**for such topics in**Differential Equations**as: First Order Differential Equations, Linear Equations of Second Order, and Laplace Transform.