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.
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.
Exponential & Logarithmic Functions
By plotting points on a graph, this video illustrates that the inverse of an 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 Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction. 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).
🔢 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.
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.
🔵 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 Español 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.