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In the Billions and Exponential Modeling
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This problem provides an opportunity to experiment with modeling real data. Populations ... More

This problem provides an opportunity to experiment with modeling real data. Populations are often modeled with exponential functions and in this particular case we see that, over the last 200 years, the rate of population growth accelerated rapidly, reaching a peak a little after the middle of the 20th century and now it is slowing down. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
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Introduction to Functions - Basics and Applications
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The purpose of this activity is to reinforce the concept of function ... More

The purpose of this activity is to reinforce the concept of function and the use of functional notation rather than to teach the concepts. This lab looks at input/output pictures to emphasize that a function has only one output for every input even through the output need not be unique. Using functional notation, students determine both range and domain values from a graph and then do the same for a variety of given functional equations. The real world applications include a piecewise function (cell phone costs) and require the student to find a variety of values as well as determining realistic domains. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Introduction to Partial Differential Equations
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Partial differential equations (PDEs) describe the relationships among the derivatives of an ... More

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222) Less

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Subject:
Functions
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Provider:
The Saylor Foundation
Provider Set:
Saylor Foundation
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Lesson: 18 Functions
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Beginning with linear functions, this lesson looks at functions of real world ... More

Beginning with linear functions, this lesson looks at functions of real world data defined by tables and graphs before moving into functions defined by equations. Function notation is introduced at the end of teh lesson and various examples are provided to get students familiar with the new notation. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 1: Linear Models
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A unit on linear modeling using tables, equations, and graphs to explore ... More

A unit on linear modeling using tables, equations, and graphs to explore different contexts (bike rentals, fuel consumption). A homework assignment is included at the end of the unit. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 21: Variation
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The lesson begins with a comparison of data tables and graphs of ... More

The lesson begins with a comparison of data tables and graphs of two functions, one directly proportional (cost of gas) and the other exponential (population), before a definition for direct variation is introduced. Direct variation is then linked to linear function (f(x)= kx)and the scaling property of direct variation is examined (i.e. a multiple of the independent variable will always correspond to that same multiple of the dependent variable). Direct variation with a power of x follows with a test for direct variation before indirect variation and indirect variation with a power of x are introduced. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
Less
Lesson 2: Intercepts (linear models)
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Finding the intercepts of a linear model graph. The intercepts are also ... More

Finding the intercepts of a linear model graph. The intercepts are also interpreted in terms of the fuel consumption problem from lesson 1 where intercepts naturally arise as the moments when the tank is full and the tank is empty. The general form of a linear equation and the intercept method of graphing are also introduced in this lesson. Homework assignment follows at end of lesson. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 3: Graphs and Equations
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This lesson begins with linear equations and inequalities in 1 variable and ... More

This lesson begins with linear equations and inequalities in 1 variable and then moves on to linear equations in 2 variables. Graphs of linear equations in 2 variables are introduced as "a picture of all its solutions." Exercises targeting the links between equations, solutions, points, and graphs follows, with the final activities focusing on use of a graphing calculator to graph equations and find coordinates. There aren't any application problems in this lesson. Less

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Subject:
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 40: Properties of Lines
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The lesson begins with horizontal and vertical lines, first looking at the ... More

The lesson begins with horizontal and vertical lines, first looking at the corresponding sets of points that comprise each, then using the points to find the slope of each type of line. Parallel and perpendicular lines are covered next, ending with applications in geometry. Less

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Subject:
Education
Mathematics
Algebra
Functions
Geometry
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 44: Nonlinear Systems
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Using a cost/revenue application problem, the lesson begins with systems involving quadratic ... More

Using a cost/revenue application problem, the lesson begins with systems involving quadratic equations. Systems with conics are introduced next along with the elimination method for solving these systems. Less

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Subject:
Education
Mathematics
Algebra
Functions
Geometry
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
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Lesson 4: Slope
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Treating slope as a rate of change of two quantities with different ... More

Treating slope as a rate of change of two quantities with different units. There are both applications based examples and non contextual examples. Slope is found both from a graph and from the relationship between two quantities given in a context. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
Less
Lesson 5: Equations of lines
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Begins with the slope-intercept equation of a line and then introduces the ... More

Begins with the slope-intercept equation of a line and then introduces the coordinate formula for slope before the introduction of the point-slope equation of a line. Linear models are then revisited. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
Less
Lesson 6: Linear Regression
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Using real world data, this lesson introduces linear regression using lines of ... More

Using real world data, this lesson introduces linear regression using lines of best fit that may calculated by hand by selecting two pints that appear to fall on the line of best fit. The lesson could also be used with a calculator to find the actual regression line. Interpolation and extrapolation are also introduced as well as scatter plots. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
Less
Lesson 7: Linear Systems
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A calculator based introduction to systems of linear equations. Systems are solved ... More

A calculator based introduction to systems of linear equations. Systems are solved using the graphing method: first by estimating the apparent intersection of the two lines and then later by using the intersect function on the calculator to find the exact solution. Inconsistent and consistent solutions are also discussed. There are both applications based problems and non applications based problems. Less

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Subject:
Education
Life Science
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
Developmental Mathematics Collection
Less
Linear Algebra II
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Linear Algebra is both rich in theory and full of interesting applications; ... More

Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212) Less

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Subject:
Algebra
Functions
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Provider:
The Saylor Foundation
Provider Set:
Saylor Foundation
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Linear Equations
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In this simulation, you will learn how to recognize a linear equation. ... More

In this simulation, you will learn how to recognize a linear equation. You can also test your knowledge in a short quiz at the end of the lesson. Less

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Subject:
Education
Mathematics
Algebra
Functions
Material Type:
Instructional Material
Interactive
Lesson Plans
Provider:
SMARTR
Provider Set:
SMARTR: Virtual Learning Experiences for Youth
Author:
Intel Corporation
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