Multivariable Calculus Version: 2011-2012

Scholastic Area that Course Is Active In: Mathematics

Strand: Algebra

Students will investigate the relationship between points, lines, and planes in three-dimensions.
Element: MMCA1.a
Represent equations of lines in space using vectors.
Element: MMCA1.b
Express analytic geometry of three dimensions (equations of planes, parallelism, perpendicularity, angles) in terms of the dot product and cross product of vectors.

Students will recognize and apply properties of matrices.
Element: MMCA2.a
Find the determinant of 2-by-2 and 3-by-3 matrices.
Element: MMCA2.b
Represent a 3-by-3 system of linear equations as a matrix and solve the system in multiple ways: the inverse matrix, row operations, and Cramer’s Rule.
Element: MMCA2.c
Apply properties of similar and orthogonal matrices to prove statements about matrices.
Element: MMCA2.d
Find and apply the eigenvectors and eigenvalues of a 3-by-3 matrix.

Students will explore functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0. a. Evaluate such functions at a point in the plane.
Element: MMCA3.a
Graph the level curves of such functions.
Element: MMCA3.b
Determine points or regions of discontinuity of such functions.

Strand: Process Standards

Students will solve problems (using appropriate technology).
Element: MMCP1.a
Build new mathematical knowledge through problem solving.
Element: MMCP1.b
Solve problems that arise in mathematics and in other contexts.
Element: MMCP1.c
Apply and adapt a variety of appropriate strategies to solve problems.
Element: MMCP1.d
Monitor and reflect on the process of mathematical problem solving.

Students will reason and evaluate mathematical arguments.
Element: MMCP2.a
Recognize reasoning and proof as fundamental aspects of mathematics.
Element: MMCP2.b
Make and investigate mathematical conjectures.
Element: MMCP2.c
Develop and evaluate mathematical arguments and proofs.
Element: MMCP2.d
Select and use various types of reasoning and methods of proof.

Students will communicate mathematically.
Element: MMCP3.a
Organize and consolidate their mathematical thinking through communication.
Element: MMCP3.b
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
Element: MMCP3.c
Element: MMCP3.c
Analyze and evaluate the mathematical thinking and strategies of others.
Element: MMCP3.d
Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.
Element: MMCP4.a
Recognize and use connections among mathematical ideas.
Element: MMCP4.b
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Element: MMCP4.c
Recognize and apply mathematics in contexts outside of mathematics.

Students will represent mathematics in multiple ways.
Element: MMCP5.a
Create and use representations to organize, record, and communicate mathematical ideas.
Element: MMCP5.b
Select, apply, and translate among mathematical representations to solve problems.
Element: MMCP5.c
Use representations to model and interpret physical, social, and mathematical phenomena.

Strand: Derivatives

Students will explore, find, use, and apply partial differentiation of functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0. a. Approximate the
Element: MMCD2.a
Approximate the partial derivatives at a point of a function defined by a table of data.
Element: MMCD2.b
Find expressions for the first and second partial derivatives of a function.
Element: MMCD2.c
Define and apply the total differential to approximate real-world phenomena .
Element: MMCD2.d
Represent the partial derivatives of a system of two functions in two variables using the Jacobian.
Element: MMCD2.e
Find the partial derivatives of the composition of functions using the general chain rule.
Element: MMCD2.f
Apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier.

Strand: Integration

Students will apply and interpret the theorems of Green, Stokes, and Gauss.
Element: MMCI2.a
Apply line and surface integrals to functions representing real-world phenomena.
Element: MMCI2.b
Recognize, understand, and use line integrals that are independence of path.
Element: MMCI2.c
Define and apply the gradient, the divergence, and the curl in terms of integrals of vectors.

Students will integrate functions of the form z = f(x, y) or w = f(x, y, z).
Element: MMCI1.a
Define, use, and interpret double and triple integrals in terms of volume and mass.
Element: MMCI1.b
Represent integrals of vectors as double and triple integrals.
Element: MMCI1.c
Integrate functions through various techniques such as changing the order of integration, substituting variables, or changing to polar coordinates.

Strand: Differential Equations

Students will use, apply, and solve linear first-order differential equations.
Element: MMDCE.a
Solve linear first-order differential equations of the form y' + p(x)y = q(x) with an integrating factor.
Element: MMCDE.b
Solve homogeneous linear first-order differential equations using a variable substitution.
Element: MMCDE.c
Solve Clairaut equations.
Element: MMCDE.d
Explore the concepts of families of solutions and envelopes.
Element: MMCDE.e
Write linear first-order differential equations that represent real-world phenomena and solve them, such as those arising from Kirchhoff's Law and mixing problems.
Element: MMCDE.f
Students will solve linear second-order differential equations of the form y" + p(x)y' + q(x)y = c using the characteristic equation where the characteristic equation has two real roots, one real root, or no real roots.

Students will use, apply, and solve linear first-order differential equations.

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