West Windsor-Plainsboro Regional School District Multivariable Calculus Honors & Accelerated Grade 12

Transcription

1 West Windsor-Plainsboro Regional School District Multivariable Calculus Honors & Accelerated Grade 12

2 Unit 1: A Brief Review of Single Variable Calculus 5 days What is a derivative in one variable? What the derivative of a function is in one variable and its applications Evaluate limits and demonstrate a clear understanding of the concept of a limit Demonstrate understanding of the derivative of a function in one variable and its applications Apply differentiation rules for functions in one variable to problems involving polynomial, rational, exponential, logarithmic, and trigonometric functions Use differentiation of functions in one variable to solve real world problems Demonstrate understanding of integration of a function in one variable and its applications Apply techniques of integration for functions in one variable to problems involving polynomial, rational, exponential, logarithmic, and trigonometric functions Use integration of functions in one variable to solve real world problems

3 Unit 2: Vectors and the Geometry of Space 25 days How can we visualize and represent shapes in three dimensional space? Represent lines and planes in three dimensional space Apply properties of vectors in two and three dimensional space to perform basic operations on vectors and vector algebra Find the dot product of two vectors, and use it to find the angle between those vectors Find the cross product of two vectors and use it to find angle measure, area, and volume Find equations of lines and planes in three dimensional space Visualize and sketch cylinders and quadric surfaces from their equations Describe points in three dimensional space using cylindrical and spherical coordinates

4 Unit 3: Vector Functions 15 days How can we use vectors to represent motion and forces? How can we use calculus to solve applications that are described by such vectors? how to use vectors and coordinate systems for three dimensional space and represent lines, planes, cylinders and quadric surfaces in space both graphically and with equations visualize and graph vector equations in three dimensional space use technology to draw space curves differentiate and integrate vector functions calculate arc length and curvature of space curves study the motion of an object in space using tangent vectors, normal vectors, and curvature use the calculus of vector equations to prove the first of Kepler s Laws of Planetary Motion

5 Unit 4: Multivariable Functions 30 days How can we explore problems where more than one variable affects an outcome and use calculus to solve such problems? how to use vectors and coordinate systems for three dimensional space and represent lines, planes, cylinders and quadric surfaces in space both graphically and with equations describe functions of two or more variables with words, tables of values, explicit formulas, graphs, and level curves evaluate limits and identify continuity for functions of two or more variables evaluate and apply partial derivatives of functions of two or more variables find tangent planes to functions in two variables at a given point use linear approximation techniques to approximate values of functions in two or more variables at specific points find and apply differentials and total differentials use the Chain Rule to differentiate functions in two or more variables use directional derivatives and gradient vectors to find the rate of change of a function in two or more variables in a specific direction use first and second derivatives to find maximum values, minimum values, and saddle points for functions of two variables use Lagrange multipliers to find the maximum and minimum values of functions in three variables subject to constraints

6

7 Unit 5: Multiple Integrals 35 days How do we evaluate integrals? Terms: integrals, iterated integrals, Jacobian transformation evaluate double integrals over rectangles, and apply them to determine the volumes of solids evaluate iterated integrals and use them to determine the volumes of solids evaluate double integrals over general regions, and apply them to determine the volumes of solids evaluate double integrals in polar coordinates, and apply them to determine the volumes of solids use double integrals for physical applications, including computing mass, electric charge, center of mass, moment of inertia, and related computations of probability density functions of two random variables use double integrals to compute the area of the surface of an object in three dimensional space evaluate triple integrals and extend the applications of double integrals to triple integrals evaluate triple integrals using cylindrical coordinates and spherical coordinates use the Jacobian transformation to integra double and triple integrals

8 Unit 6: Vector Calculus 40 days What is a vector field? Terms: vector fields, Green s Theorem, Stokes Theorem, Divergence Theorem Create and interpret vector fields Evaluate line integrals and apply them to problems involving vector fields, work, and mass Determine whether line integrals are independent of path Determine whether vector fields are conservative and apply this concept to conservation of energy Understand Green s Theorem and use it to evaluate line integrals Calculate and interpret curl and divergence of vector fields Identify, visualize, sketch and calculate the surface area of parametric surfaces Calculate surface integrals and use them to calculate flux of a vector field across a surface Use Stokes Theorem to evaluate integrals Use the Divergence Theorem to evaluate integrals

9 Unit 7: Differential Equations 15 days How can we use differential equations to solve problems in science and engineering? Terms: first and second order differential equations Solve first order linear differential equations that are separable equations Solve first order linear differential equations that are not separable equations Solve homogeneous second order linear differential equations Solve non homogeneous second order linear differential equations Use differential equations to solve problems in science in engineering Solve differential equations using the method of power series

10 Unit 8: Advanced Math Topics 15 days What advanced topics in mathematics would I like to learn more about? Identify specific areas of interest in advanced mathematics and their applications Research chosen areas of interest desired outcomes will depend on topic Summarize research in written form Share research with the class