Math Flow Chart

Home , Difference quotient, Direct comparison test, Limit comparison test, Related rates

Updated November 12, 2014 MATH FLOW CHART

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Calculus Advanced Advanced Advanced Math AB or BC Algebra II Algebra I Geometry Analysis MAP GLA-Grade 8 MAP EOC-Algebra I MAP - ACT AP Statistics

College Calculus Algebra/ Algebra I Geometry Algebra II AP Statistics MAP GLA-Grade 8 MAP EOC-Algebra I Trigonometry MAP - ACT Math 6 Pre-Algebra Statistics Extension Extension MAP GLA-Grade 6 MAP GLA-Grade 7 or or Algebra II Math 6 Pre-Algebra Geometry MAP EOC-Algebra I College Algebra/ MAP GLA-Grade 6 MAP GLA-Grade 7 MAP - ACT Foundations Trigonometry of Algebra Algebra I Algebra II Geometry MAP GLA-Grade 8 Concepts Statistics Concepts MAP EOC-Algebra I MAP - ACT

Geometry Algebra II MAP EOC-Algebra I MAP - ACT Discovery Geometry Algebra Algebra I Algebra II Concepts Concepts MAP - ACT MAP EOC-Algebra I

Additional Full Year Courses: General Math, Applied Technical Mathematics, and AP Computer Science. Calculus III is available for students who complete Calculus BC before grade 12.

MAP GLA – Missouri Assessment Program Grade Level Assessment; MAP EOC – Missouri Assessment Program End-of-Course Exam Mathematics Objectives Calculus

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Linear and Nonlinear Functions • Write equations of linear and nonlinear functions in various formats. • Graph linear and nonlinear functions. • Solve linear and nonlinear functions using a variety of methods. • Apply functions to solve real world problems.

The Derivative • Show that the difference quotient represents the average rate of change, the slope of the secant line, and the average velocity. • Demonstrate that the derivative can represent the instantaneous rate of change, the slope of the tangent line, and the instantaneous velocity. • Calculate the derivative of several types of functions. • Identify critical points, concavity, and increasing and decreasing intervals using graphs and derivatives. • Investigate and use applications of the second derivative to solve problems. • Use the derivative for business applications, implicit differentiation, related rates, and linear approximation. • Find the derivatives of trigonometric functions and use them to analyze a variety of applications.

Integration • Use the Fundamental Theorem of Calculus to find the antiderivative. • Use the anitderivative to find the area under a curve. • Find integrals using integration by parts. • Find the volume of a solid using the definite integral. • Find the integrals of trigonometric functions.

Multivariable Calculus • Use multivariable calculus to find maxima and minima and to explore three-dimensional space.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13,2015

Mathematics Objectives Advanced Placement Calculus AB

Functions, Graphs, and Limits Analysis of Graphs • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Find limits of indeterminate forms using L’Hopital’s Rule. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations.

Limits of Functions • Use properties of limits to find the limit of a sequence. • Find limits of indeterminate forms using L’Hopital’s Rule. • Evaluate indefinite and definite integrals by the method of substitution. • Use properties of limits to find the limit of a sequence. • Determine whether a sequence converges or diverges. • Find limits of indeterminate forms using L’Hopital’s Rule. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. • Construct slope fields and interpret slope fields as visualizations of various equations.

Asymptotic and Unbounded Behavior • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Solve problems involving exponential growth and decay.

Continuity as a Property of Functions • Use Euler’s Method for approximating points for a solution to an initial value problem. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Derivatives Concept of the Derivative • Construct antiderivatives using the Fundamental Theorem of Calculus. • Solve initial value problems. • Construct slope fields and interpret slope fields as visualizations of various equations.

Derivative at a Point • Construct slope fields and interpret slope fields as visualizations of various equations.

Derivative as a Function • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve initial value problems.

Second Derivatives • Find derivative and second derivatives of parametrically defined functions.

Applications and Computation of Derivatives • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve problems involving exponential growth and decay. • Evaluate indefinite and definite integrals by the method of substitution.

Integrals Interpretations and Properties of Definite Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids.

Applications of Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Calculate slopes and areas of regions in the plane determined by polar curves.

Fundamental Theorem of Calculus • Construct antiderivatives using the Fundamental Theorem of Calculus.

Techniques and Applications of Antidifferentiation • Construct antiderivatives using the Fundamental Theorem of Calculus. • Solve initial value problems. • Evaluate indefinite and definite integrals by the method of substitution. • Use integration by parts to evaluate indefinite and definite integrals. • Solve problems involving logistic population growth. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Solve problems involving exponential growth and decay.

Numerical Approximations to Definite Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Mathematics Objectives Advanced Placement Calculus BC

Limits of Functions • Use properties of limits to find the limit of a sequence. • Find limits of indeterminate forms using L’Hopital’s Rule. • Evaluate indefinite and definite integrals by the method of substitution. • Determine whether a sequence converges or diverges. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. • Construct slope fields and interpret slope fields as visualizations of various equations.

Parametric, Polar, and Vector Functions • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Convert Cartesian equations into Polar form and vice versa. • Calculate slopes and areas of regions in the plane determined by polar curves.

Derivative at a Point • Construct slope fields and interpret slope fields as visualizations of various equations.

Derivative as a Function • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve initial value problems.

Second Derivatives • Find derivative and second derivatives of parametrically defined functions.

Applications and Computation of Derivatives • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Convert Cartesian equations into Polar form and vice versa. • Calculate slopes and areas of regions in the plane determined by polar curves. • Construct slope fields and interpret slope fields as visualizations of various equations. • Use Euler’s Method for approximating points for a solution to an initial value problem. • Find limits of indeterminate forms using L’Hopital’s Rule. • Solve problems involving exponential growth and decay. • Evaluate indefinite and definite integrals by the method of substitution. • Find derivative and second derivatives of parametrically defined functions.

Fundamental Theorem of Calculus • Construct antiderivatives using the Fundamental Theorem of Calculus.

Techniques and Applications of Antidifferentiation • Construct antiderivatives using the Fundamental Theorem of Calculus. • Solve initial value problems. • Evaluate indefinite and definite integrals by the method of substitution. • Use integration by parts to evaluate indefinite and definite integrals. • Solve problems involving logistic population growth. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Solve problems involving exponential growth and decay.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Numerical Approximations to Definite Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids.

Polynomial Approximations and Series Concept of Series • Define sequences explicitly or recursively. • Use properties of limits to find the limit of a sequence. • Determine whether a sequence converges or diverges. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. • Determine the absolute convergence, conditional convergence, or divergence of a power series at the endpoints of its interval of convergence.

Series of Constants • Determine the convergence or divergence of p-series, including the harmonic series. • Analyze the truncation error of a series using the Remainder Estimation Theorem. • Use the Integral Test and the Alternating Series Test to determine the convergence or divergence of a series of numbers. • Use the nth-Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of numbers or the radius of convergence of a power series. • Determine whether a sequence converges or diverges.

Taylor Series • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. • Use derivatives to find the Maclaurin series or Taylor series generated by a differentiable function. • Approximate a function with a Taylor polynomial. • Use the nth-Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of numbers or the radius of convergence of a power series. • Analyze the truncation error of a series using the Remainder Estimation Theorem.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Mathematics Objectives Calculus III

Vectors and 3D Space Geometry 3D Coordinates • Plot points in 3D coordinate systems. • Perform algebraic operations and transformations in 3D coordinate systems. • Use graphing software to visualize 3D points and transformations of these points. 3D Vectors • Plot and compute with 3D vectors. • Use the standard basis to represent vectors. • Create a vector from two points. Lines and Planes • Use the vector definitions of lines and planes. • Formulate and solve geometric problems involving lines and planes using vectors. Dot Product • Compute the dot product of two 3D vectors and use the related theorems. • Relate the magnitude of a vector to the dot product of the vector itself. • Use the magnitude of a non-zero vector to create a unit vector with the same direction. • Find the angle between two vectors. Cross Product • Compute the cross product of two 3D vectors and use the related theorems. • Use the algebraic properties of the dot product in combination with the cross product. • Relate the magnitude of the cross product to the area of the parallelogram made by the two vectors. • Use the cross product to find the angle between two vectors. Cylinders and Quadric Surfaces • Formulate and solve geometric problems involving lines, planes, cylinders, and quadric surfaces. • Plot lines, planes, cylinders, quadric surfaces, and figures constructed from these.

Space Curves 2D and 3D Space Curves • Write parametric and vector equations of space curves. • Use technology to graph space curves. Derivatives and Integrals • Compute componentwise limits, derivatives, and integrals of space curves. • Use technology to graph space curves. • Use space curves to model particle motion. 2D Arc Length and Curvature (Space Curves continued)

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

• Compute the arc length of curves in the plane. • Compute the classical curvature and radius of curvature of curves in the plane. 3D Arc Length and Curvature • Compute the arc length of 3D space curves. • Compute the classical curvature and radius of curvature of 3D space curves. 2D and 3D Motion • Use space curves to model 2D and 3D motion. • Interpret the appropriate derivatives as velocity and acceleration. Arc Length Reparameterization • Use various function to reparameterize a space curve. • Use unit speed reparameterizations to simplify analysis of 3D space curves. Frenet Frame Fields • Create moving Frenet Apparatus to represent intrinsic geometry of a space curve. • Use unit speed reparameterization to simplify computation of Frenet Apparatus. Curvature, Torsion, and the Frenet Apparatus • Find the rate of change of the Frenet Apparatus using the curvature and torsion of the space curve. • Relate the intrinsic geometry of a space curve to its curvature and torsion. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software.

Partial Derivatives Functions of Several Variables • Construct and compute with functions from n-dim real space to m-dim real space. • Interpret geometrical representation of functions from n-dim real space to m-dim real space. Limits and Continuity • Define and compute limits for functions from n-dim real space to m-dim real space. • Define and test for continuity of functions from n-dim real space to m-dim real space. Definition and Computation • Define and compute partial derivatives for functions from n-dim real space to m-dim real space. • Define and use partial derivative rules to compute partial derivatives. Tangent Planes • Use partial derivatives of a function from R^2 to R to define the tangent plane of an implicit surface in R^3. • Use differential notation to compute tangent plane approximation of an implicit surface at a given point. The Chain Rule • Use various forms of the chain rule of a function from R^n to R^m. • Use the chain rule to compute (partial) implicit derivatives. Directional Derivatives • Define and compute the directional derivative of functions from R^2 to R and R^3 to R. • Use the directional derivative to find the direction and mazimum rate of change of functions from R^2 to R and R^3 to R. • Use the gradient operator to define and compute the directional derivative. Maxima and Minima • Use partial derivatives to find local extrema (maxima, minima) in a given direction. • Use partial derivatives to find critical points of an implicitly defined surface. • Use the second partial derivative test (Hessian determinant) to analyze the geometry of an implicitly defined surface in terms of local extrema and saddle points. Lagrange Multipliers (Partial Derivatives continued)

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

• Use Lagrange multipliers to find maxima and minima of a function with respect to a given constraint. • Apply Lagrange multipliers to solve a variety of interesting problems taken from geometry, engineering, science, and economics. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software.

Vector Fields 2D Mappings and Plots • Define a 2D vector field F from R^2 and plot it as a vector attached to each point. • Given the plot of a 2D vector field, find or match an appropriate function that represents the geometry of the field. 3D Mappings and Plots • Define a 3D vector field F from R^3 and plot it as a vector attached to each point. • Given the plot of a 3D vector field, find or match an appropriate function that represents the geometry of the field. 2D Gradient, Divergence, and Curl • Define a 2D vector field as the gradient of a scalar function f from R^2 to R. • Recognize when a 2D vector field F is or is not the gradient of a scalar function f. • Use the gradient operator to define the divergence of a vector field. • Define the curl of a 2D vector field F. 3D Gradient, Divergence, and Curl • Define 3D vector field as the gradient of a scalar function f from R^3 to R. • Recognize when a 3D vector field F is or is not the gradient of a scalar function f. • Use the gradient operator to define the divergence and curl of a vector field. • Recognize how the curl of a 2D vector field F can be viewed as the curl of a 3D vector field with the z component function equal to zero. Algebraic Properties of Gradient, Divergence, and Curl • State the basic properties of the gradient, divergence, and curl operators and their combinations. • Recognize that the curl of a gradient vector field is always zero. • Recognize that the divergence of curl vector field is always zero. • Define the Laplacian operator using the gradient operator. Geometric Properties of Gradient, Divergence, and Curl • Interpret the divergence of a vector field in terms of the expansion or contraction of the field geometry. • Interpret the curl of a vector field in terms of the rotation (small paddle wheel) about an axis at each point. Physical Interpretation of Gradient, Divergence, and Curl • Apply the gradient, divergence, and curl appropriately in physical applications. • Use the properties of the gradient to determine temperature gradients. • Use the properties of the gradient to show that conservative fields, e.g., gravitational fields, are gradients of scalar functions. • Use the properties of the divergence and curl operators to represent, interpret, and use Maxwell’s Equations for electromagnetic fields. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Multiple Integrals Double Integrals • Define the double integral of a function f from R^2 to R as the volume over a rectangular region in the plane. • Compute the double integral as a double Riemann sum over a rectangular region. Double Integrals Over a General Region • Extend definition of double integral to the volume over a general region in the plane. • Compute double integral over general region using a double Riemann sum over a rectangular region containing the general region. Iterated Integrals • Define an iterated double integral using an iterated Riemann integral. • Use Fubini’s theorem to show that iterated double integral (in either order) is equivalent to the double integral defined over a general region. • Compute double integral over general region using various iterated integrals. Double Integrals in Polar Coordinates • Define an iterated double integral using an iterated Riemann integral in polar coordinates. • Use Fubini’s theorem to show that iterated double integral (in either order) in polar coordinates is equivalent to the double integral defined over a general region in polar coordinates. • Compute double integral over general region using various iterated integrals in polar coordinates. Applications of Double Integrals • Use double integrals to compute total mass, total charge, center of mass, and moment of inertia. Triple Integrals • Define the triple integral of a function f from R^3 to R as the hyper-volume over a rectangular box in R^3 (a general region in R^3 contained in a rectangular box). • Compute the triple integral as a triple Riemann sum over a rectangular box in R^3 (a general region in R^3 contained in a rectangular box). • Define an iterated triple integral using an iterated triple Riemann integral. • Use Fubini’s theorem to show that iterated triple integral (in any order) is equivalent to the triple integral defined over a general region in R^3. • Compute triple integral over general region in R^3 using various iterated triple integrals. • Interpret hyper-volume of triple integral f(x, y, z) = 1 as volume of a general region in R^3. • Use triple integrals to compute center of mass and moments of inertia in R^3. Triple Integrals in Cylindrical Coordinates • Change back and forth from rectangular coordinates to cylindrical coordinates. • Identify geometrical settings that are natural for cylindrical coordinates. • Formulate and compute triple integrals expressed in cylindrical coordinates. Triple Integrals in Spherical Coordinates • Change back and forth from rectangular coordinates to spherical coordinates. • Identify geometrical settings that are natural for spherical coordinates. • Formulate and compute triple integrals expressed in spherical coordinates. Change of Variables • Define a change of variable in R^2 (R^3) as a transformation T from R^2 to R^2 (R^3 to R^3) such that T is a 1-1 continuously differentiable function. • Use the Jacobian matrix determinant corresponding to a change of variable transformation T to rewrite and compute double and triple integrals. • Interpret the change from rectangular coordinates to polar coordinates in double integrals as a change of variable using an appropriate Jacobian matrix determinant. • Interpret the change from rectangular coordinates to cylindrical or spherical coordinates in triple integrals as a change of variable using an appropriate Jacobian matrix determinant.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Vector Calculus Line Integrals • Define path (line) integral along a space curve in R^2 (R^3). • Interpret path (line) integral as a generalization of an arc length integral. • Define the work done along a curve in terms of a path (line) integral. • Compute path (line) integrals using various techniques of integration. Fundamental Theorem of Line Integrals • State the Fundamental Theorem of line integrals using the gradient operator and the dot product. • Interpret the Fundamental Theorem of line integrals as a generalization of the Fundamental Theorem of Calculus. • Use the Fundamental Theorem of line integrals to compute path (line) integrals of vector fields that are gradients of scalar fields (conservative vector fields) and recognize the path independence. • For vector fields that represent physical forces, interpret path integrals as the work done along the path. Green’s Theorem • State Green’s Theorem relating the path (line) integral around a simple closed curve to the double integral over the enclosed region. • Interpret Green’s Theorem as a generalization of the Fundamental Theorem of Calculus for double integrals. • Use Green’s Theorem to simplify the computation of a difficult path (line) integral using a double integral. • Use Green’s Theorem to simplify the computation of a difficult double integral using a path (line) integral. Second Version of Green’s Theorem • Restate Green’s Theorem in terms of the curl and divergence operators. • Apply this form of Green’s Theorem to flows of (incompressible) vector fields. Parametric Surfaces • Write the parametric equations of a surface in R^3 using a smooth mapping from R^2 to R^3. • Interpret the parameterization of a surface geometrically as a function that maps a 2D (flat) region of the plane to a (curved) surface in 3D space. • Use technology to visualize parameterized surfaces. • Use double integrals to compute the area of parameterized surfaces. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software. Surface Integrals • Define a surface integral for a scalar field f mapping R^3 to where the surface S is contained in the domain of f and S is parameterized. • Compute surface integrals using appropriate parameterizations and double integrals. • Compute the surface integral of a vector field F over a surface S using the normal component of F with respect to S. Stokes’ Theorem • State Stokes’ Theorem for a smooth vector field F on R^3, which relates the path (line) integral of the tangential component of F around a simple closed boundary curve C of a surface S to the surface integral of the normal component of the curl of F over the enclosed surface S. • Interpret Stokes’ Theorem as a generalization of Green’s Theorem. • Use Stokes’ Theorem to simplify the computation of a difficult path (line) integral for the vector field F. • Use Stokes’ Theorem to simplify the computation of a difficult surface integral of the flux of the vector field F through the surface.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

• Define the circulation of a vector field F about a closed curve and use Stokes’ Theorem to relate it to the magnitude of the normal component of the curl of F. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software. Divergence Theorem • State the Divergence Theorem for a smooth vector field F on R^3, which relates the surface integral of the normal component of F over the surface S, e.g., the boundary surface of a region E of R^3, to the triple integral (volume integral) of the divergence of F over E. • Interpret the Divergence Theorem as a generalization of Green’s Theorem. • Use the Divergence Theorem to simplify the computation of a difficult surface integral for the vector field F. • Use the Divergence Theorem to simplify the computation of a difficult volume integral for the vector field F. Physical Applications of Path, Surface, and Volume Integrals • Apply path, surface, and volume integrals and their related theorems to problems in fluid dynamics and electrodynamics. • Use surface and volume integrals and their related theorems to state integral forms of Maxwell’s Equations for electromagnetic fields. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software. Introduction to Differential Forms • Re-interpret differentials and their products in terms of 1-forms, 2-forms, and 3-forms. • Compute exterior products of differential forms. • Restate Stokes’ Theorem in terms of differential forms. Introduction to the Gauss-Bonnet Theorem • Define the shape operator and Gaussian curvature of a smooth, orientable patch in R^3. • Compute the shape operator and Gaussian curvature of basic geometrical shapes in R^3. • Redefine Gaussian curvature in terms of 2-forms for geometrical (metric) surfaces in R^3. • State the Gauss-Bonnet Theorem which relates the total Gaussian Curvature of a compact, orientable, geometrical (metric) surface to its Euler characteristic with respect to any rectangular decomposition of the surface. • Complete a STEM Mini-Project involving applications of these concepts and using the Maple Mathematical Software.

Second-Order Differential Equations Second-Order Linear Equations • Construct the solution of a linear homogeneous differential equation by finding a basis for the solution space of the corresponding operator polynomial. • Solve linear homogeneous differential equations satisfying various initial and boundary conditions. Non-Homogeneous Linear Equations • Use the method of undetermined coefficients to solve linear inhomogeneous differential equations satisfying various initial and boundary conditions. • Use the method of variation of parameters to solve linear inhomogeneous differential equations satisfying various initial and boundary conditions. • Interpret both methods (Methods of Undetermined Coefficients and Method of Variation of Parameters) in terms of finding a basis for the solution space of an appropriately chosen operator polynomial.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Physical Applications • Apply the methods used to solve linear differential equation to represent and solve physical problems involving simple harmonic motion, including those with various forms of damped vibration. Series Solutions • Use power series methods to solve linear differential equations. • Extend power series methods to solve nonlinear differential equations.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Mathematics Objectives Advanced Placement Computer Science A

Object-Oriented Program Design • Design a simple Java class from user requirements. • Organize a Java class using a Class Box Summary. • Use Helper methods to organize a complex task. • Use Inheritance by extending a class. • Apply Polymorphism and override methods in a superclass. • Use Inheritance and Polymorphism with the design of the three required labs: Magpie Lab, Picture Lab and Elevens Lab.

Program Implementation • Write methods and constructors for a class. • Use visibility modifiers in a class. • Use instance variables and parameters to transmit data to methods. • Use static variables and methods in a class. • Use an Interface for a set of implementing classes. • Write an abstract class. • Write recursive sorting and searching methods. • Apply these principles and techniques to the three required labs: Magpie Lab, Picture Lab and Elevens Lab.

Program Analysis • Describe the steps in the software development process. • Create appropriate test cases for if statements and loops. • Design, implement, and test recursive methods. • Apply these principles and techniques to the three required labs: Magpie Lab, Picture Lab and Elevens Lab.

Standard Data Structures • Write code to instantiate Arrays (including two dimensional Arrays) and ArrayLists, and write methods to manipulate Arrays and ArrayLists. • Instantiate and use parallel arrays to sort information. • Apply these principles and techniques to the three required labs: Magpie Lab, Picture Lab and Elevens Lab. • Use advanced data structures in the DNA01-04 Labs and in the DNA Restriction Enzyme Mapping Project.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Standard Operations and Algorithms • Apply similarities and differences between recursive and iterative solutions to a problem. • Write nested if statements, nested if-else statements, and nested loops. • Analyze Search and Sort methods using recursive and iterative algorithms. • Apply these principles and techniques to the three required labs: Magpie Lab, Picture Lab and Elevens Lab.

Computing Systems • Find and utilize the Java Virtual Machine (JVM) as a part of Object-Oriented Programming in Java. • Use a variety of integrated development environments to create Java programs. • Use the DNA01-04 Labs to apply object oriented programming techniques to create solutions to computational biology problems in order to experience STEM applications. • Complete the DNA Restriction Enzyme Mapping Project in order to experience working on a real world STEM application.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015

Mathematics Objectives Advanced Placement Statistics

Exploring Data • Construct and interpret graphical displays of data (dot plot, stem plot, histogram, cumulative frequency plot). • Investigate and interpret categorical data in bar charts and frequency tables. • Compare distributions of univariate data (in dot plots, back-to-back stem plots, parallel box plots). • Summarize distributions of univariate data. • Explore, analyze, and transform bivariate data.

Sampling and Experimentation • Recognize different methods of data collection and bias. • Plan and conduct surveys utilizing appropriate sampling methods. • Plan and conduct a variety of experiments (e.g., completely randomized experiments, matched pairs experiments, randomized comparitive experiments). • Generalize results and types of conclusions that can be drawn from observational studies, experiments, and surveys.

Anticipating Patterns • Interpret probability, and differentiate discrete random variables and their distributions. • Investigate and analyze independent random variable combinations. • Apply concepts of the normal distribution (properties, use as a model). • Compare and analyze different sampling distributions (proportions, means, t, Chi-square).

Statistical Inference • Differentiate and assess estimation methods (point estimators and confidence intervals). • Hypothesize, conduct, and analyze tests of significance.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Mathematics Objectives Applied Technical Mathematics

Calculator Skills • Use the decimal, parentheses, negative, fix, and fraction keys to simplify expressions.

Bank Accounts • Compare and contrast banking account options. • Use and maintain a checking and savings account.

Ratio and Proportion • Simplify ratios and solve proportions. • Read and compare food prices using unit price.

Percent • Compute the cost of purchases, including sales tax and tip. • Calculate cost of purchases due to sale prices, coupons, and discounts. • Calculate simple interest.

Earning and Managing Income • Determine the value of stocks and compute gains and losses. • Compute gross pay and net pay. • Analyze and manage personal budgets.

Paying Taxes • Complete basic federal tax returns by calculating exemptions, deductions, taxable income, and taxes.

Managing a Household • Determine housing costs and monthly expenses in the context of apartment leasing. • Determine the price of purchasing a car, taking into account car options, financing charges, and liability insurance.

Area and Perimeter • Calculate area and perimeter of rectangles, circles, and triangles. • Estimate the quantity of materials and the costs associated with painting a room and covering a floor.

Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015