Virginia State Standards for Mathematics: Grade 12
A.1. The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.
A.2. The student will perform operations on polynomials, including
A.3. The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form.
A.4. The student will solve multistep linear and quadratic equations in two variables, including
A.5. The student will solve multistep linear inequalities in two variables, including
A.6. The student will graph linear equations and linear inequalities in two variables, including
A.7. The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including
A.8. The student, given a situation in a real-world context, will analyze a relation to determine whether a direct or inverse variation exists, and represent a direct variation algebraically and graphically and an inverse variation algebraically.
A.9. The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.
A.10. The student will compare and contrast multiple univariate data sets, using box-and-whisker plots.
A.11. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems, using mathematical models. Mathematical models will include linear and quadratic functions.
A.12. The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations.
A.13. The student will express the square root of a whole number in simplest radical form and approximate square roots to the nearest tenth.
A.14. The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.
A.15. The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph.
A.16. The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions.
A.17. The student will compare and contrast multiple one- variable data sets, using statistical techniques that include measures of central tendency, range, and box-and whisker graphs.
A.18. The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.
G.1. The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include
G.1.a) identifying the converse, inverse, and contrapositive of a conditional statement.
G.1.b) translating a short verbal argument into symbolic form.
G.1.c) using Venn diagrams to represent set relationships.
G.1.d) using deductive reasoning.
G.2. The student will use the relationships between angles formed by two lines cut by a transversal to
G.2.a) determine whether two lines are parallel.
G.2.b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs.
G.2.c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.
G.3. The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include
G.4. The student will construct and justify the constructions of
G.5. The student, given information concerning the lengths of sides and/or measures of angles in triangles, will
G.5.a) order the sides by length, given the angle measures.
G.5.b) order the angles by degree measure, given the side lengths.
G.6. The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.
G.7. The student, given information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs.
G.8. The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.
G.8.a) Investigate and identify properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles, and diagonals;
G.8.b) Prove these properties of quadrilaterals, using algebraic and coordinate methods as well as deductive reasoning; and
G.8.c) Use properties of quadrilaterals to solve practical problems.
G.9. The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems.
G.10. The student will solve real-world problems involving angles of polygons.
G.11. The student will use angles, arcs, chords, tangents, and secants to
G.12. The student, given the coordinates of the center of a circle and a point on the circle, will write the equation of the circle.
G.13. The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems.
G.14. The student will use similar geometric objects in two- or three-dimensions to
G.14.a) compare ratios between side lengths, perimeters, areas, and volumes.
G.14.b) determine how changes in one or more dimensions of an object affect area and/or volume of the object.
AII.1. The student, given rational, radical, or polynomial expressions, will
AII.2. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include sigma and a sub n.
AII.3. The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.
AII.3.a) Add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and
AII.3.b) Write radical expressions as expressions containing rational exponents and vice versa.
AII.4. The student will solve, algebraically and graphically,
AII.5. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.
AII.6. The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.
AII.7. The student will investigate and analyze functions algebraically and graphically. Key concepts include
AII.8. The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.
AII.9. The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.
AII.10. The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.
AII.11. The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.
AII.12. The student will compute and distinguish between permutations and combinations and use technology for applications.
AII.13. The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems.
AII.14. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions.
AII.15. The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions.
AII.16. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include S and an.
AII.17. The student will perform operations on complex numbers and express the results in simplest form. Simplifying results will involve using patterns of the powers of i.
AII.18. The student will identify conic sections (circle, ellipse, parabola, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations.
AII.19. The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions.
AII.20. The student will identify, create, and solve practical problems involving inverse variation and a combination of direct and inverse variations.
T.1. The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.
T.2 The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied.
T.3. The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.
T.4. The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.
T.5. The student will verify basic trigonometric identities and make substitutions, using the basic identities.
T.6. The student, given one of the six trigonometric functions in standard form, will
T.6.a) State the domain and the range of the function;
T.6.b) Determine the amplitude, period, phase shift, and vertical shift; and
T.6.c) Sketch the graph of the function by using transformations for at least a one-period interval.
T.7. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.
T.8. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.
T.9. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
AII/T.1. The student, given rational, radical, or polynomial expressions, will
AII/T.2. The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve real-world problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include sigma and a sub n.
AII/T.3. The student will perform operations on complex numbers, express the results in simplest form using patterns of the powers of i, and identify field properties that are valid for the complex numbers.
AII/T.3.a) Add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and
AII/T.3.b) Write radical expressions as expressions containing rational exponents and vice versa.
AII/T.4. The student will solve, algebraically and graphically,
AII/T.5. The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. Graphing calculators will be used as a tool to visualize graphs and predict the number of solutions.
AII/T.6. The student will recognize the general shape of function (absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic) families and will convert between graphic and symbolic forms of functions. A transformational approach to graphing will be employed. Graphing calculators will be used as a tool to investigate the shapes and behaviors of these functions.
AII/T.7. The student will investigate and analyze functions algebraically and graphically. Key concepts include
AII/T.8. The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.
AII/T.9. The student will collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems, using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.
AII/T.10. The student will identify, create, and solve real-world problems involving inverse variation, joint variation, and a combination of direct and inverse variations.
AII/T.11. The student will identify properties of a normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.
AII/T.12. The student will compute and distinguish between permutations and combinations and use technology for applications.
AII/T.13. The student, given a point other than the origin on the terminal side of an angle, will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of the angle in standard position. Trigonometric functions defined on the unit circle will be related to trigonometric functions defined in right triangles.
AII/T.14. The student, given the value of one trigonometric function, will find the values of the other trigonometric functions, using the definitions and properties of the trigonometric functions.
AII/T.15. The student will find, without the aid of a calculator, the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting angle measures from radians to degrees and vice versa.
AII/T.16. The student will find, with the aid of a calculator, the value of any trigonometric function and inverse trigonometric function.
AII/T.17. The student will verify basic trigonometric identities and make substitutions, using the basic identities.
AII/T.18. The student, given one of the six trigonometric functions in standard form, will
AII/T.19. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.
AII/T.20. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities.
AII/T.21. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
AII/T.22. The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied.
AII/T.23. The student will find without the aid of a calculating utility the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting radians to degrees and vice versa.
AII/T.24. The student will find with the aid of a calculator the value of any trigonometric function and inverse trigonometric function.
AII/T.25. The student will verify basic trigonometric identities and make substitutions, using the basic identities.
AII/T.26. The student, given one of the six trigonometric functions in standard form [e.g., y = A sin (Bx + C) + D, where A, B, C, and D are real numbers], will (The graphing calculator will be used to investigate the effect of changing A, B, C, and D on the graph of a trigonometric functions.)
AII/T.26.a) State the domain and the range of the function;
AII/T.26.b) Determine the amplitude, period, phase shift, and vertical shift; and
AII/T.26.c) Sketch the graph of the function by using transformations for at least a one-period interval.
AII/T.27. The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. Restrictions on the domains of the inverse trigonometric functions will be included.
AII/T.28. The student will solve trigonometric equations that include both infinite solutions and restricted domain solutions and solve basic trigonometric inequalities. Graphing utilities will be used to solve equations, check for reasonableness of results, and verify algebraic solutions.
AII/T.29. The student will identify, create, and solve practical problems involving triangles.
COM.1. The student will apply programming techniques and skills to solve practical real-world problems in mathematics arising from consumer, business, and other applications in mathematics. Problems will include opportunities for students to analyze data in charts, graphs, and tables and to use their knowledge of equations, formulas, and functions to solve these problems.
COM.2. The student will design, write, test, debug, and document a program. Programming documentation will include preconditions and postconditions of program segments, input/output specifications, the step-by-step plan, the test data, a sample run, and the program listing with appropriately placed comments.
COM.3. The student will write program specifications that define the constraints of a given problem. These specifications will include descriptions of preconditions, postconditions, the desired output, analysis of the available input, and an indication as to whether or not the problem is solvable under the given conditions.
COM.4. The student will design a step-by-step plan (algorithm) to solve a given problem. The plan will be in the form of a program flowchart, pseudo code, hierarchy chart, and/or data-flow diagram.
COM.5. The student will divide a given problem into manageable sections (modules) by task and implement the solution. The modules will include an appropriate user-defined function, subroutines, and procedures. Enrichment topics might include user-defined libraries (units) and object-oriented programming.
COM.6. The student will design and implement the input phase of a program, which will include designing screen layout and getting information into the program by way of user interaction, data statements, and/or file input. The input phase will also include methods of filtering out invalid data (error trapping).
COM.7. The student will design and implement the output phase of a computer program, which will include designing output layout, accessing a variety of output devices, using output statements, and labeling results.
COM.8. The student will design and implement computer graphics, which will include topics appropriate for the available programming environment as well as student background. Students will use graphics as an end in itself, as an enhancement to other output, and as a vehicle for reinforcing programming techniques.
COM.9. The student will define simple variable data types that include integer, real (fixed and scientific notation), character, string, and Boolean.
COM.10. The student will use appropriate variable data types, including integer, real (fixed and scientific notation), character, string, and Boolean. This will also include variables representing structured data types.
COM.11. The student will describe the way the computer stores, accesses, and processes variables, including the following topics: the use of variables versus constants, variables' addresses, pointers, parameter passing, scope of variables, and local versus global variables.
COM.12. The student will translate a mathematical expression into a computer statement, which involves writing assignment statements and using the order of operations.
COM.13. The student will select and implement built-in (library) functions in processing data.
COM.14. The student will implement conditional statements that include ''if/then'' statements, ''if/then/else'' statements, case statements, and Boolean logic.
COM.15. The student will implement loops, including iterative loops. Other topics will include single entry point, single exit point, preconditions, and postconditions.
COM.16. The student will select and implement appropriate data structures, including arrays (one-dimensional and/or multidimensional), files, and records. Implementation will include creating the data structure, putting information into the structure, and retrieving information from the structure.
COM.17. The student will implement pre-existing algorithms, including sort routines, search routines, and simple animation routines.
COM.18. The student will test a program, using an appropriate set of data. The set of test data should be appropriate and complete for the type of program being tested.
COM.19. The student will debug a program, using appropriate techniques (e.g., appropriately placed controlled breaks, the printing of intermediate results, other debugging tools available in the programming environment), and identify the difference between syntax errors and logic errors.
COM.20. The student will design, write, test, debug, and document a complete structured program that requires the synthesis of many of the concepts contained in previous standards.
PS.1. The student will analyze graphical displays of univariate data, including dotplots, stemplots, and histograms, to identify and describe patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology will be used to create graphical displays.
PS.2. The student will analyze numerical characteristics of univariate data sets to describe patterns and departures from patterns, using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers.
PS.3. The student will compare distributions of two or more univariate data sets, analyzing center and spread (within group and between group variations), clusters and gaps, shapes, outliers, or other unusual features.
PS.4. The student will analyze scatterplots to identify and describe the relationship between two variables, using shape; strength of relationship; clusters; positive, negative, or no association; outliers; and influential points.
PS.5. The student will find and interpret linear correlation, use the method of least squares regression to model the linear relationship between two variables, and use the residual plots to assess linearity.
PS.6. The student will make logarithmic and power transformations to achieve linearity.
PS.7. The student, using two-way tables, will analyze categorical data to describe patterns and departure from patterns and to find marginal frequency and relative frequencies, including conditional frequencies.
PS.8. The student will describe the methods of data collection in a census, sample survey, experiment, and observational study and identify an appropriate method of solution for a given problem setting.
PS.9. The student will plan and conduct a survey. The plan will address sampling techniques (e.g., simple random, stratified) and methods to reduce bias.
PS.10. The student will plan and conduct an experiment. The plan will address control, randomization, and measurement of experimental error.
PS.11. The student will identify and describe two or more events as complementary, dependent, independent, and/or mutually exclusive.
PS.12. The student will find probabilities (relative frequency and theoretical), including conditional probabilities for events that are either dependent or independent, by applying the Law of Large Numbers concept, the addition rule, and the multiplication rule.
PS.13. The student will develop, interpret, and apply the binomial probability distribution for discrete random variables, including computing the mean and standard deviation for the binomial variable.
PS.14. The student will simulate probability distributions, including binomial and geometric.
PS.15. The student will identify random variables as independent or dependent and find the mean and standard deviations for sums and differences of independent random variables.
PS.16. The student will identify properties of a normal distribution and apply the normal distribution to determine probabilities, using a table or graphing calculator.
PS.17. The student, given data from a large sample, will find and interpret point estimates and confidence intervals for parameters. The parameters will include proportion and mean, difference between two proportions, and difference between two means (independent and paired).
PS.18. The student will apply and interpret the logic of a hypothesis-testing procedure. Tests will include large sample tests for proportion, mean, difference between two proportions, and difference between two means (independent and paired) and Chi-squared tests for goodness of fit, homogeneity of proportions, and independence.
PS.19. The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means.
PS.20. The student will identify the meaning of sampling distribution with reference to random variable, sampling statistic, and parameter and explain the Central Limit Theorem. This will include sampling distribution of a sample proportion, a sample mean, a difference between two sample proportions, and a difference between two sample means.
PS.21. The student will identify properties of a t-distribution and apply t-distributions to single-sample and two-sample (independent and matched pairs) t-procedures, using tables or graphing calculators.
DM.1. The student will model problems, using vertex-edge graphs. The concepts of valence, connectedness, paths, planarity, and directed graphs will be investigated. Adjacency matrices and matrix operations will be used to solve problems (e.g., food chains, number of paths).
DM.2. The student will solve problems through investigation and application of circuits, cycles, Euler Paths, Euler Circuits, Hamilton Paths, and Hamilton Circuits. Optimal solutions will be sought using existing algorithms and student-created algorithms.
DM.3. The student will apply graphs to conflict-resolution problems, such as map coloring, scheduling, matching, and optimization. Graph coloring and chromatic number will be used.
DM.4. The student will apply algorithms, such as Kruskal's, Prim's, or Dijkstra's, relating to trees, networks, and paths. Appropriate technology will be used to determine the number of possible solutions and generate solutions when a feasible number exists.
DM.5. The student will use algorithms to schedule tasks in order to determine a minimum project time. The algorithms will include critical path analysis, the list-processing algorithm, and student-created algorithms.
DM.6. The student will solve linear programming problems. Appropriate technology will be used to facilitate the use of matrices, graphing techniques, and the Simplex method of determining solutions.
DM.7. The student will analyze and describe the issue of fair division (e.g., cake cutting, estate division). Algorithms for continuous and discrete cases will be applied.
DM.8. The student will investigate and describe weighted voting and the results of various election methods. These may include approval and preference voting as well as plurality, majority, runoff, sequential run-off, Borda count, and Condorcet winners.
DM.9. The student will identify apportionment inconsistencies that apply to issues such as salary caps in sports and allocation of representatives to Congress. Historical and current methods will be compared.
DM.10. The student will use the recursive process and difference equations with the aid of appropriate technology to generate
DM.10.a) Compound interest;
DM.10.b) Sequences and series;
DM.10.c) Fractals;
DM.10.d) Population growth models; and
DM.10.e) The Fibonacci sequence.
DM.11. The student will describe and apply sorting algorithms and coding algorithms used in sorting, processing, and communicating information. These will include
DM.11.a) Bubble sort, merge sort, and network sort; and
DM.11.b) ISBN, UPC, Zip, and banking codes.
DM.12. The student will select, justify, and apply an appropriate technique to solve a logic problem. Techniques will include Venn diagrams, truth tables, and matrices.
DM.13.a) The Fundamental (Basic) Counting Principle;
DM.13. The student will apply the formulas of combinatorics in the areas of
DM.13.b) Knapsack and bin-packing problems;
DM.13.c) Permutations and combinations; and
DM.13.d) The pigeonhole principle.
MA.1. The student will investigate and identify the characteristics of polynomial and rational functions and use these to sketch the graphs of the functions. This will include determining zeros, upper and lower bounds, y-intercepts, symmetry, asymptotes, intervals for which the function is increasing or decreasing, and maximum or minimum points. Graphing utilities will be used to investigate and verify these characteristics.
MA.2. The student will apply compositions of functions and inverses of functions to real-world situations. Analytical methods and graphing utilities will be used to investigate and verify the domain and range of resulting functions.
MA.3. The student will investigate and describe the continuity of functions, using graphs and algebraic methods.
MA.4. The student will expand binomials having positive integral exponents through the use of the Binomial Theorem, the formula for combinations, and Pascal's Triangle.
MA.5. The student will find the sum (sigma notation included) of finite and infinite convergent series, which will lead to an intuitive approach to a limit.
MA.6. The student will use mathematical induction to prove formulas and mathematical statements.
MA.7. The student will find the limit of an algebraic function, if it exists, as the variable approaches either a finite number or infinity. A graphing utility will be used to verify intuitive reasoning, algebraic methods, and numerical substitution.
MA.8. The student will investigate and identify the characteristics of conic section equations in (h, k) and standard forms. Transformations in the coordinate plane will be used to graph conic sections.
MA.9. The student will investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions and solve equations and real-world problems. This will include the role of e, natural and common logarithms, laws of exponents and logarithms, and the solution of logarithmic and exponential equations.
MA.10. The student will investigate and identify the characteristics of the graphs of polar equations, using graphing utilities. This will include classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.
MA.11. The student will perform operations with vectors in the coordinate plane and solve real-world problems, using vectors. This will include the following topics: operations of addition, subtraction, scalar multiplication, and inner (dot) product; norm of a vector; unit vector; graphing; properties; simple proofs; complex numbers (as vectors); and perpendicular components.
MA.12. The student will use parametric equations to model and solve application problems.
MA.13. The student will identify, create, and solve real-world problems involving triangles. Techniques will include using the trigonometric functions, the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.
APC.1. The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions, using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing.
APC.2. The student will define and apply the properties of limits of functions. Limits will be evaluated graphically and algebraically. This will include
APC.2.a) Limits of a constant;
APC.2.b) Limits of a sum, product, and quotient;
APC.2.c) One-sided limits; and
APC.2.d) Limits at infinity, infinite limits, and non-existent limits. AP Calculus BC will include l'Hopital's Rule, which will be used to find the limit of functions whose limits yield the indeterminate forms: 0/0 and 8 / 8.
APC.3. The student will use limits to define continuity and determine where a function is continuous or discontinuous. This will include
APC.3.a) Continuity in terms of limits;
APC.3.b) Continuity at a point and over a closed interval;
APC.3.c) Application of the Intermediate Value Theorem and the Extreme Value Theorem; and
APC.3.d) Geometric understanding and interpretation of continuity and discontinuity.
APC.4. The student will investigate asymptotic and unbounded behavior in functions. This will include
APC.4.a) Describing and understanding asymptotes in terms of graphical behavior and limits involving infinity; and
APC.4.b) Comparing relative magnitudes of functions and their rates of change.
APC.5. The student will investigate derivatives presented in graphic, numerical, and analytic contexts and the relationship between continuity and differentiability. The derivative will be defined as the limit of the difference quotient and interpreted as an instantaneous rate of change.
APC.6. The student will investigate the derivative at a point on a curve. This will include
APC.6.a) Finding the slope of a curve at a point, including points at which the tangent is vertical and points at which there are no tangents;
APC.6.b) Using local linear approximation to find the slope of a tangent line to a curve at the point;
APC.6.c) Defining instantaneous rate of change as the limit of average rate of change; and
APC.6.d) Approximating rate of change from graphs and tables of values.
APC.7. The student will analyze the derivative of a function as a function in itself. This will include
APC.7.a) Comparing corresponding characteristics of the graphs of f, f'', and f'';
APC.7.b) Defining the relationship between the increasing and decreasing behavior of f and the sign of f ';
APC.7.c) Translating verbal descriptions into equations involving derivatives and vice versa;
APC.7.d) Analyzing the geometric consequences of the Mean Value Theorem;
APC.7.e) Defining the relationship between the concavity of f and the sign of f ''; and
APC.7.f) Identifying points of inflection as places where concavity changes and finding points of inflection.
APC.8. The student will apply the derivative to solve problems. This will include
APC.8.a) Analysis of curves and the ideas of concavity and monotonicity;
APC.8.b) Optimization involving global and local extrema;
APC.8.c) Modeling of rates of change and related rates;
APC.8.d) Use of implicit differentiation to find the derivative of an inverse function;
APC.8.e) Interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and
APC.8.f) Differentiation of non-logarithmic functions, using the technique of logarithmic differentiation. This will include AP Calculus BC will also apply the derivative to solve problems. This will include
APC.8.f.a) Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors;
APC.8.f.b) Numericalal solution of differential equations, using Euler's method;
APC.8.f.c) l'Hopital's Rule to test the convergence of improper integrals and series; and
APC.8.f.d) Geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.
APC.9. The student will apply formulas to find derivatives. This will include
APC.9.a) Derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions;
APC.9.b) Derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions;
APC.9.c) Derivatives of implicitly defined functions; and
APC.9.d) Higher order derivatives of algebraic, trigonometric, exponential, and logarithmic, functions. AP Calculus BC will also include finding derivatives of parametric, polar, and vector functions.
APC.10. The student will use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values and will interpret the definite integral as the accumulated rate of change of a quantity over an interval interpreted as the change of the quantity over f'(x)dx = f(b) - f(a). Riemann sums will use left, right, and midpoint evaluation points over equal subdivisions.
APC.11. The student will find antiderivatives directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals). AP Calculus BC will also include finding antiderivatives by parts and simple partial fractions (nonrepeating linear factors only), and finding improper integrals as limits of definite integrals. AP Calculus BC will also solve logistic differential equations and use them in modeling.
APC.12. The student will identify the properties of the definite integral. This will include additivity and linearity, the definite integral as an area, and the definite integral as a limit of a sum as well as the fundamental theorem: d/dx[integral f(t) x d(t)] = f(x)
APC.13. The student will use the Fundamental Theorem of Calculus to evaluate definite integrals, represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined.
APC.14. The student will find specific anti-derivatives, using initial conditions (including applications to motion along a line). Separable differential equations will be solved and used in modeling (in particular, the equation y'=ky and exponential growth).
APC.15. The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include
APC.15.a) The area of a region;
APC.15.b) The volume of a solid with known cross-section;
APC.15.c) The average value of a function; and
APC.15.d) The distance traveled by a particle along a line. AP Calculus BC will include finding the area of a region (including a region bounded by polar curves) and finding the length of a curve (including a curve given in parametric form).
APC.16. The student will define a series and test for convergence of a series in terms of the limit of the sequence of partial sums. This will include
APC.16.a) Geometric series with applications;
APC.16.b) Harmonic series;
APC.16.c) Alternating series with error bound;
APC.16.d) Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series; and
APC.16.e) Ratio test for convergence and divergence. For those students who are enrolled in AP Calculus BC.
APC.17. The student will define, restate, and apply Taylor series. This will include
APC.17.a) Taylor polynomial approximations with graphical demonstration of convergence;
APC.17.b) Maclaurin series and the general Taylor series centered at x = a;
APC.17.c) Maclaurin series for the functions ex, sin x, cos x, and 1/(1 - x);
APC.17.d) Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, anti-differentiation, and the formation of new series from known series;
APC.17.e) Functions defined by power series;
APC.17.f) Radius and interval of convergence of power series; and
APC.17.g) Lagrange error bound of a Taylor polynomial. For those students who are enrolled in AP Calculus BC.