Trinity Christian School Curriculum Guide

Transcription

1 Course Title: Calculus Grade Taught: Twelfth Grade Credits: 1 credit Trinity Christian School Curriculum Guide A. Course Goals: 1. To provide students with a familiarity with the properties of linear, quadratic, power, exponential, and logarithmic functions. 2. To provide students with a familiarity with sequences, limits, manipulation of algebraic expressions, solutions to equations and inequalities. 3. To provide students with a familiarity with formulas for areas and volumes for figures both with and without coordinates. 4. To introduce discrete mathematics involving logic, recursion, mathematical induction, combinatorics, graphs and circuits. B. Course Topics: The teacher will teach the following: Topic Instructional Materials Measurement Time ( in periods ) 1. Mathematical and Logical Reasoning. Identify forms and properties of logical statements; use the Law of Substitution to verify specific statements; determine whether arguments are valid or invalid; use logic to prove statements. 2. Analyzing Functions. Determine relative minima and maxima and intervals on which a function is increasing or decreasing; rewrite exponential and logarithmic expressions; identify domain, range, and minimum and maximum values of functions; determine the end behavior of functions. Quizzes and Tests 15 89

2 3. Functions, Equations, and Inequalities. Describe the sum, difference, product, quotient, and composite of two given functions; find zeros of functions using factoring or chunking; analyze the reversibility of steps used in solving equations; Intermediate Value Theorem. 4. Integers and Polynomials. Quotient-Remainder Thm.; Divide polynomials; congruence of integers with given modulus; Factor polynomials over the reals; Use the Factor Thm. to solve polynomial equations; Use Proof by Contradiction; Use Factor Search Thm. and the Fundamental Thm. of Arithmetic to determine primes and prime factorizations; Use Quotient- Remainder Thm. to solve applied problems 5. Rational Numbers and Rational Functions. Simplify rational expressions; identify numbers as rational or irrational; identify rational functions and their domains; prove properties of rational and irrational numbers; use limit notation to describe behavior of rational functions; classify discontinuities as essential or removable; apply rational expressions and rational equations. 6. Trigonometric Identities and Equations. Without a calculator, use trigonometric identities to express values of trigonometric functions in terms of rational numbers and radicals; solve trigonometric equations and inequalities algebraically; prove trigonometric identities and identify their domains; solve problems using inverse trigonometric functions; find an equation for the image of a graph under a transformation. Quizzes and Tests 16 90

3 7. Recursion and Mathematical Induction. Conjecture explicit formulas for recursively defined sequences; rewrite sums recursively; evaluate a finite or infinite geometric series; prove that a recursively defined sequence has a particular explicit formula; prove statements using the Principle of Mathematical Induction; use recursive formulas to solve problems; execute algorithms on sets of numbers. 8. Polar Coordinates and Complex Numbers. Express complex numbers in binomial, rectangular, polar, and trigonometric form; perform operations with complex numbers; convert between polar and rectangular representations of points; find powers and roots of complex numbers; find all zeros and their multiplicities of polynomials; use complex numbers to solve AC circuit problems; graph complex numbers; sketch graphs of polar equations; 9. The Derivative in Calculus. Average Rates of Change of functions; use definition of derivatives to compute derivatives; use derivatives to identify properties of functions; find rates of change in real situations; use derivatives to find velocities and acceleration; use derivatives to solve optimization problems; estimate derivatives by finding slopes of tangent lines; determine properties of derivatives from the graph of a function. Quizzes and Tests 15 Quizzes and Tests 16 Quizzes and Tests 10 91

4 10. Combinatorics. Define the essential features of counting problems; evaluate expressions indicating permutations or combinations; apply the Binomial Theorem to expand binomials; use properties of permutations and combinations to prove identities; use the Multiplication Counting Principle and permutations to solve counting problems; use combinations and the Binomial Theorem to solve counting problems; find binomial probabilities in realistic situations. 11. Graphs and Circuits. Draw graphs given sufficient information; identify parts and types of graphs; determine whether there exists a graph containing vertices with given degrees; determine whether a graph has an Euler Circuit; use graphs to solve scheduling and probability problems; use the Total Degree of a Graph Theorem to solve handshake problems; solve application problems involving circuits; use stochastic matrices to make long-term predictions; 12. Vectors. Find the magnitude and direction of twodimensional vectors; find sums, opposites, scalar products and dot products of two-dimensional vectors; find sums, lengths, scalar products, dot products, and cross products of vectors in 3-space; find the measure of the angle between two vectors; identify parallel and orthogonal vectors; use vectors in a plane to decompose motion or force into x- and y- components; use addition of vectors in a plane to solve problems involving forces or velocities; represent two-dimensional vectors in their component or polar representation; represent addition, subtraction, and scalar multiplication of vectors graphically; geometrically interpret three-dimensional Quizzes and Tests 13 92

5 vectors and their operations. 13. The Integral in Calculus. Calculate Riemann sums of functions over specified intervals; evaluate definite integrals; apply properties of definite integrals; find the distance traveled by a moving object given its variable rate; use the definite integral to solve application problems; express areas in integral notion; find areas bounded by curves; find volumes of solids. C. Student Materials: 1. Discrete Mathematics 2. TI-83-Plus Quizzes and Tests 13 D. Teacher Materials: 1. Discrete Mathematics - Teacher Edition 2. Teacher s Resource File CD-ROM version 3. TI-83-Plus E. Classical Methodology 1. Students should be able to develop problem solving skills as they are introduced to abstract thinking in mathematics. 2. Students should develop a desire to understand mathematics around us. 3. Students should see the applications of math to life situations. 93