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Written by Susan Necroto, Sanford H. Calhoun High School Rosa Quagliata, Wellington C. Mepham High School Summer 20177 Supervised by James Morris District Mathematics Chair This course is specifically designed for students who have successfully completed Algebra 2, and will provide an in-depth study of all topics essential to the study of Calculus. Students will be actively engaged in problem solving, reasoning, connecting and communicating mathematically as they explore families of functions. Special emphasis will be on the Polynomial, Exponential, Logarithmic, and Trigonometric functions from numerical, graphical, and algebraic approaches. Additional topics to be investigated include Matrices, the Polar Coordinate System,, Vectors, Permutations, Combinations, the Binomial Theorem, and Limits. The Mathematical Practice Standards apply throughout this course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sensee of problem situations. All students will take a district final examination in June. Teachers may wish to combine lessons and objectives in order to allow for additional review and/or testing within a unit. Each unit includes additional days for exams and quizzes, althoughh it is not explicitly written. The remaining time, if any, should be used to review for the final exam. Table of Contents Unit Page Chapter 1: Functions and Their Graphs..... 4 Chapter 2: Polynomial Functions.........6 Chapter 4: Trigonometry..... 8 Chapter 5: Analytic Trigonometry......9 Chapter 6: Additional Topics in Trigonometry...10 Chapter 8: Matrices and Determinants....12 Chapter 10: Topics in Analytic Geometry (Conic Sections)...14 Chapter 10: Topics in Analytic Geometry (Polar Coordinates).....15 Chapter 9: The Binomial Theorem......16 Chapter 2: Rational Functions....177 Chapter 12: Limits and an Introduction to Calculus.....18 Chapter 12: Derivatives......19 Applications of Derivatives,...200 Integrals and Applications of Integrals........211 Limits and Continuity (Precalculus H only)..22 SUPPLEMENTAL MATERIAL....24 3
Chapter 1 Functions and Their Graphs (Approximately 12 days) Chapter Overview This unit studies functions and their graphs. Students will be able to find intercepts, determine symmetry, write the equation of a line, evaluate functions, determine the domain and the range, find zeros, determine if a function is even or odd, graph Parent Functions and their transformations, find the inverse of a function, and verify that functions are inverses of each other. Essential Question Why is it important to understand how things behave? Big Ideas Patterns Intercepts Symmetry Slope Parallel Perpendicular Behavior Function Domain Range Even Functions Odd Function Vertical Shift Horizontal Shift Non-rigid Transformation Inverse Function Day 1: Intercepts and Symmetry a. Finding the x and y intercepts (Example 4, page 16) b. Testing for symmetry with x- and y-axes and the origin (Example 5, page 18) Day 2: Domain a. Finding the domain of a function (Example 7, page 44) Day 3: Domain and Range, Finding Zeros a. Finding the domain and range of a function from a graph (Example 1, page 54) b. Finding the zeros of a function (Example 3, page 56) Day 4: Even and Odd Functions a. Determining if a function is even or odd (Example 8, page 60) (Chapter 1 continued on next page) 4
Day 5: Parent Graphs (This is a review of Algebra 1 and Algebra 2) a. Students will be able to graph without using a calculator: y x 2 y x 3 y x y y 1 x x ln y x b. Identifying the parent graph without a calculator (See Exercises on page 71, # s 19-42) c. Determine the domain and range of these functions and introduce the concept of horizontal and vertical asymptotes. (Asymptotes will be discussed in greater depth in a later chapter) Day 6: Graphing Piecewise Functions a. Evaluating piecewise functions (Example 3, page 70) Day 7: Composition of Functions g x f g 2 (Example 5 page 85) a. Finding f and b. Introduce composition notation f g x c. Finding the domain of a composite function (Precalculus H students should be able to do this without a calculator.) (Example 6, page 86) Day 8: Finding the inverse of a function (this is a review of Algebra 2) a. Show the relationships between the graphs of inverse functions (reflection over y x) (See Examples on page 94) b. Find the inverse of a function algebraically (Example 6, 7 pages 96-97) Day 9: Inverse Functions a. Prove that if f and g are inverse functions, then f g x g f x x (Example 2, page 93) Students should practice verifying that f and g are inverse functions. (Exercises 19-22, page 98) b. Show that the exponential and logarithmic functions are inverses of each other. (Example 5, page 229) Precalculus H students should be fluent (without the use of the calculator) with identifying key properties and the domain and range of the parent graphs, especially exponential and logarithmic functions. 5
Chapter 2 Polynomial Functions (Approximately 10 days) Chapter Overview Students will be introduced to higher order polynomial equations. The focus is on synthetic division, Descartes Rule of Signs and also The Rational Zero Test. Students will also be able to solve for all roots of a higher order polynomial equation as well as write the polynomial equation given the roots. Students will also understand how to draw a possible rational function given the nature of the roots. Essential Questions What real world occupations use polynomials? Name an arch in the shape of a parabola? Big Ideas Classifying Shapes Continuous Continuous Repeated Zero Synthetic Division Intermediate Value Theorem Complex Number Imaginary Number Complex Conjugates (A review of factoring techniques should be included here) Day 1: Synthetic Division a. Synthetic division (Example 4 page 153) b. Using the Remainder Theorem (Example 5 page 154) c. Factoring a polynomial repeated division (Example 6 page 155) d. Applying the leading coefficient test (Example 2 page 139) e. Precalculus H only The Intermediate Value Theorem (page 143) Day 2: Descartes s Rule of Signs a. Rule of Signs (Example 9 page 173) b. The Rational Zero Test (list p/q) (page 167) Day 3: Solving a Polynomial Equation with Irrational Roots a. Use Descartes s rule of signs and the Rational Zero Test to solve a polynomial equation with irrational roots (Example 5, page 169) Day 4: Solving a Polynomial with Imaginary Roots a. Use Descartes s rule of signs and the Rational Zero Test to solve a polynomial function with imaginary roots (Example 7, page 171) (Chapter 2 continued on next page) 6
Day 5: Solving a Polynomial with Repeating Roots a. Use Descartes s rule of signs and the Rational Zero Test to solve a polynomial function with repeating roots (Example 8, page 172) Day 6: Writing the Equation of a Polynomial a. Finding a polynomial with given zeros (Example 6, page 170) b. Finding a polynomial with given fractional zeros ( Precalculus H only) Day 7: Sketching a possible polynomial given the Nature of the Roots (Exercises 129 and 137, page 180) 7
Chapter 4 Trigonometry (Approximately 8 days) Chapter Overview Students will have a better understanding of reference angles, terminal sides, special right triangles, inverse trig functions and solving an equation for all angles that make the equation true. In an effort to ease the transition from Algebra 2 to Calculus, Precalculus H students will work with angles in radian measure only. In addition, all work in the Precalculus H class involving exact trigonometric values, exponent, and graphs should be done without the use of a calculator. Essential Question Other than the temperature, what other behaviors are cyclic? Big Idea Cyclic Repeating Patterns Degree Radian Unit circle Reference angles Inverse trig functions Exact trigonometric values Day 1: Radian and Degree Measures a. Sketching and finding coterminal angles (Example 1, page 282) b. Converting from radians to degrees (Example 3, page 283) c. Converting from degrees to radians (Example 4, page 283) Day 2: Trigonometric Functions: The Unit Circle a. Evaluating trigonometric functions (Example 1, page 294) b. Evaluation trigonometric functions (Example 2, page 295) Day 3: Right Triangle Trigonometry a. Evaluate the special angles for the trig functions (page 301) b. Fundamental trig identities (page 302) * Focus on questions on page 307 such as 21 30, 31(in radians), 32(in radians), and 57 62. Day 4: Trigonometric Functions of Any Angle a. Finding the reference angle (Example 4 (a, c only), page 312) b. Using reference angles (Example 5, page 314) Day 5: Inverse Trigonometric Functions a. Evaluating inverse trigonometric functions (Example 3, page 344) b. Using inverse properties (Example 5 (b, c only), page 345) 8
Chapter 5 -- Analytic Trigonometry (Approximately 10 days) Chapter Overview Students will learn how to use the fundamental identities to evaluate trigonometric functions, simplify trigonometric expressions, develop additional trigonometric identities, and solve trigonometric equations. Big Idea Simplification Identity Trigonometric Equation Day 1 Verifying Trigonometric Identities a. Proving Identities working on one side (Example 1 5 page 381 383) b. Proving Identities working on both sides (Example 6 page 383) Days 2, 3, 4, and 5: Solving Trigonometric Equations a. Collecting like terms (Example 1, page 388) b. Extracting square roots (Example 2, page 388) c. Factoring (Example 3, page 389) d. Factoring an Equation of Quadratic Type (Example 4, page 390) e. Rewriting with a single trigonometric function (Example 5, page 390) (the focus here is on the Pythagorean identities) f. Solve equations using the calculator (Examples 49 78 page 395) Precalculus H: Solving Trigonometric Equations a. Functions of multiple angles (Examples 7 and 8, page 392) This is a fluency topic for Precalculus H students, so give them several practice problems. (Additional days may be needed) They must be able to solve these equations without the use of a calculator! Day 6 Sum and Difference Formulas a. Use the sum and difference formulas for sin, cosine and tangent (Examples 3, 6, and 7, pages 399-401) Day 7 Double Angle Formulas a. Use the double angle formulas to solve equations (Examples 1 and 3, pages 405 406) 9
Chapter 6 -- Additional Topics in Trigonometry (Approximately 14 days) Chapter Overview Students will use trigonometry to solve applied problems. Students will develop the Law of Sines and the Law of Cosines in an effort to solve problems involving oblique triangles, triangles that do not have a right angle, and they will develop a formula for finding the area of a triangle using trigonometry. Students will also use vectors to model and solve real-life problems. Big Idea Direction Oblique Ambiguous Case Vector Complex Number Magnitude Resultant Force Day 1 Law of Sines a. Use the Law of Sines to find a side or an angle. (Examples 1-3 page 430 431) Day 2 The Ambiguous Case (Examples 3, 4, and 5, pages 430-431) a. Single Solution Case b. No Solution Case c. Two Solution Case Day 3 The Area of a Triangle a. Use the Law of Sines to solve for the area of a triangle (Example 6, page 432) Day 4 Law of Cosines a. Use the Law of Cosines to find a side or an angle (Examples 1 and 2, pages 437-438) Day 5 Complex numbers a. Plot the complex number (Examples 5 10 page 476) b. Fining the absolute value of a complex number (Example 1, page 468) c. Writing a complex number in trigonometric form (Example 2, page 469) d. Writing a complex number in standard form (Example 3, page 470) Day 6 Multiply and Divide Complex Numbers and Operations a. Multiple and Divide complex numbers (Examples 4 and5, page 471) b. Operations on complex numbers I. Addition and Subtraction (Example 1, page 160) II. Multiplication (Examples 2 and 3, page 161-162) III. Division (Example 4, page 162) (Chapter 6 continued on next page) 10
Day 7 Vector Representation a. Vector Representation by directed line segments (Example 1, page 445) Day 8 Component form of a Vector a. Finding the component form of a vector algebraic and graphic (Example 2, page 446) Day 9 Operations with Vectors a. Add/Subtract and Scalar (Example 3, page 448) b. Unit Vectors (Example 4, page 449) c. Vector Operation (Example 6, page 450) Day 10 Applications with Vectors a. Finding the component form of a vector (Example 8, page 452) Day 11 Solving Problems Involving Forces* a. Find the magnitude of the resultant force when given two forces. Example: Two forces of 25 and 15 pounds act on a body so that the angle between them is an angle of 75. Find the magnitude of the resultant force to the nearest pound. * There are no practice problems in the text for this lesson see AMSCO Course 3 Integrated Mathematics for additional problems. 11
Chapter 8 -- Matrices and Determinants (Approximately 12 days) Chapter Overview This chapter covers the addition, subtraction, and multiplication of matrices, finding a determinant and an inverse of a matrix, solving systems of equations using matrices, finding the area of a triangle, collinearity, and the equation of a line using matrices. Essential Question Why is it necessary to organize information in tables? Big Idea Order Organization Order of a matrix Square matrix Scalar multiplication Matrix multiplication Identity matrix of order n Inverse of a matrix Determinate of a matrix Day 1: Operations with Matrices a. Introduce matrices and find the order of matrices (Example 1, page 570) b. Equality of matrices (Example 1, page 584) c. Addition of matrices (Example 2, page 585) d. Scalar multiplication and subtraction of matrices (Example 3, page 586) e. Solving a matrix equation (Example 6, page 588) Days 2 and 3: Finding the Product of Two Matrices a. Multiplying matrices with and without the use of the calculator. Precalculus R students will be able to multiply matrices of order 2 2 without the use of a calculator. Precalculus H students will be able to multiply matrices of order 2 2 and 3 3a without the use of a calculator. b. Introduce the identity matrix (page 591) Day 4: Finding the Inverse of a Square Matrix a. Show that two matrices are inverses of each other (Example 1 page 599) b. Formula for the inverse of a 2 2 matrix (Example 4, page 603) (Chapter 8 continued on next page) 12
Day 5 and 6: Solving a System of Equations using Matrices a. Students should be able to solve a system of equations with two variables WITHOUT a calculator. Any system larger should be solved by using a calculator. (Example 5, page 604) Day 7: Finding the Determinant of a Matrix a. Definition of the determinant of a 2x 2 matrix WITHOUT a calculator (Example 1, page 609) b. Precalculus H Find the determinant of a 3 3 matrix WITHOUT the use of a calculator. (Example 3, page 611) Day 8: Applications of the Determinant a. Finding the area of a triangle using matrices (Example 3, page 619) b. Testing for collinear points (Example 4, page 620) Day 9: Finding the Equation of a Line using Matrices a. Using matrices to write the equation of a line (Example 5, page 621) 13
Chapter 10 -- Topics in Analytic Geometry: Conic Sections (Approximately 10 days) Chapter Overview Students will have a better understanding of conic sections. The students will be introduced to a focus of a parabola and an ellipse and a directrix for a parabola and asymptotes for a hyperbola Essential Questions What shape does the path of the planets around the sun make? What roles do the figures plan in architecture? What functions do conic sections play in the real world? Big Idea Shapes Navigation Modeling Directrix Focus Tangent Foci Vertices Major axis Minor axis Center Transverse axis Conjugate axis Day 1: Circles a. Review completing the square b. Finding the equation of a circle using matrices Day 2 and 3: Introduction to Conics: Parabolas (most is a review of Algebra 2) a. Vertex at the origin (Example 1, page 735) b. Finding the focus of a parabola (Example 2, page 735) c. Finding the standard equation of a parabola (Example 3, page 736) d. Writing the equation of a parabola using matrices Day 4 and 5: Ellipse a. Finding the standard equation of an ellipse (Example 1, page 744) b. Sketching an ellipse (Example 2, page 744) c. Analyzing an ellipse (Example 3, page 745) d. Discuss eccentricity (page 146) Day 6 and 7: Hyperbola a. Finding the standard equation of a hyperbola (Example 1 page 752) b. Using asymptotes to sketch a hyperbola (Example 2 page 753) c. Find the asymptotes of a hyperbola (Example 3 page 754) 14
Chapter 10 -- Topics in Analytic Geometry: Polar Coordinates (Approximately 7 days) Chapter Overview Students will be exposed to a new coordinate plane known as the polar plane in which the radius and the angle are plotted instead of the Cartesian plane where the x and y coordinate are graphed. Students will be able to find equivalent points and reflected points in the polar plane. Students will also be able to draw polar graphs in the polar plane. Essential Question What patters do microphones release at high frequencies? What kind of coordinates will help you navigate from point A to point B? Big Idea Shapes Locations Movements Navigation Polar coordinates Polar graphs Polar equations Day 1: Polar Coordinates a. Plotting points on the polar coordinate system (Example 1, page 777) b. Multiple representations of points (Example 2, page 778) Day 2: Polar Coordinates a. Rectangular to Polar (Example 4, page 779) Day 3: Polar Coordinates a. Polar to rectangular (Example 3, page 779) Day 4: Graphs of Polar Equations a. Graphing a polar equation by point plotting (Example 1, page 783) 15
Chapter 9 The Binomial Theorem (Approximately 9 days) Chapter Overview This chapter reviews the basic principles of permutations and combinations. The Binomial Theorem is developed and students learn to find the rth term in a binomial expansion. Essential Question How does Pascal s triangle help us to expand binomials? Big Idea Patterns Pascal s Triangle Counting Principle Permutation Factorial Notation Combination Day 1 Binomial Coefficients a. Introduce Pascal s Triangle (Example 3, page 683) b. Expanding a Binomial using Pascal s Triangle (Examples 4-6, pages 684-685) Day 2 Finding a Term in a Binomial Expansion a. Finding the rth term in a binomial expansion (Example 7, page 685) Day 3 Permutations a. Find the number of permutations of n elements (Example 5, page 691) b. Distinguishable Permutations (Example 7, page 693) Day 4 Combinations a. Combinations of n Elements taken r at a time (Example 8, page 694 and Example 9, page 695) 16
Chapter 2: Rational Functions (Approximately 7 days) Chapter Overview The students will have a better understanding of how to find the domain of a rational function as well as a better understanding of a vertical asymptote. The students will be able to find the horizontal asymptote and understand that this is the behavior of the function out at infinity. Essential Question How can data be displayed to understand an outcome? Big Ideas Behavior Rational function Vertical asymptote Numerical analysis Horizontal asymptote Point of discontinuity Day 1: Finding the Domain, and Vertical and Horizontal Asymptotes of a Rational Function a. Find the domain of a rational function (Example 1, page 181) b. Find the vertical and horizontal asymptotes (Example 2, page 183) Day 2: Sketching the Graph of a Rational Function a. Graphing where the horizontal asymptote is y 0 (Examples 3 and 5, page 185 and 186) Day 3: Sketching the Graph of a Rational Function a. Graphing where the horizontal asymptote is y a (Example 4, page 185) Day 4: Sketching the Graph of a Rational Function with Common Factors a. Graphing where the graph has a point of discontinuity (Example 6, page 186) 17
Precalculus R Only *(For Precalculus H, see page ) Chapter 12 -- Limits and an Introduction to Calculus (Approximately 8 days) Overview Students will be able to determine a limit from a function and a graph. Students will be introduced to the concept of continuity. Essential Questions If interest is compounded continuously, why is it important to be able to predict the end result of one s investment? Big Idea Graphing Prediction One sided limits Limits at infinity End behavior Closeness Continuous Days 1 and 2: The Concept of a Limit a. Estimating a Limit Numerically (Example 2, page 851) b. Evaluating one-sided limits (Example 6, page 865, and Examples 7 and 8, page 866) c. Using a graph to find a limit (Example 5, page 852) d. Discuss when a limit will fail to exist (Examples 6 and 7, page 853) Day 3: Direct Substitution a. Evaluating limits algebraically (Example 9, page 856, Example 10, page 857) b. Techniques for evaluating limits (Example 1, page 861, Example 2, page 862) Day 4: Direct Substitution a. Rationalizing Technique (Example 3, page 863) Day 5: Limits at Infinity a. Evaluating limit at infinity (Example 1, page 882 and Example 2, page 883) Day 6: Continuous Functions a. Determine if a function is continuous at a point SUPPLEMENTAL Exercises for section 1.4: #1 6 determine if the graph is continuous at c-value. Page 93 exercises 39 42: Determine if a function is continuous by sketching a piece-wise curve. 18
Precalculus R Only Chapter 12: Derivatives (Approximately 10 days) Chapter Overview Students will be introduced to the definition of the derivative which will lead to the power rule. Also the students will be introduced to the product, quotient and chain rule to evaluate and find derivatives of complex functions. The students will understand the idea of a tangent line and normal line to the graph using derivatives. Essential Questions If the demand of a product increased, what will happen to the supply of the product? Big Ideas Rates of change Velocity Slope Definition of the Derivative Power rule Product rule Acceleration Approximation Quotient rule Chain rule Tangent line Normal line Day 1: Definition of the Derivative a. Evaluating a limit from calculus (Example 9, page 867) b. Introduce the Power Rule Day 2: Slope of a Graph a. Visually approximating the slope of a graph (Examples 1 and 2 page 872, Examples 3 and 4, page 874 and Example 5, page 875) b. Use definition of the derivative or power rule to calculate the slope Day 3: Product Rule (SUPPLEMENTAL page 127) Day 4: Quotient Rule (SUPPLEMENTAL page 129) Day 5: Chain Rule (SUPPLEMENTAL page 137) Day 6: Chain with Product and Quotient (SUPPLEMENTAL EXERCISES for Section 2.4) Day 7: Equation of the Tangent Line a. See exercises on page 878 # 43 50 b. See exercise on page 879 # 55 70 Day 8: Equation of the normal line a. Use the same Exercises on page 878 #43-50. (Change instructions to find the equation of the normal line.) 19
Precalculus R Only Applications of Derivatives (Approximately 8 days) Overview Students will be introduced to finding the maximum, minimum, intervals of increasing, intervals of decreasing, point of inflection, intervals of concave up and intervals of concave down of a function using the number line test. Students will also be introduced to perimeter, area and volume problems using calculus. Essential Question Why is it important for a company to maximize the amount of production while minimizing the cost of the item? Big Idea Optimization Increasing Decreasing Extrema Points of inflection Concave up Concave down Day 1: Increasing and Decreasing Functions a. Discuss the connection between increasing and decreasing and f (x). b. Discuss maximum and minimum points. (SUPPLEMENTAL pages 179-186) Days 2 and 3: The First Derivative Test a. Use the first derivative test to determine max and min points. b. Use the first derivative test to determine intervals of increasing and decreasing. (SUPPLEMENTAL pages 179-186) Days 4 and 5: Concavity and the Second Derivative Test a. Define concavity b. Discuss the connection between concavity and f (x). c. Define a point of inflection d. Discuss the second derivative test e. Use the second derivative test to find points of inflection and intervals of concave up and concave down. (SUPPLEMENTAL pages 189-194) Day 6: Optimization Maximize/Minimize - Area/Perimeter (SUPPLEMENTAL pages 213-214 and EXERCISES for Section 3.7) Day 7: Optimization Volume of a box (SUPPLEMENTAL pages 213-214 and EXERCISES for Section 3.7) 20
Precalculus R Only Integrals and Applications of Integrals (Approximately 8 days) Overview The students will be introduced to finding the anti-derivative of simple functions. Students will understand the difference between an indefinite integral and a definite integral. Students will also be introduced to finding the area under one and two curves. Essential Question What is the total distance traveled from point A to point B? Big Ideas Distance Anti-derivative Area Indefinite Integral Definite integral Days 1 and 2: Finding Anti-derivatives a. Use the formula to find an antiderivative (SUPPLEMENTAL pages 251-253 and EXERCISES for Section 4.1) Day 3 and 4: Evaluating Definite Integrals and Area a. Evaluate a definite integral (SUPPLEMENTAL pages 282 and 289) b. Use an integral to find the area under a curve. Day 5: Find the Area of a Region Between Two Curves a. Determine the area between two curves (SUPPLEMENTAL Section 6.1 pages 403-406 and EXERCISES for Section 6.1) 21
Precalculus H Only The precalculus H Students will Oscillating Behavior Example 8 page 854 Limits Unit AP Calc Style (Approximately 13 days) Day 1 Graphs with Limits Day 2 Evaluating Limits Algebraically Day 3 Evaluating Limits at Infinity Day 4 Sketching a Piecewise Curve Day 5 Continuity at an Endpoint and Intermediate Value Theorem Day 6 An Application of Continuous Functions Day 7 Trig Limits Day 8 Practice Day on all different limits (AP Style) Day 9 Some Interesting Limit Problems Day 10 More on Limits with graphs 22