Document Sample

Course Title: Advanced Precalculus, Level 5 Grade: 11 Length of Course: One Year (5 credits) Prerequisites: Algebra, Geometry, Algebra 2 Description: For the serious cyclist, keeping focus on the road ahead is essential. Precalculus instruction should use a similar approach. In keeping with this thought we concentrate on those topics which are essential for success in calculus, emphasizing depth rather than breadth. A central theme of this course is functions as models of change. Functions express the way one variable quantity is related to another quantity. This course emphasizes that functions can be grouped into families and that functions can be models for real world data. Once introduced a family of functions is compared and contrasted with other families of functions. Linear, exponential, power, logarithmic, trigonometric and rational functions are covered in depth. Recognizing that for some, precalculus can be a capstone course, inclusion of quantitative literacy topics such as combinatorics, probability, sequences and series has been provided. This class we will be guided by two principles. The first is The Rule of Three which requires that every topic be presented geometrically, numerically and algebraically. The second guiding principle is The Way of Archimedes which states that formal definitions and procedures evolve from the investigation of practical problems. The problems we consider come from the both the Natural and Social Sciences, as well as Business arenas and are generally understood to be important. The graphing calculator is a marvelous tool which this course employs both as a problem solver and an exploratory tool to anticipate upcoming concepts. The National Council of Teachers of Mathematics states: “Some mathematics becomes more important because technology requires it; some mathematics becomes less important because technology replaces it. Some mathematics becomes possible because technology allows it.” Throughout our studies appropriate use of technology is incorporated. This course strives to give students a proper balance between the mastery of skills and the comprehension of key concepts. With that in mind, this curriculum guide clearly defines the learning objectives for each unit in terms of the key skills and key concepts that must be mastered within each unit. Evaluation: Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as a common departmental Midterm and Final Exam. Assessments will equally emphasize measurement of the degree to which required skills have been mastered as well as how well key concepts have been understood. Text: Precalculus with Trigonometry, Concepts and Applications, Paul A. Foerster, Key Curriculum Press 2003 Reference Texts: Advanced Mathematics, Richard G. Brown, Houghton-Mifflin 1992 Advanced Precalculus Grade 11 Unit 1: Functions & Mathematical Models Topic Learning Objectives: Key Definitions, Skills and Concepts 1.1 Functions: What is a function? What are the four ways a function can be represented? What are the domain and range of a Algebraically, function? Numerically, Graphically and Skills check, ability to: Verbally Given the graph of a function, be able to answer questions regarding function values at specific inputs, find its domain and range Given a table of values for a function, be able to graph and find domain and range Given an equation of a function, be able to graph, find function values at specific inputs and find domain and range Given anecdotes, be able to sketch appropriate graphs Concept check: Given that altitude is a function of time, is it always, sometimes or never true that time is a function of altitude? What is the difference between interpolation and extrapolation? Why is it important to recognize these differences? Does all real life data form a function? 1.2 Kinds of Functions What is the mathematical definition of a function? How do you use Euler notation to describe a function? What are some of the functions families that have already been studied? How do we use Boolean logic to restrict the domain of a function on a graphing calculator? What is the vertical line test? Skills check, ability to: Given the equation of a function and a restricted domain, create a graph on the calculator and state its range and intercepts Given the sketch of a function, identify the family to which it most likely belongs Given two quantities that are related, create a reasonable sketch Use the Vertical Line Test to determine if a sketch is that of a function Concept check: Explain how the vertical line test is a graphical interpretation of the Golden Rule of Functions. Grade 11 Advanced Precalculus, page 2 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 1.3 Dilations and What are dilations and translations? How are these accomplished graphically and algebraically? Translations of Function Graphs Skills check, ability to: Draw the graph of f(x c), f(cx), cf(x), and f(x) c, given the graph of f Given the equation of f(x), write the equations for g(x) if g(x) is a dilation and/or translation of f(x) Concept check: What is the benefit of understanding about transformations of functions? What are the real world applications of composition of functions? If c > 0, why does f ( x c) shift the graph of f (x) to the right and not to the left as one might expect? 1.4 Composition of What is composition of functions? What is the domain of a composed function? How can the graphing calculator Functions be used to display composed functions? Skills check, ability to: Evaluate f(g(c)) given rules, graphs or tables for f and g Find f(g(x)) given rules for f and g Find the domain of f(g(x)) given domains for f and g Express a complicated function as the composition of easier functions Concept check: Is the function f(g(x)) the same as g(f(x))? Why or why not? If f(x) is linear and g(x) is linear, must f(g(x)) also be linear? Why or why not? Grade 11 Advanced Precalculus, page 3 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 1.5 Inverse of What is the definition of an inverse function? Do all functions have inverse functions? How can you tell if a Functions function will have an inverse that is also a function? What is the horizontal like test? How can you graphically construct inverses? What is a one-to-one function? How can you algebraically construct inverses? What are parametric equations? Skills check, ability to: Test for whether a given function is one-to-one and the existence of an inverse function (horizontal line test) Verify whether two functions are inverses Find the inverse of a one-to-one function (algebraic, graphical, and numerical method) and its corresponding domain and range Use the parametric mode on a graphing calculator to plot a function and its inverse Concept check: How can a function that is not one-to-one, have an inverse function? In the expression f 1 (x), is -1 an exponent? Can a discrete function that is not always monotonic have an inverse? Why or why not? Identify real world examples where change should most accurately be described using a parameter 1.6 Reflections, How are reflections over the coordinate axes accomplished using transformations? What are even and odd Absolute Values and functions? What is the Greatest Integer Function? What is a piecewise function? Other Transformations Skills check, ability to: Given the graph of f(x), sketch f(-x). –f(x), f ( x) and f( x ) Sketch the Greatest Integer Function and identify the places where it has a step discontinuity Sketch piecewise functions Concept check: Is every function even or odd? What is the visual impact of a function that is even/odd? Create an anecdote that would result in a step function Grade 11 Advanced Precalculus, page 4 Advanced Precalculus Grade 11 Unit 2: Periodic Functions and Right Triangle Problems Topic Learning Objectives: Key Definitions, Skills and Concepts 2.2 Measurement of How are angles drawn on the Cartesian Plane? What is standard position of an angle? What does a negative Rotation angle measure mean? What are quadrantal angles? What are coterminal angles? What is a reference angle? Skills check, ability to: Given an angle in degrees, both positive and negative, correctly sketch it in standard form, identify its quadrant, and its reference angle Correctly interpret subdivisions of a degree (minutes/seconds) Concept check: True/False: The reference angle of 90º is 0º True/False: The position of the terminal ray of an angle drawn in standard position is unique to the angle measure Find a formula for all angles coterminal with a given angle 2.3 Sine and Cosine What is a periodic function? What are the definitions of the Sine and Cosine Functions on the Unit Circle? Functions Skills check, ability to: Correctly find the sine & cosine of an angle using a calculator Correctly find the sine and cosine of an angle given a point on its terminal ray Correctly sketch the graph of the sine and cosine function and identify its domain and range Correctly sketch transformations of the sine and cosine functions Give the exact values of the sine and cosine functions for the quadrantal angles Give the exact values of the sine and cosine functions for the special angles (30º, 60º, 45º, 120º, etc) Concept check: Explain why the function vales for the sine function change sign where they do Where is the sine function increasing/decreasing? Why is this happening? State the quadrants where one would expect to see a positive/negative sine. Explain how you arrived at your answer. Why is range of both the sine and cosine functions [-1, 1]? Why doesn’t the radius of the circle influence the values of the sine and cosine functions? Draw the BOX diagram for the sine and cosine functions showing input, output and rule True/False: If A and B are angles of a triangle such that A > B, then cos A > cos B. Justify your answer. Grade 11 Advanced Precalculus, page 5 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 2.4 Values of the Six What are the definitions of the six trig functions? Trigonometric Functions Skills check, ability to: Correctly state the exact values of the six trig functions of special and quadrantal angles Correctly state the values of the six trig functions of an angle given a point on its terminal ray Correctly identify the domain and range of the six trig functions Correctly state the values of five trig functions of an angle given the quadrant in which its terminal side lies and the value of one of its trig functions Concept check: Explain where and why each of the six trig functions are increasing/decreasing State the quadrants in which one would expect to find positive/negative functions. Justify your answer. Which of the following does not represent a real number: A) sin 30º B) tan 45º C) cos 90º D) csc 90º E) sec 90º Before calculators became common classroom tools, students used trig tables to find trigonometric ratios. Below is a simplified trig table for angles between 40º and 50º. Without using a calculator, can you determine which column gives sine values, which gives cosine values and which gives tangent values? Justify your answers. Angle ? ? ? 40º 0.8391 0.6428 0.7660 42º 0.9004 0.6691 0.7431 44º 0.9657 0.6947 0.7193 46º 1.0355 0.7193 0.6947 48º 1.1106 0.7431 0.6691 50º 1.1917 0.7660 0.6428 Grade 11 Advanced Precalculus, page 6 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 2.5 Inverse Trig What is the principal branch of the sine and cosine functions? What does the notation sin 1 (x) = mean? Functions and Triangle What is an angle of elevation/depression? Problems Skills check, ability to: Correctly use a calculator to find sin 1 (c) Correctly solve right triangle problems that require finding sides and/or angles Concept Check: Why must we use the principal branch of the periodic functions to develop an inverse function? Why isn’t the entire sine function invertible? State the domain and range of the inverse sine and cosine functions. Justify your answer Draw the BOX diagram for the inverse sine and cosine functions showing input, output and rule Why is sin 1 (2) undefined? What inputs to the inverse sine BOX result in an undefined output? Why? Why isn’t cos 1 (-.6) a negative number? To get a rough idea of the height of a building, John paces off 50 feet from the base of the building, and then measures the angle of elevation to the top of the building to be 58º. About how tall is the building? Justify your choice. A) 31 feet B) 42 feet C) 59 feet D) 80 feet E) 417 feet Grade 11 Advanced Precalculus, page 7 Advanced Precalculus Grade 11 Unit 3: Applications of Trigonometric and Circular Functions Topic Learning Objectives: Key Definitions, Skills and Concepts 3.2 General Sinusoidal What is a sinusoidal graph? What do the terms period, amplitude, cycle, frequency and phase shift mean relative Graphs to the graph of a sinusoid? What is concavity? What are points of Inflection? What are critical points? Skills check, ability to: Correctly graph and state pertinent information given the equation of a sinusoid Correctly state pertinent information and the equation given the graph of a sinusoid Correctly identify the equation and graph a sinusoid given pertinent information Concept check: True/False: The equation of a sinusoid is unique. Why or why not? 3.3 Graphs of Tangent, What are the graphs of the remaining trig functions? Where do they have asymptotes? What is the Quotient Cotangent, Secant and Property for Tangent and Cotangent? Cosecant Functions Skills check, ability to: Correctly graph and state pertinent information given the equation of any trig function Correctly state pertinent information and the equation of any trig function given its graph Correctly identify the equation and graph any trig function given pertinent information Correctly identify domain and range for all the trig functions Correctly sketch transformations for all the trig functions Concept check: Explain how the domain and range of the trig functions are established Are the discontinuities seen in the graphs of the trig functions step discontinuities? Why or why not? Explain how one could accurately draw the graph of the Cosecant function given the graph of the Sine function. What is the amplitude of the Tangent function? Explain. Grade 11 Advanced Precalculus, page 8 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 3.3 Graphs of Tangent, The graph of y = csc has the same set of asymptotes as the graph of y = Cotangent, Secant and A) sin Cosecant Functions B) tan (cont’d) C) cot D) sec E) csc 2 The graph of y = sec never intersects the graph of y = A) B) 2 C) csc D) cos E) sin If k 0, what is the range of the function y = k csc ? A) [-k, k] B) (-k, k) C) (- , -k) (k, ) D) (- , -k) [k, ) 1 1 E) (- , ] [ ,) k k Grade 11 Advanced Precalculus, page 9 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 3.4 Radian Measurement of What is a radian? How does one convert from radians to degrees and degrees to radians? What is the Angles relationship between radian measure and arclength? Skills check, ability to: Correctly convert between degrees and radians Express special angles and quadrantal angles as radians Correctly find the trig functions of angles expressed as radians Concept Check: True/False: The radian measure of all three angles in a triangle can be integers. Justify your answer. If the perimeter of a sector is 4 times its radius then the radian measure of the central angle of the sector is: A) 2 B) 4 2 C) 4 D) E) Impossible to determine without knowing the radius A central angle in a circle of radius r has a measure of x radians. If the same central angle was drawn is a circle of radius 2r, then its radian measure would be: x A) 2 x B) 2r C) x D) 2x E) 2rx Grade 11 Advanced Precalculus, page 10 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 3.5 Circular Functions What is a circular function and how is it different from a trig function? What is the argument of a circular function? Skills check, ability to: Correctly find the arclength of a circle subtended by an angle in radians Correctly sketch the graph of circular functions (both original and transformed) given its equation Correctly find the equation of a circular function given its graph Concept Check: Draw the BOX diagram for the circular functions showing input, output and rule. Be specific as to the input. Explain why the word wrapping is appropriate. True/False: The values of sin x and sin (x + 2 ) are always the same. Justify your answer. The period of the function f(x) = 210sin(420x+840) is A) 840 B) 420 C) 210 210 D) 420 E) A sinusoid with amplitude 4 has minimum value of 5. Its maximum value is _______. The graph of y = f(x) is a sinusoid with period 45 passing thru the point (6,0). Which of the following can be determined from the given information? I. f(0) II. f(6) III. f(96) A) I only B) II only C) I and III only D) II and III only E) I, II and III Grade 11 Advanced Precalculus, page 11 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 3.6 Inverse Circular What is the Arccosine relation and how is it different from the Inverse Cosine function? Relations: Given y, Find x Skills check, ability to: Given the equation of a circular or trigonometric function and a particular function value correctly find the value of x or either graphically, numerically or algebraically Concept Check: Graph each of the following functions and interpret the graph to find the domain, range and period of each function. Which of the three functions has points of discontinuity? Are the discontinuities removable or nonremovable? A) y = sin 1 (sin x) B) y = cos 1 (cos x) C) y = tan 1 (tan x) 3.7 Sinusoidal What is a mathematical model? Functions as Mathematical Models Skills check, ability to: Correctly interpret real world data to form a mathematical model Use this model to answer questions about the data Use this model to make reasonable predictions about the future Concept check: Grade 11 Advanced Precalculus, page 12 Advanced Precalculus Grade 11 Unit 4: Trigonometric Function Properties, Identities and Parametric Functions Topic Learning Objectives: Key Definitions, Skills and Concepts 4.2 Pythagorean, What are the Pythagorean, Reciprocal and Quotient Properties? Reciprocal and Quotient Properties Skills check, ability to: Correctly express one trig function in terms of another or others using the appropriate properties Concept check: On the assumption that one knows sin 2 x + cos 2 x = 1, explain how the other forms of the Pythagorean Identity can be derived. Explain how knowing the Quotient Property will allow one to determine the asymptotes of the Tangent and Cotangent functions. Graph the functions y = sin 2 x and y = -cos 2 x in the same viewing window. Describe the apparent relationship between the two graphs and verify it with a trigonometric property. Grade 11 Advanced Precalculus, page 13 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 4.3 Identities and What is the difference between an equation and identity? How can identities help to transform a complicated trig Algebraic expression into a simpler one? Transformations of Expressions Skills check, ability to: Correctly transform a trig expression into another, simpler, trig expression Correctly demonstrate algebraically that a given equation is an identity Concept check: Why is it “illegal” to work on both sides of an equation in an effort to prove it is an identity? True/False: All trig equations are identities. x2 1 x2 1 Consider the equation 2 . The left hand side of the equation is not defined when x = 1, x 1 x 1 while the right hand side is defined for all x. What impact does this observation have on the status of the equation as an identity? f ( x) If f(x) = g(x) is an identity and = k, which of the following must be false? g ( x) A) g(x) 0 B) f(x) = 0 C) k=1 D) f(x) – g(x) = 0 E) f(x) · g(x) > 0 True/False: sin = tan cos for all real numbers. Grade 11 Advanced Precalculus, page 14 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts Solving Trig Equations How does one solve a trig equation? Are there multiple answers to a trig equation? Are there trig equations with (NOTE: This material no solutions? Are there extraneous solutions to trig equations? is covered in Section 4.4. However, use of Skills check, ability to: an alternative text is Correctly solve a trig equation given a domain algebraically and graphically recommended) Correctly solve a trig equation without a given domain Correctly solve a quadratic trig equation algebraically and graphically Correctly solve a trig equation with a complicated argument algebraically and graphically Correctly solve an equation containing a trig expression and an algebraic expression using a graphing calculator (i.e. sin x = x) Concept check: When it is reasonable to use a graphing calculator to solve a trig equation? How can you solve a trig equation exactly using a graphing calculator? How can you extend the ideas of solving a trig equation to solving a trig inequality? 4.5 Parametric What is a parametric function? How do you graph a parametric function? How do you convert between Functions parametric and rectangular functions? Skills check, ability to: Correctly graph a pair of parametric functions by hand and using the graphing calculator Correctly use the Pythagorean identity to eliminate the parameter from a pair of parametric equations Identify a pair of parametric equations as an ellipse or circle Given the graph of an ellipse or circle, find the correct parametric equation Concept check: Why do you think parametric equations have been introduced at this time? Grade 11 Advanced Precalculus, page 15 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 4.6 Inverse Trig In what ways are the graphs of inverse trig relations and inverse trig functions the same/different? Relations Graphs Skills check, ability to: Correctly sketch all six inverse trig functions Correctly create the BOX diagram of all six inverse trig function identifying input, output and rule Correctly state the domain and range of all six inverse trig functions Correctly identify the quadrants that correspond to the range for the six inverse trig functions Correctly evaluate functions composed of trig and inverse trig functions (i.e. sin (tan 1 (-1)) Correctly evaluate functions composed of a trig function and its inverse function (i.e. sin (sin 1 (4)) Concept check: If f(x) = x+3, f ( f 1 (7)) = 7 and f 1 (f(7)) =7, why is sin (sin 1 (4)) 4? Under what circumstances will sin (sin 1 (x)) = x? Grade 11 Advanced Precalculus, page 16 Advanced Precalculus Grade 11 Unit 5: Properties of Combined Sinusoids Topic Learning Objectives: Key Definitions, Skills and Concepts 5.2 Composite What does the graph of y = A cos x + B sin x look like? Is it periodic? Is it sinusoidal? How can the linear Argument and Linear combination of cosine and sine be written as a single cosine function with a phase shift? What is the expansion of Combination Properties cos (A-B)? Skills check, ability to: Correctly express a linear combination of cosine and sine as a single cosine function with a phase shift Correctly express a single cosine function with a phase shift as a linear combination of cosine and sine Correctly solve trig equations involving a linear combination of sine and cosine Concept check: If f is a trig function and g is a trig function, is the new function f + g always periodic? Always sinusoidal? Under what circumstances will f + g be sinusoidal? True/False: Cosine distributes over subtraction (that is cos (A-B) = cos A – cos B) Grade 11 Advanced Precalculus, page 17 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 5.3 Other Composite What is the Odd-Even Property for the trig functions? What is the Cofunction Property for the trig Functions? Argument Properties What are the expansions of sin (A B), cos (A B) and tan (A B)? Skills check, ability to: For the six trig functions be able to express f (-x) in terms of f(x) and f(90º - x) in terms of f(x) Correctly expand sin (A B), cos (A B) and tan (A B) Correctly use the expansions to verify identities and solve trig equations Concept check: True/False: If cos A + cos B = 0, then A and B are supplementary angles. Justify your answer. If cos A cos B = Sin A sin B, then cos (A + B) = A) 0 B) 1 C) cos A + cos B D) sin A + sin B E) cos A cos B + sin A sin B Exactly evaluate sin 15º Assume A, B, and C are the three angles of some triangle. Prove sin (A + B) = sin C Grade 11 Advanced Precalculus, page 18 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 5.5 The Sum and What are the sum to product properties? Product Properties Skills check, ability to: Correctly transform the sum or difference to a product of sines and/or cosines with positive arguments Concept check: 1 Prove the following identity sin u sin v =(cos (u-v) – cos (u+v)). This is called the product-to-sum formula. 2 uv u v Using the product-to-sum formula prove cos u - cos v = -2 sin sin . This is called the sum to 2 2 product formula. 5.6 Double and Half How can sin 2A, cos 2A and tan 2A be expressed as functions of sin A, cos A and tan A? Argument Properties A A A How can sin , cos and tan be expressed as functions of sin A, cos A and tan A? 2 2 2 Skills check, ability to: A Correctly find the exact values of functions of 2A and , given the function value of one trig function of A 2 and a domain for A Correctly use the expansions to verify identities and solve equations Concept check: Explain how the other forms of cos 2A can be derived if one knows cos 2A = cos 2 A – sin 2 A Recall that we could write exact values of sin and cos when had a reference angle of 0, 30, 45, 60 and 90 degrees. Explain how you could now find exact values for = 15º and = 75º Grade 11 Advanced Precalculus, page 19 Advanced Precalculus Grade 11 Unit 6: Triangle Trigonometry Topic Learning Objectives: Key Definitions, Skills and Concepts 6.2 Oblique Triangles: What is an oblique triangle? What is the Law of Cosines and when is its use appropriate? Law of Cosines Skills check, ability to: Correctly utilize the Law of Cosines to “solve” a triangle Concept check: True/False: If ABC is any triangle with sides and angles labeled in the usual manner, then b2 c 2 2bc cos A . Justify your answer. 6.3 Area of a Triangle What is the trigonometric formula for the area of a triangle? What is Hero’s Formula? Skills check, ability to: Correctly compute the area of a triangle given two sides and the included angle Correctly compute the area of a triangle given three sides Concept check: True/False: If a, b and are two sides and the included angle of a parallelogram, then area of the parallelogram is ab sin . Justify your answer. 6.4 Oblique Triangles: What is the Law of Sines and when is its use appropriate? Law of Sines Skills check, ability to: Correctly use the Law of Sines to “solve” a triangle Concept check: True/False: The perimeter of a triangle with two 10 inch sides and two 40º angles is greater than 36. Justify your answer. Grade 11 Advanced Precalculus, page 20 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 6.5 The Ambiguous When would one expect to find multiple solutions to a triangle? How, in the process of solving a triangle would Case one know that multiple solutions are possible? Skills check, ability to: Correctly find the measure of the third side in a triangle given the measure of two sides and the non included angle Concept check: What is the triangle inequality and what is its application to the Law of Sines? Which of the following three triangle parts do not necessarily fix a triangle: A) AAS B) ASA C) SAS D) SSA E) SSS Navigation Problems What is a course? What is a bearing? How is a compass reading expressed in surveying? (NOTE: Navigation problems are covered Skills check, ability to: in Section 6.6, however Correctly use information of on the course of a ship or plane to solve problems related to its travel use of an alternative Correctly use surveying information to solve problems text is recommended. See Brown text: Concept check: Applications of Trig to Navigation and Surveying) Grade 11 Advanced Precalculus, page 21 Advanced Precalculus Grade 11 Topic Learning Objectives: Key Definitions, Skills and Concepts 6.7 Real World What problems can be solved by the creation of a triangle and appropriate techniques to “solve” the triangle? Triangle Problems Skills check, ability to: Correctly formulate and solve real world problems that are appropriate to this material Concept check: Vectors (NOTE: Vector What is a vector? When are two vectors equal? What is the magnitude of a vector? What is vector subtraction? Addition is covered in What is multiplication of a vector by a scalar? How are vectors added geometrically? How are vectors Section 6.6, however represented algebraically? How are vectors added algebraically? use of an alternative text is recommended. Skills check, ability to: See Brown text: Correctly solve problems relating to the geometric interpretation of vectors Geometric and Correctly solve problems relating to the algebraic interpretation of vectors Algebraic Representation of Concept check: Vectors) Grade 11 Advanced Precalculus, page 22 Advanced Precalculus Grade 11 Unit 7: Polar Coordinates and Complex Numbers Polar Coordinates What is the polar coordinate system and how is it used to plot points? What is the procedure to convert between (NOTE: Polar rectangular coordinates and polar coordinates? How are polar equations graphed? coordinates are covered in Section 13.2, Skills check, ability to: however use of an Correctly plot a point given in polar coordinates alternative text is Correctly covert between rectangular and polar points and equations recommended. See Correctly graph polar equations by hand and on the graphing calculator Brown text: Polar Coordinates) Concept check: True/False: Polar coordinates are unique. Justify your answer. True/False: If r 1 and r 2 are not 0, and if (r 1 , ) and ( r 2 , + ) represent the same point in the plane, then r 1 = -r 2 . Justify your answer. 13.3 Intersection of When and where do two polar curves have an actual intersection? Polar Curves Skills check, ability to: Correctly identify actual points of intersection between two polar curves by hand and on the graphing calculator Concept check: Grade 11 Advanced Precalculus, page 23 Advanced Precalculus Grade 11 13.4 Complex Numbers How are complex numbers expressed in polar form? How are complex numbers multiplied/divided in polar in Polar Form form? What is DeMoivre’s Theorem? How can you find all the roots of complex numbers? Skills check, ability to: Correctly express complex numbers in polar form Correctly multiply/divide complex numbers in polar form Correctly find powers of complex numbers in polar form Correctly find all the roots of complex numbers Concept check: Consider the number 4 + 3 i . You need to raise this number to the 5th power. What are the three options you have for performing this operation, and which will be the easiest? Grade 11 Advanced Precalculus, page 24 Advanced Precalculus Grade 11 Unit 8: Properties of Elementary Functions 7.2 Identifying What are the graphical features of linear, constant, quadratic, power and exponential functions? When are each Functions from increasing/decreasing/neither? When is each concave up/down? What are real world examples of each? What Graphical Patterns are the procedures for finding a specific equation of each? Skills check, ability to: Correctly identify the type of function from its sketch Correctly find, by hand and using the regression feature on the graphing calculator, the equation of the function given specific information Concept check: What kinds of transformations can make an exponential function appear to be a power function (if viewed over a specific domain)? True/False: Every exponential function is strictly increasing. Justify your answer. Refer to the expression f( a, b, c ) = a · b c (So, for example f( 2, 3 ,x ) = 2 · 3 x , an exponential function) If b = x, state the conditions on a and c under which the expression f ( a, b, c ) is a quadratic function. If b = x, state the conditions on a and c under which the expression f ( a, b, c ) is a decreasing linear function. If c = x, state the conditions on a and b under which the expression f ( a, b, c ) is an increasing exponential function. If c = x, state the conditions on a and b under which the expression f ( a, b, c ) is a decreasing exponential function. Grade 11 Advanced Precalculus, page 25 Advanced Precalculus Grade 11 7.3 Identifying What numerical patterns are exhibited by linear, quadratic, power and exponential functions? What is direct and Functions form inverse variation? Numerical Patterns Skills check, ability to: Correctly identify a function given its data table Correctly find function values given an initial value and the type of function Solve direct and inverse variation problems Concept check: Is it possible for more than one function to fit the same set of data? If this is the case, how would you decide which function is a better model? Grade 11 Advanced Precalculus, page 26 Advanced Precalculus Grade 11 7.4 Logarithms: What is a common/natural logarithm? How do you convert between exponential form and logarithmic form? Definition, Properties, What are the properties of Logarithms? What is the Change of Base Formula? How do you solve equations and Equations involving logarithms? How do you solve exponential equations? Skills check, ability to: Convert between exponential form and logarithmic form Express sum/differences of logs as a single logarithm Express products of constants and logs as a single logarithm Correctly calculate the log of any base using the Change of Base Formula Correctly solve logarithmic and exponential equations Correctly identify extraneous roots of logarithmic equations Concept check: Explain why it is possible for a logarithmic equation to have extraneous roots. Determine if each of the following statements are true or false and then explain why you answered as you did: log 5 = 2.5 log 2 log 5 > log 2 log 5 = log 10 – log 2 log 5 = 1 – log 2 log 5 < log 10 Describe how to transform the graph of f(x) = ln x into the graph of g(x) = log 1 x e True/False: The logarithm of a positive number is positive. Justify your answer. True/False: If $100 is invested at 5% annual interest for 1 year, there is no limit to the final value of the investment if it is compounded sufficiently often. Justify your answer. Grade 11 Advanced Precalculus, page 27 Advanced Precalculus Grade 11 7.5 Logarithmic What is the graphical pattern exhibited by a logarithmic function? What is the domain/range of a logarithmic Functions function? What is the numerical pattern exhibited by a logarithmic function? What is the relationship between an exponential function and a logarithmic function? Skills check, ability to: Identify a logarithmic function from its graph and its data table Correctly find, by hand and on the graphing calculator, a specific equation of a logarithmic function Correctly identify real world illustrations of logarithmic functions Correctly sketch a logarithmic function that has been transformed Concept check: Graph the equations y 1 = log(x), y 2 = log (10x) and y 3 = log (100x). How do the graphs compare and why is this happening? Immediately following the gold medal performance of the US Woman’s gymnastic team in the 1996 Olympics, an NBC commentator, John Tesh, said of one of the team members: “Her confidence and performance have grown logarithmically.” He clearly thought this was an enormous compliment. Is it a compliment? Is it realistic? 7.6 Logistic Functions What is a logistic function and when is its use appropriate? What are real world illustrations of logistic growth? for Restrained Growth What is the domain/range of a logistic function? What is the significance of the inflection point found in a logistic function? Skills check, ability to: Correctly find, by hand and on the graphing calculator, a specific logistic function that accurately fits data Correctly identify the carrying capacity from anecdotal information Concept check: Is all growth logistic? Compare and contrast logistic and exponential functions. Grade 11 Advanced Precalculus, page 28 Advanced Precalculus Grade 11 Unit 9: Sequences and Series 14.2 Arithmetic, What is a sequence? How are sequences categorized? How are sequences defined (explicitly v. recursively)? Geometric and Other What does the graph of a sequence look like? What is its domain/range? What is meant by the limit of an Sequences infinite sequence? Skills check, ability to: Correctly identify the type of sequence given some consecutive terms Correctly create a formula, both explicit and recursive, for a sequence given some consecutive terms Correctly use a formula to calculate additional terms of the sequence Correctly generate a sequence on the graphing calculator Given a sequence, correctly identify the term number of any term Correctly use sequences to solve problems Identify the Fibonacci sequence Correctly use a graphing calculator in sequence mode to produce the graph of a sequence Correctly identify the limit of an infinite sequence Correctly calculate limits in the form: lim f (n) n Correctly identify when a limit does not exist Concept check: True/False: If a geometric sequence contains all positive terms, then the new sequence formed by taking the log of all the terms in the original sequence is arithmetic. Justify your answer. Grade 11 Advanced Precalculus, page 29 Advanced Precalculus Grade 11 14.3 Series and Partial What is a series and a partial sum of a series? What is sigma notation and how is it used to indicate partial sums? Sums What is meant by a converging/diverging series? Under what circumstances can an infinite series have a sum? Skills check, ability to: Correctly calculate partial sums of arithmetic/geometric series Correctly evaluate an expression given in sigma notation Correctly write a partial sum of an arithmetic/geometric series in sigma notation Correctly use pattern recognition to write a partial sum in sigma notation Use a graphing calculator to evaluate an expression given in sigma notation Correctly demonstrate when a series converges Correctly find the number of terms in a series given S n Correctly calculate the sum of an infinite geometric series Concept check: Can an infinite arithmetic series have a sum? Why or why not? Do all infinite geometric series have sums? Can infinite series, other than geometric converge? Why or why not? If two infinite geometric series have the same sum are they necessarily the same series? Explain. Can the sum of a convergent infinite geometric series be less than its first term? Explain. When basketball player Patrick Ewing was signed by the NY Knicks, he was given a contract for $30 million: $3 million a year for 10 years. Of course, since much of the money was to be paid in the future, the team’s owners did not have to have all $30 million available on the day of the signing. How much money would the owners have to deposit in a bank account on the day of the signing in order to cover all future payments? Assuming the account was earning interest, the owners would have to deposit much less than $30 million. (Assume the account was earning 5% interest per year). (Answer: $24.3 million) Now suppose Patrick Ewing’s contract guaranteed him and his heirs an annual payment of $3 million forever. How much would the owners need to deposit in an account today in order to provide these payments? (Answer: $63 million) Grade 11 Advanced Precalculus, page 30 Advanced Precalculus Grade 11 Mathematical Induction What is mathematical induction and how is it used? (NOTE: This material is not covered in the Skills check, ability to: Foerster text. Use of an Correctly formulate and prove a hypothesis using mathematical induction alternative text is recommended) Concept check: Students often time do not believe that an inductive proof has actually proven anything. Explain why a well constructed inductive proof is valid. Grade 11 Advanced Precalculus, page 31 Advanced Precalculus Grade 11 Unit 10: Combinatorics 15.1 Venn Diagrams What is a Venn Diagram and how is it used to illustrate union/intersection of sets and thereby aid in counting? (See Brown text: Combinatorics Skills check, ability to: Chapter) Correctly illustrate sets having no intersection and a finite intersection Correctly illustrate wholly contained subsets Use Venn Diagrams to solve counting problems Use Venn Diagrams to illustrate DeMorgan’s Laws Concept check: Is union distributive over intersection? Justify your answer. Is complement distributive over intersection? Justify your answer. 15.2 The What are these principles and how do they aid in solving counting problems? What are mutually exclusive Multiplication, events? What is the meaning of factorial? Addition and Complement Principles Skills check, ability to: Correctly use these principles to solve counting problems Correctly interpret English language questions into problems that can be solved using these principles Correctly construct a branch diagram Use a graphing calculator to evaluate factorials Concept check: 15.3 Permutations and What is a permutation? What is a combination? What English words indicate each? Combinations Skills check, ability to: Correctly evaluate permutations and combinations by hand and using a graphing calculator Correctly solve problems using permutations and combinations Correctly interpret English language into problems involving permutations and combinations Concept check: There are many ways one could justify that 0! = 1 and not 0. Choose one rationale and explain it thoroughly. You have a fresh carton containing one dozen eggs and you need to choose two for breakfast. Give a counting argument based on this scenario to explain why 12 C2 12 C10 . Grade 11 Advanced Precalculus, page 32 Advanced Precalculus Grade 11 15.4 Permutations with How does the formula for permutations differ when all the elements are not distinguishable? How is the formula Repetition; Circular for linear permutations different from circular permutations? Permutations Skills check, ability to: Correctly evaluate the number of permutations when there is repetition of elements Correctly evaluate the number of circular permutations Concept check: MIXED This is practice in determining which of the many combinatoric principles should be applied to solve problems. COMBINATORICS EXCERCISES Skills check, ability to: Concept check: Grade 11 Advanced Precalculus, page 33 Advanced Precalculus Grade 11 15.5 The Binomial What is the binomial theorem and how is it related to the study of combinatorics? Theorem: Pascal’s Triangle Skills check, ability to: Correctly expand powers of binomials Correctly find individual terms of a binomial expansion Correctly find binomial coefficients using either combinations or Pascal’s Triangle Concept check: There are two ways to expand a binomial (repeated multiplication and the theorem). Expand (x + x) 4 and show that the results are the same. Which method do you prefer? Just by looking at patterns in Pascal’s Triangle, guess the answers to the following questions: A) What positive integer appears the least number of times? B) What number appears the greatest number of times? C) Is there any positive integer that does NOT appear in Pascal’s Triangle? D) If you go along any row alternately adding and subtracting the numbers, what is the result? E) If p is a prime number, what do all the interior numbers along the p th row have in common? F) Which rows have all even numbers? G) Which rows have all odd numbers? Grade 11 Advanced Precalculus, page 34 Advanced Precalculus Grade 11 Unit 11: Probability 16.1 Introduction to What is an experiment, an event, and a sample space? What is the difference between experimental and Probability (See Brown theoretical probability? What information is communicated by quantifying the probability of an event? text: Probability Chapter) Skills check, ability to: Correctly identify and list the sample space for an experiment Correctly identify events that have a probability of zero and one Correctly evaluate the probability of either of two events (mutually exclusive or not) Concept check: True/False: The sample space for rolling two dice and considering the sum is {2,3,4,5,6,7,8,9,10,11,12}. Why or why not? 16.2 Probability of How do you compute the probability of two events occurring together? What is conditional probability? What Events Occurring are independent events? Together Skills check, ability to: Correctly find the probability of two events occurring together Correctly determine if two events are independent Create a branch diagram including probabilities Concept check: 16.3 The Binomial What is the binomial probability theorem and when is its use appropriate? How can the binomial probability Probability Theorem theorem be used to approximate probabilities when trials are not independent? Skills check, ability to: Correctly determine when the use the binomial probability theorem is appropriate Correctly use the binomial probability theorem Use the Binomial probability theorem to approximate probabilities when trials are not independent Concept check: Grade 11 Advanced Precalculus, page 35 Advanced Precalculus Grade 11 16.4 Probability How can combinatorics be used to solve probability problems? Problems Solved with Combinations Skills check, ability to: Correctly use combinations to solve probability problems Concept check: Probability problems can be solved in many ways. Create a problem and show multiple ways to arrive at your solution. Comment upon the merits of each method. 16.5 Conditional What are the formulas related to conditional probability? How can conditional probability be used to judge the Probability accuracy of a prediction? Skills check, ability to: Correctly create a branch diagram including probabilities Correctly use conditional probability to compare two events, one that can be determined with complete confidence and one that is more difficult to determine Concept check: How is conditional probability used to determine the accuracy of the SAT test? Why is this information important? How is conditional probability used to determine the accuracy of a medical test? What is the impact of false positives/false negatives? Grade 11 Advanced Precalculus, page 36 Advanced Precalculus Grade 11 16.6 Expected Value How can one find the expected value of a given random experiment? Skills check, ability to: Correctly calculate the expected value of a random experiment Correctly identify when a game of chance is fair Concept check: Gladys has a personal rule never to enter the lottery (picking 6 numbers from 1 to 46) until the payoff reaches $4 million. When it does reach $4 million, she buys ten different $1 tickets. Assume that the payoff is exactly $4 million. 1 A) What is the probability that Gladys holds the winning ticket? (Answer: , which is 46 C6 1 .00000010676 ) 9,366,819 B) Fill-in the probability distribution for Gladys’s possible payoffs in the table below (Note that we subtract $10 from the $4 million, since Gladys has to pay for her tickets even if she wins) Value Probability -10 +3,999,990 C) Find the expected value of the game for Gladys. D) In terms of the answer to part B, explain to Gladys the long-term implications of her strategy. Grade 11 Advanced Precalculus, page 37 Advanced Precalculus Grade 11 Unit 12: Limits and Introduction to Calculus 19.1 Limits Are all limits calculated as n ? What is a working definition of a limit? How can the graph of a function be used to calculate a limit? What is a continuous function? Skills check, ability to: Correctly evaluate limits in the form: lim f ( x) , lim f ( x) , lim f ( x) x c x x Correctly evaluate one sided limits Correctly evaluate limits of piecewise functions Identify intervals where a function is continuous Concept check: True/False: If lim f ( x) L , then f(c) = L. Justify your answer. x c True/False: If lim f ( x) L , and f is continuous at x = c, then f(c) = L. Justify your answer. x c Draw a function f defined on the interval [-4, 5] with the following constraints: lim f ( x) 6 , lim f ( x) 5 , lim f ( x) 8 , f(2) does not exist, lim f ( x) does not exist x 0 x 2 x 2 x 4 Grade 11 Advanced Precalculus, page 38 Advanced Precalculus Grade 11 19.2 Graphs of Rational What is a rational function? Do all rational functions have asymptotes? How do you determine if a rational Functions function has an asymptote or a hole? What is a slant asymptote? Skills check, ability to: Correctly sketch a rational function Concept check: Holes are often called removable discontinuities while asymptotes are called nonremovable discontinuities. Explain why these names make sense. x2 9 Compare the graph of f ( x) and g(x) = x + 3. x 3 Are the domains equal? Does f have a vertical asymptote? Explain why the graphs appear identical. Are the functions identical? x2 Let f ( x) Which of the following is true about the graph of f? x5 A) There is no vertical asymptote B) There is a horizontal asymptote but no vertical asymptote C) There is a slant asymptote but no vertical asymptote. D) There is a vertical asymptote and a slant asymptote. E) There is a vertical asymptote and a horizontal asymptote. 20.1 The Slope of A What is meant by the slope of a curve? Is this constant as it is with a line? What is the derivative of a function? Curve Skills check, ability to: Find the slope of a curve at a given point Find the function which represents the derivative of a function at all points Write the equation of a line tangent to a curve at a given point Concept check: True/False: All functions have derivatives everywhere. Justify your answer. What geometric characteristics determine if a function is differentiable? Grade 11 Advanced Precalculus, page 39 Advanced Precalculus Grade 11 Recommended Unit Sequencing and Pacing Guide Timeframe Q1 Unit 1: Functions and Mathematical Models 1.1 Functions: Algebraically, Numerically, Graphically, and Verbally 1.2 Kinds of Functions 1.3 Dilations and Translation of Function Graphs 1.4 Composition f Functions 1.5 Inverse Functions 1.6 Reflections, Absolute Values, and Other Transformations Unit 2: Periodic Functions and Right Triangle Problems 2.1 Introduction to Periodic Functions 2.2 Measurement of Rotation 2.3 Sine and Cosine Functions 2.4 Values of Six Trigonometric Functions 2.5 Inverse Trigonometric Functions and Triangle Problems Unit 3: Applications of Trigonometric and Circular Functions 3.1 Sinusoids: Amplitude, Period and Cycles 3.2 General Sinusoidal Graphs 3.3 Graphs of Tangent, Cotangent, Secant and Cosecant Functions 3.4 Radian Measure of Angles 3.5 Circular Functions 3.6 Inverse Circular Relations: Given y, Find x 3.7 Sinusoidal Functions As Mathematical Models Unit 4: Trigonometric Function Properties, Identities and Parametric Functions 4.1 Introduction to the Pythagorean Property 4.2 Pythagorean, Reciprocal and Quotient Properties 4.3 Identities and Algebraic Transformation of Expressions Grade 11 Advanced Precalculus, page 40 Advanced Precalculus Grade 11 Timeframe Q2 Unit 4: Trigonometric Function Properties, Identities and Parametric Functions 4.4 Arcsine, Arctangent, Arccosine and Trigonometric Equations 4.5 Parametric Functions 4.6 Inverse Trigonometric Relation Graphs Unit 5: Properties of Combined Sinusoids 5.1 Introduction to Combination of Sinusoids 5.2 Composite Arguments and Linear Combination Properties 5.3 Other Composite Arguments 5.6 Double and Half Argument Properties Unit 6: Triangle Trigonometry 6.1 Introduction to Oblique Triangles 6.2 Oblique Triangles: Law of Cosines 6.3 Area of a Triangle 6.4 Oblique Triangles: Law of Sines 6.5 The Ambiguous Case 6.7 Real-World Triangle Problems Vectors Midterm Grade 11 Advanced Precalculus, page 41 Advanced Precalculus Grade 11 Timeframe Q3 Unit 7: Polar Coordinates, Complex Numbers 13.1 Introduction to Polar Coordinates 13.2 Polar Equations and Other Curves 13.3 Intersection f Polar Curves 13.4 Complex Numbers in Polar Form Unit 8: Properties of Elementary Functions 7.1 Shapes of Function Graphs 7.2 Identifying Functions from Graphical Patterns 7.3 Identifying Functions from Numerical Patterns 7.4 Logarithms: Definition, Properties, and Equations 7.5 Logarithmic Functions 7.6 Logistic Functions for Restrained Growth Unit 9: Sequences and Series 14.1 Introduction to Sequences and Series 14.2 Arithmetic, Geometric and Other Sequences 14.3 Series and Partial Sums Grade 11 Advanced Precalculus, page 42 Advanced Precalculus Grade 11 Timeframe Q4 Unit 10: Combinatorics 15.1 Venn Diagrams 15.2 The Multiplication, Addition and Complement Principles 15.3 Permutations and Combinations 15.4 Permutations with Repetition; Circular Permutation Mixed Combinatoric Exercises 15.5 The Binomial Theorem; Pascal’s Triangle Unit 11: Probability 16.1 Introduction to Probability 16.2 Probability of Events Occurring Together 16.3 The Binomial Probability Theorem 16.4 Probability Problems Solved With Combinations 16.5 Working With conditional Probability 16.6 Expected Value Unit 12: Limits and Introduction to Calculus 19.1 Limits of Functions 19.2 Graphs of Rational Functions 20.1 The Slope of A Curve Final Grade 11 Advanced Precalculus, page 43

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 5 |

posted: | 12/4/2011 |

language: | English |

pages: | 43 |

OTHER DOCS BY stariya

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.