Estimated Pre Calculus Pacing Timeline


 Duane Knight
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1 Estimated Pre Calculus Pacing Timeline School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to time based on data to meet the needs of your students as some concepts may take less time and some may take more time. Estimated Timeframe for Instruction and Unit Name Assessment Unit 1: Prerequisite Information Chapter 0 (7 days) August 16 August 24 Unit 2: Functions Chapter 1 (12 days) August 25  September 10 Unit 3: Power, Polynomial, and Rational Functions Chapter 2 (12 days) Optional Unit 4: Exponential and Logarithmic Functions Chapter 3 (7 days) Unit 5: Trigonometric Functions Chapter 4 (17 days) September 13 September 28 September 29 October 7 October 8 November 2 Note: Quarter 1 ends on October 15 Unit 6: Graphs of Trigonometric Functions Chapter 4 (10 days) November 3 November 17 Unit 7: Trigonometric Identities and Equations Chapter 5 (23 days) November 18 December 22 Includes review and exams Note: Quarter 2 ends on December 22 Unit 8: Conics and Parametric Equations Chapter 7 (15 days) January 10 January 31 Unit 9: Vectors Chapter 8 (15 days) February 1 February 22 Unit 10: Polar Coordinates and Complex Numbers Chapter 9 (17 days) February 23 March 25 Note: Quarter 3 ends on March 11 Unit 11: Sequences and Series Chapter 10 (20 days) March 28 April 22 Unit 12:Limits and Derivatives Chapter 12 (15 days) April 25 May 13 Review/Exams (11 days) May 17 June 1 Note: Quarter 4 ends on June 1 Page 1 of 26
2 Unit Title: (1) Preparing for Precalculus Chapter 0 Number of Days: 7 days Know: Understand: Do: Sets have notations to denote elements, subsets and components. The same operations that are used on real numbers can be used on complex numbers. Quadratic functions can be used to model data to analyze real life situations. There are specific rules for performing operations with exponents and radicals. Systems of equations and matrices can be used to model and solve real world equations. There are procedures to follow for solving equations and inequalities both algebraically and graphically. Use set notation to denote elements, subsets, and complements. Find intersections and unions of sets. Perform operations with imaginary numbers and complex numbers. Graph and solve quadratic equations. Simplify expressions in radical and exponential form. Solve systems of equations and inequalities. Perform operations and solve real life problems with matrices. Page 2 of 26
3 Unit Title: (1) Preparing for Precalculus Chapter 0 Number of Days: 7 days Key Learning: There are procedures to follow for solving equations and inequalities both algebraically and graphically. Unit Essential Question: What skills will prepare me for this course? Sets MA.912.S.3.2 MA.912.D How do I work with set notation and find intersections and unions of sets? set, subset, universal set, empty set, compliment Operations with Complex Numbers MA.912.A How do I perform operations with pure imaginary numbers and complex numbers? imaginary number, complex number, complex conjugates Quadratic Functions and Equations MA.912.A How do I solve and graph quadratic equations? parabola, quadratic equation, quadratic formula nth Roots and Real Exponents MA.912.A How do I simplify expressions in radical and exponential form? nth root, principal root Systems of Linear Equations and Inequalities MA.912.A How do I solve systems of equations and inequalities both algebraically and graphically? systems of equations, independent, dependent, inconsistent, system of inequalities Matrix Operations MA.912.D.8.1 MA.912.D How do I perform operations with matrices? matrix, element, dimensions, zero matrix, scalar Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 3 of 26
4 Unit Title: (2) Functions Chapter 1 Number of Days: 12 days Know: Understand: Do: Graphs of common functions can be shifted, reflected, and stretched and will help in sketching a wide variety of functions by hand. Combinations of functions can help model and analyze real world situations. Functions and their graphs can be analyzed to predict their future behavior. Write sets using setbuilder and interval notation. Determine if relations are functions. Identify and evaluate functions and state their domains. Find domain, range, intercepts, and symmetry of a given relation. By analyzing the graph of a function the future behavior of a function can be indicated. Relations have inverses that can be found algebraically and graphically. Identify odd and even functions. Determine where a relation is continuous and discontinuous. Approximate zeros using the Intermediate Value Theorem. Determine the end behavior of a function. Apply end behavior to real world situations. Determine the intervals of which functions are increasing and decreasing. Identify extrema by hand and by using a graphing calculator. Page 4 of 26
5 Apply extrema to real world situations. Use real world situations to find the average rate of change. Describe and graph various transformations to a given parent function. Graph a piecewisedefined function. Perform various operations with functions. Compose and decompose functions. Determine whether or not an inverse function exists algebraically and graphically. Find and verify inverse functions algebraically and graphically. Page 5 of 26
6 Unit Title: (2) Functions Chapter 1 Number of Days: 12 days Key Learning: Functions and their graphs can be analyzed to predict their future behavior. Unit Essential Question: How do I analyze graphs to predict future function behavior? Functions and Their Graphs MA.912.A How do I identify and evaluate functions and state their domains? 2. How do I identify and graph functions? setbuilder notation, interval notation, implied domain, piecewise defined function, relevant domain, zeros, roots, even function, odd function Behaviors of Graphs and Limits MA.912.A.5.6 MA.912.C How do I use limits to determine the continuity of a function and determine end behavior of functions? 4. How do I find the average rate of change and determine whether the function is increasing, decreasing or constant? continuous, limit, discontinuous, infinite, jump, end behavior, increasing, decreasing, constant, maximum, minimum, extrema Parent Functions and Transformations MA.912.A How do I describe and graph transformations of functions? parent, step, greatest integer function Function Operations and Composition of Functions MA.912.A.2.7 MA.912.A How do I perform operations and find compositions of functions? composition Inverse Relations and Functions MA.912.A How do I solve problems involving functions and their inverses? inverse relation, inverse function, onetoone Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 6 of 26
7 Unit Title: (3) Power, Polynomial, and Rational Functions Chapter 2 Number of Days: 12 days Know: Understand: Do: The graph of any function gives a good indication of the future behavior of a function. Polynomial and rational functions are useful in real life situations. Describe domain, range, intercepts, end behavior, continuity and where the function is increasing and decreasing. Solve radical and exponential equations. Polynomial division can help rewrite polynomials that are used to model real life problems. The zeros of polynomial functions yield important information to model real life problems. Apply the Leading Coefficient Test. Find the number of possible real zeros and the turning points. Graph higherdegree polynomial functions. Divide polynomials by long and synthetic division. Apply the Remainder Theorem in evaluating a polynomial. Apply the Factor Theorem to determine if one polynomial is a factor of another. Find all possible rational zeros of a function using the Rational Zero Theorem. Use the Fundamental Theorem of Algebra and the Linear Factorization Theorem to assist in finding zeros. Find all asymptotes of a rational function. Graph a rational function. Solve rational equations. Page 7 of 26
8 Unit Title: (3) Power, Polynomial, and Rational Functions Chapter 2 Number of Days: 12 days Key Learning: Polynomial and rational functions are useful in real life situations. Unit Essential Question: How are polynomial and rational functions used in real life? Exponential and Radical Functions MA.912.A How do I graph and analyze exponential and radical functions? 2. How do I solve exponential and radical equations? exponential function, radical function, extraneous solutions Polynomial Functions MA.912.A.4.5 MA.912.A How do I graph polynomial functions and use them to model realworld data? polynomial function, Leading Coefficient Test, multiplicity Zeros of Polynomial Functions & The Remainder Theorem MA.912.A.4.6 MA.912.A How do I use the Remainder Theorem to evaluate a polynomial? 5. How do I find real and complex zeros of a polynomial function? synthetic division, lower bound, upper bound Rational Functions MA.912.A.5.6 MA.912.C How do I analyze and graph rational functions? 7. How do I solve rational equations? rational function, asymptote, holes Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 8 of 26
9 Unit Title: (4) Exponential and Logarithmic Functions (Optional) Chapter 3 Number of Days: 7 days Know: Understand: Do: Exponential and logarithmic functions are inverses of one another. Real world situations can be solved using exponential and logarithmic functions. Sketch and analyze graphs of exponential and logarithmic functions. Properties of logarithms will help in solving exponential and logarithmic equations. Model real life data using exponential and logarithmic graphs. Evaluate logarithms using various methods. Solve exponential and logarithmic equations using Change of Base Formula and properties of logarithms. Solve real world problems using exponential and logarithmic equations. Page 9 of 26
10 Unit Title: (4) Exponential and Logarithmic Functions (Optional) Chapter 3 Number of Days: 7 days Key Learning: Real world situations can be solved using exponential and logarithmic functions. Unit Essential Question: How can real world situations be solved using exponential and logarithmic functions? Exponential and Logarithmic Functions (Optional) MA.912.A How do I evaluate, sketch and analyze exponential and logarithmic functions? , 32 exponential function, logarithmic function, natural base, natural log Using Properties of Logarithms in Solving Exponential and Logarithmic Equations (Optional) MA.912.A.8..2 MA.912.A How do I apply properties of logarithms to solve exponential and logarithmic equations? , 34 continuous compound interest Real World Applications (Optional) MA.912.A How do I solve real world problems using exponential and logarithmic equations? & additional resources Additional Information: This unit is optional. These benchmarks are not listed in the Precalculus State Benchmarks, however are necessary for preparation of Calculus. Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 10 of 26
11 Unit Title: (5) Trigonometric Functions Chapter 4 Number of Days: 17 days Know: Understand: Do: There are six trigonometric functions related to right triangles. Angles can be measured using degrees and radians. There are a variety of methods to solve for sides and angles of triangles and ways to find the area of triangles. Find the values of the six trigonometric ratios in any quadrant. Find the missing side lengths and angle measures of a right triangle. Use angle of elevation and depression to solve real life problems. There is a relationship between the unit circle and real numbers. Trigonometric functions are periodical. The Laws of Sines and Cosines are used to solve triangle measures. There are two additional methods of finding area in any triangle. Convert between DMS and decimal degree form. Convert between degrees and radians. Find and draw coterminal angles. Find arc length. Find angular and linear speed. Find the area of a sector. Evaluate a trig function given a point. Evaluate trig functions of a quadrantal angle. Find and use reference angles to find trig values. Find exact values using the unit circle. Apply the Law of Sines and the Law of Cosines. Find the area of oblique triangles. Page 11 of 26
12 Unit Title: (5) Trigonometric Functions Chapter 4 Number of Days: Key Learning: 17 days There are a variety of methods to solve for sides and angles of triangles and ways to find the area of triangles. Unit Essential Question: How do I solve for sides and angles of triangles and find their areas? Right Triangle Trigonometry MA.912.T.2.1 MA.912.T How do I find values of trigonometric functions in right triangles? trigonometric functions, inverse trigonometric functions, angles of elevation and depression Degrees and Radians MA.912.T How do I convert degree measures of angles to radian measure and vice versa? 3. How do I use angle measures to solve real world problems? initial side, terminal side, standard position, radian, coterminal angles, linear and angular velocity, sector Trigonometric Functions on the Unit Circle MA.912.T.1.2 MA.912.T How do I find values of trigonometric functions for any angle? 5. How do I find values of trigonometric functions using the unit circle? quadrantal angle, reference angle, unit circle, periodic function, period The Law of Sines and the Law of Cosines MA.912.T.2.3 MA.912.T How do I solve oblique triangles using Law of Sines and Law of Cosines? 7. How do I find the area of an oblique triangle? oblique triangle, Law of Sines and Cosines, ambiguous case, Heron s Formula Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 12 of 26
13 Unit Title: (6) Graphing Trigonometric Functions Chapter 4 Number of Days: 10 days Know: Understand: Do: Trigonometric functions are periodical. Trigonometric functions and their graphs can be used to model real life data. Each number in a trigonometric equation represents different transformations. Inverse functions can be evaluated using various methods. Trigonometric functions and their graphs can be used to model real life data. Find amplitude, period, frequency, x intercepts and phase shift of sine and cosine functions. Find period, xintercepts and asymptotes of tangent functions. Graph all trigonometric functions including reciprocal and inverse functions. Evaluate inverse functions. Use inverse trigonometric properties to find exact values. Evaluate compositions of trigonometric functions. Page 13 of 26
14 Unit Title: (6) Graphing Trigonometric Functions Chapter 4 Number of Days: 10 days Key Learning: Trigonometric functions and their graphs can be used to model real life data. Unit Essential Question: How do trigonometric functions and their graphs model real world data? Graphing Sine and Cosine Functions MA.912.T How do I graph sine and cosine functions? amplitude, frequency, phase shift, vertical shift, midline Graphing Other Trigonometric Functions MA.912.T.1.4 MA.912.T How do I graph tangent and reciprocal functions? damped trigonometric function Inverse Trigonometric Functions MA.912.T.2.3 MA.912.T How do I evaluate and graph inverse trigonometric functions? arcsine function, arccosine function, arctangent function Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 14 of 26
15 Unit Title: (7) Trigonometric Identities and Equations Chapter 5 Number of Days: 23 days Know: Understand: Do: Fundamental trigonometric identities can be used to simplify and rewrite trigonometric expressions. Techniques for solving algebraic equations can be applied to solve trigonometric equations. Identities can be used to rewrite trigonometric functions in more convenient forms. Rewriting equations using trigonometric identities helps to solve reallife problems. Derive and prove trigonometric identities. Simplify trigonometric expressions using the trigonometric identities. Use sum, difference, doubleangle, and halfangle formulas to find exact values and to establish identities. Solve trigonometric equations using exact values and decimal approximations. Rewriting equations helps to solve reallife problems. Solve trigonometric equations using a graphing calculator. Page 15 of 26
16 Unit Title: (7) Trigonometric Identities and Equations Chapter 5 Number of Days: 23 days Key Learning: Rewriting equations using trigonometric identities helps to solve reallife problems. Unit Essential Question: How do I rewrite equations using trigonometric identities and solve reallife problems? Trigonometric Identities MA.912.T.1.8 MA.912.T How do I use basic trig identities to find trig values, simplify, and rewrite trig expressions? trigonometric identity, cofunction, oddeven identities Verifying Trigonometric Identities MA.912.T.1.8 MA.912.T How do I verify a trigonometric identity? verify an identity Solving Trigonometric Equations MA.912.T.1.8 MA.912.T How do I solve trig equations using identities or algebraic techniques? Sum and Difference Identities MA.912.T.3.2 MA.912.T How do I use sum and difference identities to evaluate and solve trigonometric equations? sum and difference identities Multiple Angle and ProducttoSum Identities MA.912.T.3.3 MA.912.T How do I use halfangle and doubleangle identities for sine, cosine and tangent functions? halfangle and doubleangle identities Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 16 of 26
17 Unit Title: (8) Conic Sections and Parametric Equations Chapter 7 Number of Days: 15 days Know: Understand: Do: Conic sections are the intersection of a plane and a double cone. Each conic section has its own equation. Conic sections and parametric equations can be used to model and solve many types of real life problems. Write an equation of a parabola. Identify the vertex, focus, directrix, and axis of symmetry and then graph the equation of a parabola. The equations can be used to graph the conic section and identify important features of the section. Graphs can be analyzed to find the equation of the conic section. Parametric functions consist of two functions, each of which is a function of another variable, the parameter. The parameter allows you to determine where an object was at a given time. These equations define a plane curve. Write an equation of a circle. Identify the center and radius and then graph an equation of a circle. Write the equation of an ellipse. Identify the center, vertices, covertices, foci, length of the major and minor axis, and then graph the equation of an ellipse. Write an equation of a hyperbola. Identify the center, vertices, foci, and asymptotes of a hyperbola and then graph the equation of a hyperbola. Graph parametric equations by hand and using a graphing calculator. Rewrite a rectangular equation into parametric form and vice versa. Solve real world problems using projectile motion. Page 17 of 26
18 Unit Title: (8) Conic Sections and Parametric Equations Chapter 7 Number of Days: 15 days Key Learning: Conic sections and parametric equations can be used to model and solve many types of real life problems. Unit Essential Question: How do I solve and model real life situations using conic sections and parametric equations? Parabolas MA.912.A.9.1 MA.912.A How do I analyze, graph and write equations of parabolas? conic, parabola, focus, directrix Ellipses and Circles MA.912.A.9.1 MA.912.A How do I analyze, graph and write equations of ellipses and circles? ellipse, foci, major axis, minor axis, center, vertices, covertices, eccentricity Hyperbolas MA.912.A.9.1 MA.912.A How do I analyze, graph and write equations of hyperbolas? 4. How do I identify the types of conic sections? hyperbola, transverse axis, conjugate axis Parametric Equations MA.912.D.10.1 MA.912.D How do I graph parametric equations and solve problems related to projectile motion? parametric equation, parameter, orientation, parametric curve Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 18 of 26
19 Unit Title: (9) Vectors  Chapter 8 Number of Days: 15 days Know: Understand: Do: Vectors represent a specific direction and magnitude. Vectors are used to analyze numerous aspects of every day life. Vectors are directed line segments that demonstrate the applications of physics. Graph vectors and find the position vector. Perform operations with twodimensional and threedimensional vectors including addition, subtraction, and scalar multiplication. Express a vector in component form. Vectors can be used in both twodimensions and threedimensions. Find the magnitude and direction of a vector. Find a unit vector in the same direction as a given vector. Find the dot product of two vectors. Find the angle between the two vectors. Determine whether two vectors are parallel, orthogonal, or neither. Solve applied problems involving vectors. Find the cross product of twodimensional and threedimensional vectors. Plot points in threedimensional space. Find the distance and midpoint between two points in space. Apply the geometric properties of cross products of vectors in space. Page 19 of 26
20 Unit Title: (9) Vectors Chapter 8 Number of Days: 15 days Key Learning: Vectors are directed line segments that demonstrate the applications of physics. Unit Essential Question: How are vectors used to demonstrate the application of physics problems? Introduction to Vectors MA.912.D.9.1 MA.912.D How do I represent and operate with vectors geometrically? 2. How do I solve vector problems? vector, initial point, terminal point, standard position, direction, magnitude, equivalent vectors, opposite vectors, resultant, zero vector, components, rectangular components Vectors in the Coordinate Plane MA.912.D.9.2 MA.912.D How do I represent and operate with vectors in the coordinate plane? 4. How do I write a vector as a linear combination of unit vectors? component form, unit vector, linear combination Dot Products and Vector Projections MA.912.D.9.1 MA.912.D How do I find the dot product of two vectors, and use the dot product to find the angle between them? dot product, orthogonal Vectors in Three Dimensional Space MA.912.D.9.1 MA.912.D How do I plot points and vectors in the threedimensional coordinate system? 7. How do I express algebraically and operate with vectors in space? threedimensional coordinate system, zaxis, octant, ordered triple, Dot and Cross Products of Vectors in Space MA.912.D.9.1 MA.912.D How do I find dot products of an angle between vectors in space? 9. How do I find cross products of vectors in space, and use cross products to find area and volume? cross product, torque, parallelepiped, triple scalar product Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 20 of 26
21 Unit Title: (10) Polar Coordinates and Complex Numbers Chapter 9 Number of Days: 17 days Know: Understand: Do: Many types of graphs are simpler to graph in polar form than rectangular form. These graphs are identified with special names. Applications of polar equations of circles model real life problems. Polar equations, more so than a rectangular equation, allow us to graph illustrations of what we see in nature. Graph points on a polar grid. Find multiple representations of polar coordinates. Graph polar equations. Find the distance between polar coordinates. Methods for converting between polar and rectangular forms of points and equations offer a different mathematical perspective. Find symmetry, zeros and maximum rvalues. Identify and graph classic curves. Convert between rectangular and polar form of coordinates and equations. Find the absolute value of a complex number. Convert between polar form and rectangular form of a complex number. Find the product and quotient of complex numbers in polar form. Apply De Moivre s Theorem. Page 21 of 26
22 Unit Title: (10) Polar Coordinates and Complex Numbers Chapter 9 Number of Days: Key Learning: 17 days Polar equations, more so than a rectangular equation, allow us to graph illustrations of what we see in nature. Unit Essential Question: When is it more appropriate to use polar equations rather than rectangular equations? Polar Coordinates MA.912.T.4.1 MA.912.T How do I graph points with polar coordinates? polar coordinate system, pole, polar axis, polar coordinates, polar equation, polar graph Graphs of Polar Equations MA.912.T How do I graph polar equations? 3. How do I identify and graph classical curves? limacon, cardioid rose, rose, lemniscates, spiral of Archimedes Polar and Rectangular Forms of Equations MA.912.T.4.2 MA.912.T How do I convert between polar and rectangular coordinates and equations? Complex Numbers and De Moivre s Theorem MA.912.T.4.4 MA.912.T How do I convert complex numbers from rectangular to polar form and vice versa? 6. How do I find products, quotients, powers and roots of complex numbers in polar form? complex plane, absolute value of a complex number, polar form of a complex number, trigonometric form of a complex number Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 22 of 26
23 Unit Title: (11) Sequences and Series Chapter 10 Number of Days: 20 Days Know: Understand: Do: Sequences and series are useful in identifying patterns. These patterns can be defined using formulas for arithmetic or geometric sequences. Using the formulas for sequences and series simplifies the solving of real life problems. Identify sequence and series notations and patterns. Find partial sums using sigma notation. Mathematical induction can be used to prove statements involving positive integers. Understand the use of factorial. Recognize, write, and find nth terms and partial sums of arithmetic and geometric sequences. Link arithmetic sequences to linear equations. Use arithmetic and geometric sequences to model and solve real life problems. Link geometric sequences to exponential functions. Find sums of finite and infinite geometric sequences and use to solve real life problems. Use mathematical induction to prove statements. Page 23 of 26
24 Unit Title: (11) Sequences and Series Chapter 10 Number of Days: 20 days Key Learning: Using the formulas for sequences and series simplifies the solving of real life problems. Unit Essential Question: How do I solve real world problems using the formulas for sequences and series? Sequences, Series, and Sigma Notation MA.912.D What different types of sequences exist? 2. How do I use sigma notation to represent and calculate sums of series? finite and infinite sequences, recursive sequence, explicit sequence, Fibonacci sequence, converge, diverge, finite and infinite series, partial sum, sigma notation Arithmetic Sequences and Series MA.912.D How do I find the nth term and arithmetic mean of arithmetic sequences and series? 4. How do I find the sum of n terms of an arithmetic series? arithmetic sequence, common difference, arithmetic means, arithmetic series Geometric Sequences and Series MA.912.D How do I find the nth term and geometric mean of geometric sequences and series? 6. How do I find the sum of n terms on a geometric series? geometric sequence, common ratio, geometric mean, geometric series Mathematical Induction MA.912.D How do I use mathematical induction? principal of mathematical induction, anchor step, inductive hypothesis, inductive step Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 24 of 26
25 Unit Title: (12) Limits and Derivatives Chapter 12 Number of Days: 15 days Know: Understand: Do: Limits provide useful information about the behavior of a function. Methods for finding the slopes of tangent lines can help describe the behavior of a function. Limits can be used to predict the values of a function as x gets infinitely large or infinitely small. Techniques for evaluating limits can be used to solve problems dealing with rates of change, cost, and area. Estimate limits of functions at fixed values and at infinity. Evaluate limits of polynomial and rational functions at selected points using the properties, substitution, factoring, and rationalizing. Evaluate limits of polynomials and rational functions at infinity. The continuity or types of discontinuity of a function on an interval or at a point provide important information about its behavior. Find limits of sequences. Find instantaneous rates of change by calculating slopes of tangent lines. Find average and instantaneous velocity. Page 25 of 26
26 Unit Title: (12) Limits and Derivatives Chapter 12 Number of Days: 15 days Key Learning: Techniques for evaluating limits can be used to solve problems dealing with rates of change, cost, and area. Unit Essential Question: How can evaluating limits be used to help solve problems involving rates of change, cost and area? Estimating Limits Graphically MA.912.C.1.1 MA.912.C.1.4 MA.912.C How do I estimate limits of functions at fixed values and at infinity? OneSided Limit, Two Sided Limit, Unbounded Behavior, Oscillating Behavior Evaluating Limits Algebraically MA.912.C.1.2 MA.912.C.1.3 MA.912.C How do I evaluate limits of polynomial and rational functions at selected points and at infinity? direct substitution, indeterminate form Derivatives MA.912.C How do I find the instantaneous rates of change by calculating derivatives? 4. How do I use the product and quotient rules to calculate derivatives? tangent line, instantaneous rate of change, difference quotient, instantaneous velocity Additional Information: Language Arts Benchmarks used in each concept: LA , LA , LA , LA , LA Glencoe McGrawHill, PRECALCULUS, 2011, is the Pasco County adopted textbook. Page 26 of 26
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