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AP Calculus Curriculum Guide Grade 12 |
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Objectives |
Core Curr. |
Instructional |
Assessment |
Resources |
GEPA HSPA Terra Nova |
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CHAPTER P Carlestan Plane & Function (10 days) The student will be able to: |
References and Materials Major Text: Calculus, 5th Edition,Larson, Ronald E., Robert P. Hostetler, and Bruce H. Edwards, Lexington, MA, DC Heath & Co., 1994 |
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Section 1 ! Review the real numbers, the number line and the Cartesian Plane. |
4.1 4.4 4.8 |
Define whole #, integer, rational and irrational & real. Non-negative order, bounded and unbounded solutions of problems. | Problems, p. 8-9. Discuss interest, profit, production. Word problems related to topics |
Reference Books: Texts Calculus-Vol I, 6th Edition, Anton, Howard, NY, John Wiley & Sons, Inc. 1998 |
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Section 2 ! Do the Cartesian Plane, Distance & Midpoint Formulas Equations of Circle. |
4.3 4.5 4.13 |
Review formulas - Importance of Pythagorean Theorem, Multiple movements of circles. | Problems p. 15-16. Problems on Federal Debt, Life Expect Satellite Communication, Building Design, René Descartes, p. 10, Pierre DeFermate, p. 14 | Calculus Firm Graphical, Numerical, and Symbolic Points of View, Ostebee, Arnold & Paul Zorn, NY, Saunders College Publishing, 1997 | |
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Section 3 ! Do graph of equation, Intersection & Symmetry.
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4.2 4.3 4.16 |
Define, identify and apply concepts of functions with respect to domain, range, intercepts, symmetry, asymptotes, zeros, and odd & even functions. | Problems p. 23-24 Including Break-even Analysis, Cost of Living, Agriculture and Saturated Steam | Calculus-Concepts & Contents, Stewart, James, Pacific Grove, CA; Brooks/Cole Publishing Co., 1998 | |
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Section 4 ! Do lines in a plane.
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4.1 4.5 4.11 |
Define & discuss concepts of slope (+, -, 0, and no) Equations of lines [slope-intercept form, two point, etc.], sketching graphs & parallel and ¦ lines. | Problems p. 31-33, Including-Temperature Conversion, Reimbursement Expenses, Career Choice, Depreciation and Apartment Rental, Career Interview, p. 33 | Calculus, 7th Edition, Varberg, Dale & Edwin Purcell, Upper Saddle River, NJ, Prentice Hall, 1997 | |
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Section 5 ! Do functions.
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4.2 4.11 4.13 |
Identify graphs of functions. Apply the algebra of functions by finding sum, difference, produce, quotient, composition, and inverse where they exist. Describe transformations of graphs. | Exercise, pp. 42-44 Including 2 essay questions (23 & 24), Ripple, Automobile, Aerodynamics and Volume | ||
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Section 6 ! Review trigonometric functions. |
4.3 4.5 4.8 4.16 |
Define and discuss angles and degree measure, radians measure, evaluating trigonometric functions, solving equations and graphing. | Exercise, pp. 53-57-Math models-instrumentation, height of a mountain, machine shop calculation, musical sound and blood pressure | ||
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Review Section ! Review Chapter P
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4.1 4.18 |
Go over all aspects - do representative sample of problems, pp. 57-59 |
Test Chapter P Book and text driven test |
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CHAPTER 1 Limits and Their Properties (10 days) |
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| ! Introduce Limits. |
4.4 4.5 4.10 4.15 |
Explain the tangent line problem, introduction to limits, limits that fail to exist, and a formal definition of a limit. |
Exercises p. 68-69-omit E-S proofs-Show graphing calculator interpretation Peter Gustave Durchlet, p. 65 Augustin, Louis Cauchy p. 66 |
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| ! Do properties of algebraic and trigonometric function limits. |
4.2 4.5 4.15 |
Use properties of limits to calculate the limit of a sum, difference, product, and quotient of functions including trigonometric functions. |
Career Interview, p. 75 Problems, pp. 74-75 |
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| ! Evaluate limits. |
4.1 4.5 4.15 4.16 |
To find a strategy for finding limits including cancellation and rationalization techniques. Introduce Squeeze Theorem Use technology to test. | Problems, pp. 82-83 including free-falling object | ||
| ! Do continuity and one sided limits. |
4.4 4.5 4.11 4.15 |
Discuss continuity at a point and on an open interval, one-sided limits and continuity on closed interval, properties of continuity and the intermediate value theorem. |
Charles’ Law & Absolute Zero Technology, p. 88 Exercises, pp. 92-95 including 65-68 essays; include problems on Salary Contract, and Telephone Rate |
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| ! Do Infinite Limits. |
4.3 4.5 4.15 |
Define infinite limits use concept of infinite limits to find vertical asymptote of functions. | PP. 101-Problems including Boyles Law, moving paddles average speed | ||
| ! Review Chapter 1. |
4.5 4.18 |
Go over all concepts Chapter 1 including word problems. |
Test Chapter 1 Include word problems, pp. 102-103 Test #1 AC |
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| CHAPTER 2 Differentiation (23 days) |
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| Do the derivative and tangent sine problems. | 4.3 4.7 4.15 |
Find the slope of a tangent line and write the equation of a tangent line using the limit process. Discuss different ability and continuity. | Isaac Newton p. 105 PP. 112-113 Calculator Technology using definition of derivative |
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| Do basic differentiation rules and rates of change. | 4.4 4.5 4.16 4.17 |
Define and compute the derivative of a function using constant rule, power rule, constant multiple rule, sum and difference rules. Derivatives of sine and cosine functions. Introduce rates of change. | Rate of Change p. 121, Technology p. 120, pp. 123-126 including essay questions 53 & 54, vertical motion, stopping distance, profit and inventory management | ||
| Do product and quotient rules and higher order derivatives. | 4.3 4.5 4.13 4.15 |
Define the derivative of a function using the product and quotient rules. Derivative of additional trigonometric functions. Higher order derivatives (2nd, 3rd, etc.) Compute successive derivatives of functions. | PP. 134-136-Including Boyle’s Law Acceleration, Population, Growth and Inventory. Career Interview p. 136 | ||
| Do the Chain Rule. | 4.3 4.5 4.15 |
Use the Chain Rule to differentiate composite functions-both algebraic and trigonometric application of the General Power Rule. | PP. 143-145 Including Doppler effect, harmonic motion, pendulum, wave motion and circulatory system | ||
| Do Implicit Differentiation. | 4.3 4.4 4.15 |
Define and demonstrate differences between implicit and explicit functions. Find the derivative of implicity-define functions. |
Technology uses-Isaac Burrow & Kappa Curve, p. 150 PP. 151-152-including Orthogonal Projections and Optical Illusions |
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| Do related rates. | 4.3 4.5 4.7 4.15 |
Define the derivative of a function in variety of ways including rates of change of the function and instantaneous velocity. Apply the concepts of average and instantaneous rates of change of a function. | PP. 158-161-Including area, volume, depth, construction, air traffic control, base bale, mechanical design and electricity | ||
| Review chapter. | 4.2 4.18 |
Go over all possible materials in review exercises, pp. 161-163. Do additional work in past AP problems. | Test Chapter 2-Teacher’s Resource Guide and previous AP questions | ||
| CHAPTER 3 Applications of Differentiation (25 days) |
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| Do extreme on an interval. | 4.3 4.5 4.13 4.15 |
Apply the Extreme Value Theorem to find the maximum and minimum value of a function on a closed interval. Define and explain concept of critical numbers. | PP. 170-172-Including technology, inventory cost, lawn sprinkler and honey comb | ||
| Do Rolle’s Theorem and the Mean Value Theorem. | 4.1 4.5 4.15 4.16 |
State and apply Rolle’s and Mean-Value Theorem. Apply concepts to finding tangent lines and instantaneous rate of change. | P. 175 Joseph-Louis LaGrange PP. 177-178 Including Render Costs, vertical motion and sales |
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| Do increasing and decreasing functions and the first derivative test. | 4.3 4.5 4.11 4.15 |
Use the first Derivative Test to find the intervals on which a function is increasing and decreasing and to determine relative extreme of a function. |
PP. 186-188, Problem including Drug Concentration, Electrical Resistance and Rainbow’s |
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| Do concavity and the Second Derivative Test. | 4.3 4.5 4.11 4.15 |
Use the Second Derivative Test to determine intervals of concavity of a function and to locate inflection prints. Use the Second Derivative Test to analyze relative extreme of a function. | PP. 194-196 Multi-problem including Beam Deflection, Electric Field Intensity, Average Cost, and Engine Design | ||
| Do Limits at Infinity. | 4.1 4.5 4.6 4.13 |
Calculate limits at infinity and use the concept of limits at infinity to determine horizontal asymptote of functions. Technology-graphing calculators showing limiting processes. | P. 200 Maria Agnesi PP. 203-204 including Average Cost, Relativity and Demographics Problems from AP Tests |
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| Do a summary of curve sketching. | 4.3 4.5 4.15 4.16 |
Use information about intervals of increase and decrease, relative extreme, intervals of concavity, and inflection points to sketch the graph of a function. Use information about the derivative of a function to determine a sketch of the graph of the function and determine the concavity of the function. | Technology Problems AP Problems PP. 211-212-Including 3 essay questions with calculator |
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| Do Optimization Problems. | 4.3 4.5 4.11 4.15 |
Use derivative to solve applied minimum and maximum problems including topics of area, volume, distance, and rates of growth. Apply derivative to problem solutions involving speed, velocity, and acceleration. | AP Test Problems PP. 218-223-problems include chemical reaction, traffic control, area, volume, illumination, maximum and minimum time, force, and function |
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| Do Newton’s Method. | 4.2 4.3 4.5 4.13 |
Use Newton’s method to approximate the zeros of a function. Define iterations and explain possible failure. Explain algebraic solutions of polynomial equations. | PP. 228-230-Essay problem #19 Medicine and Advertising Cost problems; Career Interview, p. 230 |
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| Do differential. | 4.1 4.5 4.8 4.15 |
Use differentials to obtain linear approximations. Discuss error propagation | P. 234 Gottfried Leibniz PP. 236-237-problem using differentials in Ohm’s Law, Environment, Area and Projectile Motion |
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| Review chapter. | 4.2 4.18 |
Review all aspects of chapter. Graphing with and without calculator.
Using 2nd derivative to find graph. PP. 244-247-all problems |
Test Chapter 3-Teachers Resource Book Tests 3 A-C AP Test Problems |
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| CHAPTER 4 Integration (30 days) |
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| Do antiderivative and indefinite integration. | 4.3 4.7 4.8 4.15 4.18 |
Define and differentiation or integration. Compute simple integrals using basic rules. Use the antiderivative to solve problems involving motion along a straight line. | PP. 256-258 Multiple problem dealing with vertical motion, acceleration and marginal cost. Some AP type questions | ||
| Introduce area under the curve. | 4.1 4.7 4.9 4.15 |
Define Sigma notation and summation formulas. Understand the concept of area under the curve using upper and lower sum. Understand the concept of area under a curve using Riemann sum over equal subdivisions. | P. 261 Archimedes PP. 267-270 Approximations Compute program for "Monte Carlo Method". |
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| Do Riemann Sums and Definite Integrals. | 4.1 4.5 4.11 4.15 |
Compute Riemann sums using left end points, right endpoints, and midpoints as evaluation points. Use the limit of Riemann sum to calculate a definite integrals. | P. 271 George Riemann P. 277-279 |
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| Do the Fundamental Theorem of the Calculus. | 4.3 4.4 4.15 |
Use the First Fundamental Theorem of Calculus to evaluate definite integrals. Calculate antiderivatives using substitution of valuables and change of limits. Use the Mean Value Theorem for integrals to find the average value of a function on an interval. | PP. 289-291-Evaluates the definite integral with calculator. Work with depreciation, average profit and operating cycle. | ||
| Do integration by substitution. | 4.1 4.5 4.15 4.17 |
Use the Second Fundamental Theorem of Calculus to find derivatives. Calculate antiderivatives using substitution of variables and change of limits. Integration of even and odd functions. | PP. 301-303, Miscellaneous problems including cash flow, marginal cost,
sales and electricity Additional AP Problems |
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| Do numerical integration. | 4.4 4.5 4.8 4.15 |
Use the Trapezoidal Rule to approximate area under a curve. Use Simpson’s Rule and Error Analysis. | PP. 311-313, Look at area problems with trapezoidal rule and Simpson’s rule | ||
| Review chapter. | 4.2 4.17 4.18 |
Review problems on pages 313-315. | Test-Chapter 4 Resource Book and AP type questions |
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| CHAPTER 5 Logarithmic, Exponential, and other Transcental Functions (Differentiation) (15 days) |
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| Do the Natural Logarithmic Function and Differentiation. | 4.3 4.5 4.8 4.16 |
Review the properties of the natural log function and the exponential
function. Differentiate the logarithmic function. Define letter ‘e’. |
PP. 324-326-Problems include graphing, sum and difference, etc. as well as sound intensity, home mortgage and boiling point | ||
| Do inverse functions. | 4.3 4.5 4.11 4.15 |
Define inverse functions. Review domain and range. Discuss existence of inverse function. Find the derivative of the inverse of a function. | PP. 341-343, Assorted problems and with without calculator | ||
| Do differentiation of exponential functions. | 4.2 4.3 4.5 4.17 |
Define the Natural Exponential Function. Differentiate exponential functions. Do properties of natural exponential function. The normal density curve. | PP. 350-351 problems through 66 with essay questions AP Questions |
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| Do bases other than ‘E’. | 4.3 4.5 4.8 4.15 |
Discuss bases other than ‘e’. Define log function to base a derivative for bases other than ‘e’. Application of exponential functions. Review compound interest formulas. |
PP. 359-360 problems to 59 Graphing, derivatives, depreciation, compound interest, and learning theory |
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| Do Inverse Trigonometric Functions and Differentiation. | 4.4 4.5 4.13 4.15 |
Define inverse trig functions. Review limitations. Find derivative of inverse trigonometric functions. Review base differentiation rules for elementary functions. | Galileo, p. 376 PP. 377-378 Assorted problems, falling objects, rising balloon, etc. |
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| CHAPTER 5 Logarithmic, Exponential, and other Transcental Functions (Integration) (5 days) |
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| Do the Natural Logarithmic Function and Integration. | 4.2 4.5 4.15 |
Use log rules for integration. Use the integrals of trigonometric functions. | PP. 333-334-Miscellaneous problems-population growth, heat transfer and
average price P. 334 Career Interview |
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| Exponential Functions-Integration. | 4.2 4.5 4.15 4.16 |
Do integral of exponential functions. Find areas bounded by exponential functions. | PP. 351-problems 67-98 | ||
| Do Differential Equations. | 4.2 4.4 4.5 4.11 |
Apply the antiderivative to solving problems such as exponential growth and decay. Differential equation. | PP. 367-369-Set up equations and solve. Do word problems on radioactive
decay, compound interest, and learning curve. Write essay problem #39 |
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| Review chapter. | 4.2 4.17 4.18 |
Go over questions from 5.2-5.4 and 5.6- integration section do selective problem from review set. | Test Integration portion of Chapter 5 - Multiple Choice, Essay questions from previous AP tests | ||
| CHAPTER 6 Applications of Integration (15 days) |
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| Do Area of Region between Two Curves. | 4.3 4.5 4.7 4.15 |
Use the definite integral to find the area under a curve. Use the definite integral to find the area between two curves. Discuss representative rectangle. Working from a bar graph. | PP. 409-411, as many types of problems as possible. AP type questions |
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| Do Volume: Dis |
4.3 4.5 4.7 4.16 |
Find the volume of a solid of revolution using the disc and washer methods. Solid of revolution with hole. Find the volume of a solid with known cross sections. | PP. 420-423-various types of problems-mostly word AP type questions. Set up interval- alternate solution by calculator |
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| Do Volume: Shell | 4.1 4.5 4.7 4.16 |
Find the volume of solid revolution using the shell method. Compare and contrast disc and shell methods. Do integration by calculator. |
PP. 429-430-representative sample of problems (at most 10) to show concepts | ||
| Review three sections of chapter. | 4.2 4.17 4.18 |
Go over problems from review section. | Test-Resource Book and AP type questions | ||
| TEST PREPARATION (15 days or more if possible) |
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| 4.1, 4.2 4.3, 4.4 4.5, 4.15 4.16, 4.17 4.18 |
Take multiple choice type problems from 1988 & 1993 released exams.
Use other multiple choice questions from review book. Work out all the open ended questions from 1993-1997. |
Problems given for homework graded according to College Board Standards | |||
| POST TEST CHAPTER 7 Integration Techniques and L’Hopital’s Rule (10 days) |
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| Do Basic Integration Rules. | 4.3 4.5 4.16 |
Use more than one basic integration rule to evaluate integrals beak a quotient into two parts to evaluate integrals. Use trigonometric identities to change one integral into a form that can be evaluated directly. |
PP. 476-478, Multiple problems of each type using calculator for
solutions
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| Do Integration by Parts. |
4.1 |
Evaluate a integral using simple integration by parts. Do repeated integration by parts. Use tabular form. | PP. 486-488, Do approximately 8 problems of varying degrees of difficulty | ||
| Do partial fractions. | 4.3 4.13 4.15 |
Define method of partial fractions. Discuss decomposition of factors. Linear factors. |
Career Interview, p. 508 John Bernoulli, p. 509 P. 517, 7-8, 27-34 |
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| Do Indeterminate Forms and L’Hopital’s Rule. | 4.3 4.5 4.13 4.15 |
Define indeterminate forms, i.e.: 0/0, , 1 , etc. Use L’Hopital’s Rule to evaluate limits of indeterminate forms. |
P. 526 Guillaumi L’Hopital P. 532, 1-40 |
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| Review chapter. | 4.2 4.3 4.18 |
Review sections of Chapter 7. Go over problems in review exercises. | Test Chapter 7, Use BC questions | ||
NOTE SPECIAL EDUCATION MODIFICATIONS SUGGESTIONS. SEE IEP FOR SPECIFIC ACCOMMODATIONS.
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