Mrs. Regina Ramey
Boone County Honors Academy
Calculus Graphical, Numerical, Algebraic, 3rd edition; Finney, Demana, Waits, Kennedy
3-ring binder, Pencil, Notebook, Graph Paper, and Graphing Calculator
Before studying calculus, all students should complete four courses of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on) and know the values of the trigonometric functions of the numbers 0, pi/6, pi/4, pi/3, pi/2, and their multiples.
Students should be able to:
§ work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.
§ understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems.
§ understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and should be able to use integrals to solve a variety of problems.
§ understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.
§ communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.
§ model a written description of a physical situation with a function, a differential equation, or an integral.
§ use technology to help solve problems, experiment, interpret results, and verify conclusions.
§ determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
§ develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.
1. Use graphical and numerical evidence to estimate limits and identify situations where limits fail to exist.
2. Apply rules to calculate limits.
3. Use the limit concept to determine where a function is continuous.
1. Use the limit definition to calculate a derivative, or to determine when a derivative fails to exist.
2. Calculate derivatives (of first and higher orders) with pencil and paper, without calculator or computer algebra software, using:
(a) Linearity of the derivative;
(b) Rules for products and quotients and the Chain Rule;
(c) Rules for powers, trigonometric and inverse trigonometric functions, and for logarithms and exponentials.
3. Use the derivative to find tangent lines to curves.
4. Calculate derivatives of functions defined implicitly.
5. Interpret the derivative as a rate of change.
6. Solve problems involving rates of change of variables subject to a functional relationship.
7. Approximate functions by using linearization (differentials).
III. Applications of the Derivative
1. Find critical points, and use them to locate maxima and minima.
2. Use critical points and signs of first and second derivatives to sketch graphs of functions:
(a) Use the first derivative to find intervals where a function is increasing or decreasing.
(b) Use the second derivative to determine concavity and find inflection points.
(c) Apply the first and second derivative tests to classify critical points.
3. Use Differential Calculus to solve optimization problems.
4. Apply the Mean Value Theorem.
IV. The Integral
1. Find antiderivatives of functions; apply antiderivatives to solve separable first-order differential equations.
2. Use the definition to calculate a definite integral as a limit.
3. Apply the Fundamental Theorem of Calculus to evaluate definite integrals and to differentiate functions defined as integrals.
4. Calculate elementary integrals with pencil and paper, without calculator or computer algebra software, using:
(a) Linearity of the integral;
(b) Rules for powers (including exponent -1) and exponentials, the six trigonometric functions and the inverse sine, tangent and secant;
(c) Simple substitution.
V. Transcendental Functions
1. Use the relation between the derivative of a one to one function and the derivative of its inverse.
2. Calculate with exponentials and logarithms to any base.
3. Calculate derivatives of logarithmic, exponential and inverse trigonometric functions; interpret and apply such derivatives as usual.
4. Use logarithmic differentiation.
5. Use models describing exponential growth and decay.
TENTATIVE PACING GUIDE
Chapter 1 Prerequisites for Calculus 11 days
Chapter 2 Limits and Continuity 10 days
Chapter 3 Derivatives 30 days
Chapter 4 Applications of Derivatives 28 days
Chapter 5 The Definite Integral 26 days
Chapter 6 Differential Equations and Mathematical Modeling 22 days
Chapter 7 Applications of Definite Integrals 21 days
This timeline gives approximately 15 days to “review” the course before the exam.
Put your knowledge to the test: The AP Calculus AB Exam
assesses your mastery of
The AP Calculus AB Exam is 3 hours and 15 minutes. The 105-minute, 45-question multiple-choice section tests your proficiency on a wide variety of topics. The 90-minute, six-problem free-response section gives you the chance to demonstrate your ability to solve problems using an extended chain of reasoning.
The multiple-choice section of the exam has two parts. For Part A, you'll have 55 minutes to complete 28 questions without a calculator. For Part B, you'll have 50 minutes to answer 17 questions using a graphing calculator. For more information, see the calculator policy for the AP Calculus Exams.
Unlike other multiple-choice tests, random guessing can hurt your final score. While you don't lose anything for leaving a question blank, one quarter of a point is subtracted for each incorrect answer on the test. But if you have some knowledge of the question and can eliminate one or more answers, it's usually to your advantage to choose what you believe is the best answer from the remaining choices.
The free-response section tests your ability to solve problems using an extended chain of reasoning. You'll have 45 minutes for each of the two parts of the free-response section. In Part A, you'll answer three questions using a graphing calculator. In Part B, you'll answer three questions without a calculator. During the second timed portion of the free-response section (Part B), you are permitted to continue work on problems in Part A, but you are not permitted to use a calculator during this time. For more information, see the calculator policy for the AP Calculus Exams.
The multiple-choice and free-response sections each account for one-half of your final exam grade. Since the exams are designed for full coverage of the subject matter, it is not expected that all students will be able to answer all the questions.
AP Grade Reports are sent in July to the college you designated on your answer sheet, to you, and to your high school. Each report is cumulative and includes grades for all the AP Exams you have ever taken, unless you have requested that one or more grades be withheld from a college or canceled.
Karl's Calculus Tutor http://www.karlscalculus.org/
Ask Dr. Math http://mathforum.org/dr.math/dr-math.html
TI Graphing Calculators http://education.ti.com/educationportal/
Tutorials for the Calculus Phobe http://www.calculus-help.com/funstuff/phobe.html
Mr. Calculus http://users.adelphia.net/~sismondo/index.html
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