Analysis / AP Calculus Course Outline

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1 Analysis / AP Calculus Course Outline Unit 1: Pre-Calculus Review (1-1.5 weeks) Lines 1. Slope as rate of change 2. Parallel and perpendicular lines 3. Equations of lines Functions and graphs 1. Functions 2. Domain and range 3. Families of function 4. Piecewise functions 5. Composition of functions Exponential and logarithmic functions 1. Exponential growth and decay 2. Inverse functions 3. Logarithmic functions 4. Properties of logarithms Trigonometric functions 1. Graphs of basic trig functions a. Domain and range b. Transformations c. Inverse trig functions Unit 2: Limits and Continuity (1.5 weeks) Estimating limits numerically and graphically Rates of Change Limits at a point 1. Properties of limits 2. Two-sided 3. One-sided Limits involving infinity 1. Asymptotic behavior 2. End behavior 3. Properties of limits 4. Visualizing limits Continuity 1. Continuous functions 2. Discontinuous functions a. Removable discontinuity b. Jump discontinuity c. Infinite discontinuity Instantaneous rates of change

2 Unit 3: The Derivative (2.5 weeks) Definition of the derivative o Understanding the derivative: graphically, numerically, and analytically o Approximating rates of change from graphs and tables of data o Use the calculator to approximate the values of derivatives and definite integrals obtained through analytical means in order to verify that the answers are reasonable Differentiability 1. local linearity 2. Numeric derivatives using the calculator 3. Differentiability and continuity Derivatives of algebraic functions Derivative rules when combining functions Applications of velocity and acceleration Derivatives of trigonometric functions The Chain rule Implicit derivatives 1. Differential method 2. y method Derivatives of inverse trigonometric functions Derivatives of logarithmic and exponential functions Unit 4: Applications of the Derivative (2 weeks) Extreme values 1. Local (relative) extrema 2. Global (absolute) extrema Using the derivative 1. Mean value theorem 2. Rolle s theorem 3. Increasing and decreasing functions Analysis of graphs using the first and second derivatives 1. Critical values 2. First derivative test for extrema 3. Concavity and points of inflection 4. Second derivative test for extrema Optimization problems Linearization problems and Newton s Method Related rates Unit 5: The Definite Integral (1.5 weeks) Approximating areas o Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions that are represented analytically, graphically, and by tables of data 1. Riemann sums

3 2. Trapezoidal rule 3. Definite integrals The Fundamental Theorem of Calculus (part 1) Definite Integrals and antiderivatives 1. The Average Value Theorem The Fundamental Theorem of Calculus (part 2) Unit 6: Differential Equations and Mathematical Modeling (2 weeks) Antiderivatives Integration using u-substitution Separable differential equations 1. Growth and decay 2. Slope fields Use programs in the calculator to show slope fields draw a solution curve on a slope field 3. General differential equations Unit 7: Applications of Definite Integrals (1.5 weeks) Summing rates of change Particle motion Areas in the plane Volumes 1. Volumes of solids with known cross-sections 2. Volumes of solids of revolution a. Disk method and Shell Method Unit 9: Infinite Series (1.5 weeks) Geometric series Power series: 1. Term-by-term differentiation and integration to find power series of new functions Taylor s series / Maclaurin series Lagrange form of the remainder Tests for convergence/divergence: 1. nth term test 2. Direct comparison 3. Ratio test 4. Integral test 5. Limit comparison test 6. Alternating series test (Leibniz s theorem) Radius and interval of convergence Additional explanation of the Analysis / AP Calculus standards can be found at:

4 Text: Calculus 6th Edition by Larson, Hostetler, and Edwards Calculus Graphical, Numerical, Algebraic by Finney Demana Waits Materials: Lined notebook paper, binder, composition notebook, NOTECARDS (a bunch), pencil, and graphing calculator (recommend TI-83 or 84). Grading: The HCHS math department grades each 9 weeks are based on these percentages. Tests and major assessments 40% Homework / Classwork 35% Quizzes 25% Final GRADE: (at the end of the semester) First Nine-Week Average...40% Second Nine-Weeks Average...40% Final Exam (NO EXEMPTIONS )...20% Your average must be 70 or above at the end of each nine weeks to be scaled by 10%. Grade Scale: A: % B: 89-80% C: 79-70% F: (UNACCEPTABLE) 69% - below On major formal assessments, rubrics will be utilized for measuring proficiency performance and comprehension of the standards. The categories will broadly cover mathematical concepts, reasoning, errors, terminology, notation, and neatness & organization. Justification of Answers: Students will be asked to justify their answers on homework, quizzes, and tests, and I prefer that they write the justifications in complete sentences. We talk about the amount of work they need to show and the correct way to justify their work on a variety of problems. We do work problems on the board and students are expected to mathematically explain the concepts and procedures involved. Late work will not be accepted. Graphing Calculator / Technology: Each student has access to a TI Graphing calculator both in and out of the classroom setting. Calculators are used extensively in the classroom during the exploration and discovery process. Students are encouraged to use the calculator in order to enhance their visual understanding of concepts taught in class. Students are instructed how to use the table feature of the calculator and are expected to represent, solve, and numerically compare functions and their behaviors. All graphing calculator AP problems for the previous 10 years (and many earlier ones as well) are worked, discussed, and reviewed in class. Laboratory assignments where data is generated and analyzed are performed during each semester. Students are encouraged to make conjectures concerning functions, their derivatives, their integrals, and other aspects of functions and check these conjectures with a graphing calculator. Many features of the calculator are used to interpret results gained from analytical and numerical work. Many conclusions reached by analytical and numerical means are supported with the graphing calculator. Because the AP Exam is segmented into calculator and non-calculator sections, teaching methods which can be applied to both parts are used. The calculator is presented in the classroom as a

5 useful tool, but students are also encouraged to employ other methods which can be used in the noncalculator section of the AP Exam. In earlier courses you have learned about functions. Functions express the way one variable quantity, such as distance you travel, is related to another quantity, such as time. In calculus you will learn the algebraic formulas for variable rates that will tie together the math you have learned in earlier courses. Graphing calculators help you understand the concepts even before you develop the formulas. In this way you will be able to work calculus problems from the real world starting on day one. During the first few weeks, we spend extra time familiarizing students with their graphing calculators. Later, once you understand the concepts, the formulas will give you time-efficient ways to work these problems. The instruction support graphically will become an often-used tool because it is so quick to do and the graph the calculator produces gives students confidence in their conjectures and answers. The graphing calculator is used to help students develop an intuitive feel for concepts before they are approached through typical algebraic techniques. I use the calculator as a tool to illustrate ideas and topics. I stress the four required functionalities of graphing technology: 1. Finding a root 2. Sketching a function in a specified window 3. Approximating the derivative at a point using numerical methods 4. Approximating the value of a definite integral using numerical methods The time saved by utilizing technology for problem solving and learning concepts can be used to develop ability to write about mathematics. Students are also frequently asked to use the calculator to approximate answers found algebraically to see if they are reasonable. You will be asked to keep a written journal recording the concepts and techniques you have been learning, and verbalize things you have not yet mastered. Therefore you will learn calculus in four ways: algebraically, graphically, numerically, and verbally. This will enable you to relate the various representations to each other. At times you may feel you are becoming submerged in details. When that happens, just remember that calculus involves 4 major concepts: Limits Derivatives Integrals (one kind) Integrals (another kind) If you are absent the day of the test or quiz, you will take it the day you return. Also, when absent you are responsible for getting your missed assignments. me and I will send you the assignment for the day. Please note that homework may be graded for completeness and/or accuracy!! Do not let yourself get behind. I am glad to help students during class and after school. Study Groups: Much of calculus depends on an understanding of a concept taught in a previous lesson. Students are encouraged to form study groups and tutor themselves, this builds understanding. Additionally, math tutoring is available before and after school, please make arrangements with teacher.

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