CURRICULUM MAP: 10097.map

Calculus (AP) 500

TIME FRAME:
GRADE: 11-12
CONTACT:

         MAP LEVEL:

25.1 MATHEMATICS - ALG REASONING: PATTERNS & FUNCTS --- 25.1.1.9.1 --- 25.1.1.9.2 --- 25.1.1.9.3 --- 25.1.1.9.11 --- 25.1.2.9.3 --- 25.1.2.9.5
25.1 MATHEMATICS - ALG REASONING: PATTERNS & FUNCTS --- 25.1.2.9.7 --- 25.1.3.9.2 --- 25.1.1.0.3 --- 25.1.1.1.1 --- 25.1.1.1.4 --- 25.1.1.3.2
25.1 MATHEMATICS - ALG REASONING: PATTERNS & FUNCTS --- 25.1.1.4.2 --- 25.1.1.4.3 --- 25.1.1.5.1 --- 25.1.1.5.2 --- 25.1.1.6.1 --- 25.1.1.9.11
25.2 MATHEMATICS - NUMERICAL & PROP REASONING --- 25.2.2.9.1 --- 25.2.2.9.3
25.3 MATHEMATICS - GEOM & MEASUREMT --- 25.3.1.9.2 --- 25.3.2.9.2 --- 25.3.2.9.4 --- 25.3.2.9.5



1) What are limits? How are they analyzed? Why are they important to the study of calculus?

2) What is a derivative? What techniques are used to find derivatives? How is a derivative related to a tangent line? How can the derivative concept be applied to real world problems?

3) What is integration, and how does it relate to differentiation? What are techniques used to find an anti-derivative, and what does the answer represent? How can the integral concept be applied to real world problems?

4) What is the stylistic focus of the AP exam when dealing with limits, derivatives, and integrals?



This course focuses on the following concepts:
Analysis of Graphs
Limits of Functions
Asymptotic and Unbounded Behavior
Continuity as a Property of Functions
Concept of Derivative
Derivative at a Point
Derivative as a Function
Higher Order Derivatives
Techniques of Differentiation
Applications of Derivatives
Interpretations and Properties of Definite Integrals
Applications of Integrals
Fundamental Theorem of Calculus
Techniques of Antidifferentiation
Applications of Antidifferentiation
Numerical Approximations to Definite Integrals
Introduction to Sequences
Introduction to Series



Students will develop the ability to:

-- Determine the relationship between the geometric and analytic representations of functions and how to use calculus to explain and predict both local and global behavior,
-- Explain the limiting process,
-- Calculate limits using algebra,
-- Estimate limits from graphs or tables of data,
-- Explain asymptotes in terms of graphical behavior,
-- Describe asymptotic behavior in terms of limits involving infinity,
-- Compare relative magnitudes of functions and their rates of change,
-- Explain what makes a function continuous or discontinuous,
-- Relate continuity to limits,
-- Present the derivative graphically, numerically, and analytically,
-- Relate the derivative to instantaneous rate of change,
-- Use the definition of the derivative to find the derivative,
-- Explain the relationship between differentiability and continuity,
-- Find the derivative at a point,
-- Identify where there are vertical tangents and removable discontinuities,
-- Approximate rate of change from graphs and tables,
-- Relate the graph of a function to the graph of its derivative,
-- Translate verbal descriptions into equations involving derivatives and vice versa,
-- Relate the graph of a function to its second derivative graph,
-- Identify points of inflection,
-- Analyze curves in terms of increasing and decreasing intervals and concavity,
-- Find maximums and minimums through equation analysis,
-- Model rates of change including related rate problems,
-- Differentiate implicitly,
-- Differentiate power, exponential, logarithmic, trigonometric, and inverse trigonometric functions,
-- Differentiate sums, products, and quotients of functions,
-- Differentiate using the chain rule,
-- Compute Riemann sums using left, right, and midpoint evaluation points,
-- Explain the definite integral as a limit of Riemann sums over equal subdivisions,
-- Explain the definite integral as the change of the quantity over the interval,
-- Explain the properties of definite integrals,
-- Solve application problems that involve the area of a region, the volume of solids of revolution using the disk and shell methods, the volume of a solid with known cross sections, the average value of a function, and the distance covered by a particle along a line,
-- Use the Fundamental Theorem of Calculus to represent and evaluate definite integrals,
-- Solve separable differential equations and use them in modeling,
-- Use Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values,
-- Use derivatives of basic functions to antidifferentiate,
-- Use u-substitution of variables to antidifferentiate,
-- Use integration by parts to integrate,
-- Use properties of trigonometric powers to integrate,
-- Use trigonometric substitution to integrate,
-- Use partial fractions to integrate,
-- Analyze the behavior of sequences and series.



All students will:

-- Have daily exposure to mechanical, theoretical, and application-based AP-style questions,
-- Utilize the graphing calculator extensively to supplement pencil and paper techniques.




Students will be assessed by:

1. Daily homework assignments, which will acount for 10% of their final grade

2. In-class pop quizzes, one per week

3. Take-home quizzes, six questions per week

4. Unit tests (comprised of 50% multiple choice and 50% free response) designed from released AP exam questions and graded in a manner identical to the AP exam every one to two weeks

5. Projects that are 100% application-based

6. Final exam that covers an entire semester's material. This final exam will account for 20% of their final grade.



Projects that involve hands-on experience with differentation and integration (e.g.related rates, optimization, area, or volume) and research into areas not specifically covered by the curriculum (e.g. math history).



1. Course Materials: unit packets made from available released AP exam questions (used in place of a traditional text from January - May)

2. Course Text - Calculus with Analytic Geometry - 6th Edition - Larson & Hostetler (used in May and June)

3. Supplementary texts

4. Graphing calculator

5. Graphing calculator projector