Advanced Placement Calculus (AB) Rev 4/16/02



CHAPTER 1-FUNCTIONS

Students will be able to identify R -- Plotting points and lines Quizzes 2.2.11f

the properties of elementary functions --Slope of a line Use of Graphing Calculator 2.4.11b, c, e

and apply these to analytic geometry -- Sketching quadratic and cubic equations Test 2.5.11 b,c,d

-- Distance between points 2.8.11e,g,h,j,

-- Describing a line k,l,m,n,

point / slope form o,p,q, r

slope / intercept form 2.9.11i

two point form

standard form

-- Absolute value function

algebraically

geometrically

-- Functions and use functional notation

including domain and range

-- Zeros of a function

-- Sums, Differences, products, and quotients of

functions

-- Composite functions

-- Odd and Even functions

-- Inverse functions

-- Set and interval notation

-- Trigonometric and Inverse Trigonometric Functions

-- Radian measure

--Trigonometric identities

-- Parametrically defined functions

CHAPTER 2-LIMITS & CONTINUITY

Students will be able to define and L -- Definition and computation of the limit of an algebraic Quizzes 2.2.11f

use the concepts of limits expression in one variable Use of Graphing 2.4.11b, c

(Sections 2.1 - 2.3, 4.5) -- Limit notation and calculations Calculator 2.5.11 b,c,d

-- Limits of sums, differences, products, and quotients Test

Students will analyze the -- Limits of composite functions

Continuity of a function -- Use of limits to analyze horizontal and vertical

(Sections 2.5-2.6 ) asymptotes

-- Use of L’Hopital’s Rule to determine the limit of

an expression of indeterminate form (Sec 4.5)

L -- Definition and determination of continuity at a point c Quizzes 2.2.11f

-- Determination of the domain for which a given Use of Graphing 2.4.11b, c

function is continuous Calculator 2.5.11 b,c,d

-- Removable and nonremovable discontinuity Test

CHAPTER 3 & 4 THE DERIVATIVE

Students will be able to use the L -- Definition of the derivative of a function as a limit Quizzes 2.2.11f

definition of the derivative and (Section 3.1 - 3.2) graded assignments 2.4.11b, c

the rules of differentiation. -- Use of the definition to compute the derivative Test 2.5.11 b,c,d

of various functions (Section 3.3) Use of Graphing 2.8.11s,t

-- Differentiability of a given function Calculator

-- Use of the derivative to find the slope of a tangent line

-- Derivatives of sums, differences,

products and quotients

-- Derivatives of a composite function

the “Chain Rule” (Section 3.5)

-- Derivative of an inverse function

(Section 4.1)

-- Recognition of a relation requiring implicit differentiation

(Section 3.6)

-- Computation of derivatives implicitly

-- Definition of “differential” (Section 3.8)

-- Derivatives in differential notation

-- Increment of a function

-- Exponential functions and their derivatives (Sections 4.2 - 4.3)

-- Logarithmic functions and their derivatives

-- Trigonometric functions and their derivatives (Section 3.4)

-- Inverse trigonometric functions and their derivatives

(Section 4.4)

CHAPTER 5-DERIVATIVE APPLICATION

Students will be able to apply the L -- Equation of Tangent and Normal Lines Quizzes 2.2.11f

concept of derivatives to -- Linear Approximation (Section 3.8) Use of Graphing 2.4.11b, c

curve sketching and modeling -- Related Rates (Section 3.7) Calculator 2.5.11 b,c,d

--Average and Instantaneous Rates of Change graded assignments 2.8.11 t

-- Curve analysis: including... (Sections 5.2-5.3,5.5) Test 2.9.11 i

vertical and horizontal asymptotes 2.10.11a,b

intercepts

intervals of increase/decrease/ concavity

extrema (relative and absolute)

inflection points

points of nondifferentiability

-- Applied Extrema Problems (Section 5.6)

-- Rolle’s Theorem Section (5.8)

-- Mean Value Theorem (for derivatives)

-- The Intermediate Value Theorem

-- Newton’s Method (Section 5.7)

CHAPTER 6-INTEGRATION

Students will be able to use L -- Integration of xa(power rule) (Section 6.2) Quizzes 2.2.11f

Techniques of Integration -- Integration using u substitution (Section 6.3) graded assignments 2.4.11b, c

-- Integration of Trigonometric Functions Use of Graphing Calc. 2.5.11 b,c,d

Test

Students will be able to solve problems L -- Definition and computation the indefinite integral Quizzes 2.4.11b, c

using the indefinite integral -- Initial Value Problems and the constant graded worksheets 2.5.11 b,c,d

of integration (Section 6.2) Test

-- Integrating separable differential equations (Section 9.1)

-- Rectilinear Motion-Derivative (Sec 5.4)

-- Rectilinear motion-Integral (Sec 6.7)

-- Euler’s method and Slope fields (Sec 9.2)

-- Exponential Growth/Decay (Sec 9.3)

-- Newtons Law of Cooling (Sec 9.3)

Students will be able to utilize L -- Estimation of the area under a curve using a finite number Quizzes 2.2.11f

limits as a means of of rectangles (left hand/ right hand) (Section 6.4) graded worksheets 2.4.11b, c

determining the area under a curve -- Using limits, determining the exact area under a curve Use of graphing calc. 2.5.11 b,c,d

(Section 6.5) 2.9.11 i

-- Estimation of the area using the trapezoidal and midpoint rules 2.10.11d

Students will apply the Definite L -- The Fundamental Theorem of Calculus Quizzes 2.4.11b, c

Integral to analyze functions (Section 6.6, 6.8) graded worksheets 2.5.11 b,c,d

-- Functions defined by integration Test 2.10.11c

--Integrals involving ln(x) and exp(x) (Section 6.9)

CHAPTER 7-APPLICATIONS OF

THE DEFINITE INTEGRAL

Students will apply the Definite L -- Area under a curve (or between curves) Quizzes 2.4.11b, c

Integral to solve Practical Problems (Section 7.1) graded worksheets 2.5.11 b,c,d

-- Volumes of solids (Section 7.2 - 7.3) Test 2.9.11i

of rotation around a horizontal of vertical axis 2.10.11 e

(Disk/Washer method)

formed by perpendicular slices to a given axis

-- Displacement and distance traveled in a specific

time interval (Section 6.7)

-- Average Value Problems

TOPICS TAUGHT AFTER THE AP EXAM

CHAPTER 8-PRINCIPALS OF

INTEGRAL EVALUATION

-- Volumes of solids (Section 7.2 - 7.3)

(Cylindrical Shell Method)

-- Integration by Parts (Section 8.2)

-- Integration by trig. substitution (Section 8.3-8.4)

-- Integation by completing the square

ADDITIONAL ASSESSMENT: AP FREE RESPONSE QUESTIONS

Below is a list of the released AP Free Response Questions (FRQ) that are assigned, solved, discussed and tested. Students are assigned the FRQs over a 2-3 week period in the time frame indicated. Students work collaboratively both in class and out to solve the problems. Students are then quizzed on selected problems. The quizzes are strictly graded according to the AP 9 point rubric.

|Topic |AB Curriculum |Assignment Time Line |

| | | |

|Local Linearity & Implicit Differentiation |1995 AB 3 |Early October |

|Local Linearity & Implicit Differentiation |2000 AB/BC 5 |  |

|Local Linearity & Implicit Differentiation |2006 AB6 (After ch 4) |  |

| | | |

|Related Rates |1994 AB 5 |Late October |

|Related Rates |1995 AB 5 |  |

|Related Rates |2002B AB 6 |  |

|Related Rates |2005B AB 5 |  |

| | | |

|Curve Sketching/Applied Max&Min |1990 BC 3 |December |

|Curve Sketching/Applied Max&Min |1993 AB 4 |  |

|Curve Sketching/Applied Max&Min |2000 AB 3 |  |

|Curve Sketching/Applied Max&Min |2001 AB 4 |  |

|Curve Sketching/Applied Max&Min |2004 B AB4 |  |

|Curve Sketching/Applied Max&Min |2005 AB 4 |  |

|Curve Sketching/Applied Max&Min |2006B AB 2 |  |

|Curve Sketching/Applied Max&Min |2006B AB 3 |  |

| | | |

|Slope Fields/Euler's/Diffeq |1998 AB 4 |Early March |

|Slope Fields/Euler's/Diffeq |1998 BC 4 |  |

|Slope Fields/Euler's/Diffeq |2000 BC 6 |  |

|Slope Fields/Euler's/Diffeq |2003 AB/BC 5 |  |

|Slope Fields/Euler's/Diffeq |2005 AB 6 |  |

|Slope Fields/Euler's/Diffeq |2005B AB 6 |  |

|Slope Fields/Euler's/Diffeq |2006 AB5 |  |

|Slope Fields/Euler's/Diffeq |2006 B AB5 |  |

| | | |

|Accumulation/Fundamental Theorem |2003 AB 6 |Late March |

|Accumulation/Fundamental Theorem |2003 AB 4 |  |

|Accumulation/Fundamental Theorem |2002 AB/BC 2 |  |

|Accumulation/Fundamental Theorem |2000 AB 4 |  |

|Accumulation/Fundamental Theorem |1999 AB/BC 5 |  |

|Accumulation/Fundamental Theorem |2004B AB 2 |  |

|Accumulation/Fundamental Theorem |2005 AB 2 |  |

|Accumulation/Fundamental Theorem |2006 AB 3 |  |

| | | |

|Rectilinear Motion |1998 AB 3 |Early April |

|Rectilinear Motion |2000 BC 2 |  |

|Rectilinear Motion |2002 AB 3 |  |

|Rectilinear Motion |2003 AB 2 |  |

|Rectilinear Motion |2005B AB 3 |  |

|Rectilinear Motion |2006 AB/BC 4 |  |

|Rectilinear Motion |2006B AB 6 |  |

|Approximation |1996 AB 3 |  |

|Approximation |2002 AB 6 |  |

|Average Value |2004AB/BC1 |  |

|Average Value |2005 AB 3 |  |

|Average Value |2005 AB 5 |  |

|Average Value |2006 AB/BC 2 |  |

|Average Value |2006B AB 4 |  |

| | | |

|Area/Volume/Cross Sectional Volume |2002 AB 1 |Late April |

|Area/Volume/Cross Sectional Volume |2002B AB 1 |  |

|Area/Volume/Cross Sectional Volume |2003 AB 1 |  |

|Area/Volume/Cross Sectional Volume |2004B AB6 |  |

|Area/Volume/Cross Sectional Volume |2005 AB 1 |  |

|Area/Volume/Cross Sectional Volume |2005B AB1(cross sec) |  |

|Area/Volume/Cross Sectional Volume |2006 AB/BC 1 |  |

ADDITIONAL ASSESSMENT: END OF THE YEAR REVIEW

Approximately 2 weeks prior to the exam, students are assigned the 1997 and 1998 Multiple Choice exams. Students are encouraged to complete these exams for review. The students are required to complete the 2003 exam (both parts). Students are tested on selected problems from both parts of the 2003 exam.

ADDDITIONAL ASSESSMENT: VOLUME PROJECT

AB Calculus – Volume Project

How much volume and surface area is in that piece of candy or fruit you’re eating?

Directions:

➢ Before Cutting the fruit or candy – measure it!

➢ Put the appropriate section of the fruit or candy on the graph paper and create a region bounded by the outline of it and the x-axis such that when the region is revolved around the x-axis, the entire piece of fruit or candy is formed.

➢ Find 12 points on the outline of the fruit. Record them and label them on the graph.

➢ Enter the points into a calculator lists such that all x values are in L1 and y values in L2. You may also use Excel or another computer software package.

➢ Determine the best regression model for the outline of the data using technology. A piecewise function may be used.

➢ Set up integral(s), which would determine the volume of the solid form if the region is revolved around the x-axis or y-axis.

➢ Use technology to evaluate the definite integral.

➢ Determine the units used in this experiment and convert to cubic centimeters, if necessary. How much is in that little piece of candy or fruit?

➢ Try to confirm your answer using another method of finding the volume.

➢ Determine the surface area as well.

o [pic] is the formula for the surface area of a volume of revolution.

Part II - Create a Lab Report That Contains The Following:

1. Objective

2. Materials Used

3. Procedure – Describe the process for finding the volume and surface area

4. Data & Calculations

5. Error Analysis – sources for error & compare to if you calculated the volume and surface area with formulas or used differentials to estimate the area.

6. Conclusion – explain how you could better the approximation for the area and volume.

Paper Guidelines – Does Your Paper

1. Clearly summarize the problem to be solved?

2. Provide a paragraph which explains how the problem will be approached?

3. State the answer in a few complete sentences which stand on their own?

4. Give a precise and well-organized explanation of how the answer was found, including:

• algebraic support

• graphical support

• numerical support

5. Clearly label diagrams, tables, graphs, or other visual representations of the math?

6. Define all variables, terminology, and notation used?

7. Clearly state the assumptions which underlie the formulas and theorems, and explain how each formula or theorem is derived, or where it can be found?

8. Give acknowledgment where it is due?

9. Use correct spelling, grammar, and punctuation?

10. Contain correct mathematics?

11. Solve the problem(s) that were originally asked?

Total Point Value – 25 points

TECHNOLOGY

Both the students and the instructor use the TI-83 graphing calculator extensively. Students analyze data and functions graphically and numerically. They are asked to analyze and interpret the results of what they see happening on tables, graphs, and scatter plots. Students use graphing calculators to graph functions within an arbitrary viewing window, to find roots and points of intersection, to determine numeric derivatives, and to approximate definite integrals. The limitations of technology (i.e. absolute value of x – nderiv at x = 0 should be undefined) and how to store intermediate values are stressed. The algorithms utilized by the graphing calculator software are also related to various topics throughout the course (Newton’s method, numeric integration and Rieman Sums, numeric differentiation and symmetric difference quotient.) Students are required to complete assignments that are both calculator active and non-calculator active.

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