AP Calculus 

Textbook:

            Calculus

            Sixth Edition

            Larson, Hostetler and Edwards

            Houghton Mifflin

 

General Description:

             Calculus is a yearlong college level math course open to seniors.  Students who take calculus in high school generally advance faster through their math courses in college.  Calculus is explored through the interpretation of graphs and tables.  Derivatives are interpreted as rates of change and local linear approximation. Integration of functions allows for the analysis of the area under given curves. Applications are given for many fields including biology, business, chemistry, economics, engineering, finance and physics.  Calculus incorporates the use of computers and graphing calculators.  Graphing calculators are an integral part of the AP Calculus course and proficiency with the calculators is expected on the AP test.  The AP test is given in May to determine if the student can receive college credit for the course.

 

Objectives:

1.         Students will be able to determine the area of irregular two and three- dimensional figures.

2.         Students will be able to solve problems involving average or constant rates.

3.         Students will be able to create arithmetic and geometric sequences to fit a given set of conditions.

4.         Students will be able to understand the concept of asymptotes, and find the equations of vertical and horizontal asymptotes.

5.         Students will be able to find limits graphically, numerically, and analytically.

6.         Students will be able to determine where a graph is increasing and decreasing.

7.         Students will be able to find maxima and minima on an interval.

8.         Students will be able to use optimization to solve business and economic problems.

9.         Students will be able to use integration to find the area between two curves, volume, surface area and arc length.

 

Outline:

            Chapter P        Preparation for Calculus

            Chapter 1        Limits and Their Properties

            Chapter 2        Differentiation

            Chapter 3        Applications of Differentiation

            Chapter 4        Integration

            Chapter 5        Logarithms, Exponential and Other Transcendental Functions

            Chapter 6        Applications of Integration

            Chapter 7        Integration Techniques

 

Homework:

            Many supplemental materials are covered both in and out of school.  Students will spend time both in class and at home doing practice AP problems.

Overnight assignments are due at the beginning of the next day class period.  Absent students will turn in work the day after their return.  Each day’s assignments will be left on the teacher’s voice mail.

 

Class rules:

            Everyone receives the classroom rules on the first day of school to keep in their notebooks.  Notebooks will be collected periodically and checked for the rules.  The class will follow the school-wide final exam policy.

 

Materials:

            Students should bring the following materials to class each day:

                        Textbook

                        Notebook

                        Pencil

                        Calculator – not required but strongly recommended

                        Graph Paper

 

Grading Policy:

            The district grading scale will be used in this class.  The grade will be determined in the following manner:

                        Homework                                          20%

                        Notebook                                            20%

                        Classroom Participation                      20%

                        Tests and Quizzes                               40%

 

Prerequisite course:              Pre-Calculus and Trigonometry

 

Unit Plan 

Calculus

12^th grade

Mrs. Abbett

Unit 5 – The Integral

22 days            Jan. 26 – Mar. 1

            Students start this chapter by using Riemann sums to approximate the area of a region bounded by a continuous function.  They are able to see the inaccuracy of this method by doing the same problem with half of the students using left hand endpoints and half of the students using right hand endpoints.  We then look at integration first as an antiderivative and then as a method to determine area more accurately.  We look at different ways to rewrite functions so that they will be more easily integrated.

            The first application of an antiderivative that we investigate is vertical motion.  Using gravity as the acceleration of an object moving vertically and labeling acceleration as f’’(x), we can use integration to determine velocity and height.  Then we start finding the area of objects with known area formulas by integration.  We move on to functions with graphs that are unknown curves and find the area between the graph and the x axis by using definite integrals.

            Once the students have learned the basic ways of integrating a function, other methods are explored.  We discover how to integrate by change of variables and substitution.  We also find area by using the Trapezoidal Rule and Simpson’s Rule.

            1.         Antiderivatives and Indefinite Integration.

            2.         Vertical Motion Problems.

            3.         Reimann Sums and Area.

            4.         Definite Integrals.

            5.         The Fundamental Theorem of Calculus.

            6.         Integration by Substitution and Change of Variables.

            7.         The Trapezoidal Rule and Simpson’s Rule.

     

College Board AP Calculus Standards and Goals:

Interpretations and properties of definite integrals

1.  Definite integral as a limit of Riemann sums
2.  Definite integral of the rate of change of a quantity over an interval interpreted

as the change of the quantity over the interval.

1. Basic properties of definite integrals (examples include additivity and linearity).
2. Use of the Fundamental Theorem to evaluate definite integrals
3. Use of the Fundamental Theorem to represent a particular antiderivative, and

the analytical and graphical analysis of functions so defined.

1. Numerical approximations to definite integrals. Use of Riemann sums (using

left, right, and midpoint evaluation points) and trapezoidal sums to approximate

definite integrals of functions represented algebraically, graphically, and by tables

of values.

Daily Plans

Day 1 – Definition of an Antiderivative

Introduce the antiderivative as the inverse of the derivative.  Give each student a function to find the derivative of, with the only difference in the functions being the constant term.  Share the results with each other, and brainstorm why each student got the same answer.  Demonstrate how to find an antiderivative.  Discuss the effects of taking the antiderivative  on  the graph of the function, and how there is an infinite number of constants that can complete the function. Do examples on the smart board and using the graphing calculators.

Assign:            Page 248, #6 – 30 even.

Day 2 – Integrals of  Trig Functions

Recall the derivatives of each of the six basic trig functions.  Practice simplifying expressions using different trig identities in pairs.  Define the integral of sine and cosine, and have the students determine the others.  Use these integrals and simplification techniques to find the integral of various trig functions. 

Assign:            Pages248-249, #31-42.

Day 3 – Solving Differential Equations

Show how to set up dy/dx = f(x) as y = F(x) + C.  Put in the given point to find the value of C.  Check answers using the graphing calculators.  Do examples as a whole group.

Assign:            Page 249, #46-60 even.

Day 4 – Sigma Notation

Recall sigma notation and finding the sum of a function.  Have each of the students make a problem using the correct notation.  Rotate the problems so that each student will solve someone else’s problem.  Have students discuss their results with each other.  Now, give an expanded version of a sum and ask the students if they can determine the correct sigma notation, in essence, work backwards.  Do other examples on the smart board.

Assign:            Pages 260-261, #2 – 22 even

Day 5 – Limits of Sums

Introduce Lower and Upper Sums of a given region.  Show the difference in the sums when using the right or left endpoints of the subintervals.  Extend the concept to the Limit at each the lower and upper sums.  Do examples as a whole group.

Assign:            Page 261, #24 – 38 even

Day 6 – Area of a Region

Do investigation to find the area between the x axis and a given parabola.  Have one student estimate it by dividing it into 4 regions, one student six regions, and the other student eight regions.  Compare the results of each student.  Use the results to find the area of a general region: any isosceles triangle, rectangle, semi-circle, etc.

Assign:            Page 262, #42 – 50 even

Day 7 – Definite Integrals

Introduce and explain the symbols used to represent a definite integral.  Show graphically what it represents.  Recall the parabola from yesterday and do the problem together using a definite integral.  Compare the answer with yesterday’s result.  Do examples on smart board and check the solutions on the graphing calculators.

            Assign:            Page 271 – 272, # 2 – 24 even

Day 8 – Evaluating Definite Integrals

Extend yesterday’s lesson to include trig functions, rational functions and radical functions.  Practice as a whole group setting up a definite integral to represent the area under a given curve.  Discuss how to set up a pair of definite integrals when part of the graph is below the x axis.

Assign:            Page 283, 6 – 30, 34 – 42 even.

Day 9 – Second Fundamental Theorem Of Calculus

Define the Second Fundamental Theorem of Calculus.  Show how it relates to the definite integrals that we have been using the last two days.  Discuss the appropriate times to use the Theorem and demonstrate on the smart board how it is used.

            Assign:            Page 285, #50, 52, 74 – 84 even

Day 10 – AP Practice Problems

Day 11 – Unit 5 Mid – Chapter Review

Day 12 – Unit 5 Mid – Chapter Test

Day 13 – Integration by Substitution

Recall f(g(x)) and g(f(x)).  Take a function that is composed of two simpler functions and discuss ways to break it apart into f(x) and g(x).  Find f(g(x)) and g(f(x)) to determine if the students solution is correct.  Substitute their solutions into the integral to make it easier to integrate.  Do examples on smart board.

            Assign:            Page 296, #2 – 26 even

Day 14 – Finding the Equation by Integration

Recall yesterday’s lesson.  Extend it by looking for an exact function when given a point.    Discuss how to use substitution when finding a definite integral and how to tell when substitution is not the best option.  Have each student look at a problem and determine the method they would use to integrate it.  Discuss their solutions and see if the others agree with their choice.

Assign:            Page 296, #34 – 54 even

Day 15 – Area by Substitution

This is the last extension of  substitution.  Students should look at a given graph, set up the definite integral, determine the method of integration to use, and evaluate the integral to get the area.  Include examples that involve radicals and trig functions.  Show students how to enter an integral into the graphing calculators.  Use the calculators for one example and then do an example by hand.  Check the solutions on the graphing calculators.

Assign:            Page 297, #56 – 68 even

Day 16 – Trapezoidal Rule

Discuss ways that we know so far to estimate the area under the curve.  Introduce the Trapezoidal Rule and have students discuss how it may be more accurate than dividing an interval into rectangular subintervals.  Practice using the Trapezoidal Rule to find an integral as a whole class.

Assign:            Page 304, #2 – 10 even

Day 17 – Simpson’s Rule

Introduce Simpson’s Rule for estimating an integral.  Have students find the ways that Simpson’s Rule and the Trapezoidal Rule are similar and how they are different.  Have each student evaluate an integral using each method and present their findings to the rest of the class.

Assign:            Page 304, #12 – 20 even

Day 19 – Error formulas and Applications

Discuss area problems that might not be able to be solved by using a definite integral: irregular shapes, impossible to reach boundaries, etc.  Show how the Trapezoidal Rule and Simpson’s Rule can be applied to these problems.  Then Analyze the error involved in the estimation.

Assign:            Pages 304 – 305, #22, 24, 38, 40, 42

Day 20 – AP practice problems

Day 21 – Unit 5 Review

Day 22 Unit 5 Test