AP® Calculus AB

AP® Calculus AB

Course Syllabus

Philosophy

The design of my course is driven by two major goals: student success on the AP® Exam and student preparation for subsequent college calculus courses. To have a better understanding of the study of calculus, a student must see the bigger picture. Successful calculus students will not only master the individual concepts of limits, derivatives, indefinite integrals and definite integrals, but they will be able to connect these concepts while applying them to the physical world.

Teaching Strategies

The course is well-supported. The students not only have access to lecture notes, a text book, teacher supplements and TI-84 calculators, but they also have access to APEX software. Due to small class sizes, every AP® calculus student has his or her own individual account with APEX. Teachers and students can allow APEX to do as much or as little as they wish. When the basic skills are being taught in the first part of the course, students rely more on teacher-centered activities and text-based resources. The second part of the course is hybrid in nature and in practice. The students become much more independent, trusting the skills that were previously mastered. Conversations are more student driven and teacher facilitated. The students become very excited with this process! While APEX offers a substantial amount of objective practice and quiz problems, it is still necessary to incorporate supplementary skill-building activities and labs, as well as short-answer and open response test questions related to real world applications.

Course Requirements

In addition to conceptual understandings, the following course goals will also be satisfied:

· Students should be able to work with functions represented in a variety of ways; graphical, numerical, analytical, or verbal. Students should understand the connections among these. [C-3]

This should be a continuation of multiple representation techniques taught in previous courses. We use multiple representations on the first day of the precalculus review and everyday after that. It is important to understand that tables and graphs alone cannot prove an idea; one must describe what is being shown in these models.

· Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems. [C – 4]

Students will often be asked to watch a tutorial on APEX and explain what they saw to another student or to the teacher. It is often with an oral explanation that a teacher can tell if a student really understands what is going on. Only in a few instances does APEX require a written explanation of a problem; however, many of the supplementary worksheets and labs that I use do require well-written explanations.

· Technology should be used regularly by students and teachers to reinforce the relationships among multiple representations of functions, to confirm written work, to implement experimentation, and to assist in interpreting results. [C – 5]

TI-84 calculators play an extremely important role in calculus. Coming into calculus, students must be comfortable with representing functions algebraically, graphically, and with at table and moving among the three with ease. After learning the concepts and basic skills, students will be able to use calculators to approximate the derivative at a point by numerical methods, as well as the definite integral. In addition to these skills, students will also have access to a couple of optional calculator programs. These programs have been proven to save time, but are not mandatory skills.

APEX also brings a lot of technological support to the course. The graphics used in various explanations, particularly with related rates, volumes of solids of revolution, and cross-section problems, allow students to observe superior visual representations of these 3-dimensional ideas.

Grading Policy

Although I feel that it is extremely important to reward students for excellent work in the classroom, our society is more objective-minded and test-oriented; therefore, I use a weighted grading system in which chapter tests and weekly quizzes are given the most emphasis. After all, the students are preparing for the AP® Exam. At the end of the first semester, all students will take a mid-term exam that will, alone, count for 20% of the semester grade. The mid-term is of “AP® format”, and acts as a great preparation tool. Marking period grades are calculated as follows:

Tests & Quizzes60%

Classroom Activities and Labs 30%

Homework

10%

Student Activities

In addition to the text book and APEX, many of the student assignments and labs come from supplementary resources. I have included several of the supplements as evidence of these additional curricular requirements. It is often through these various lab exercises that the students connect individual skills into cohesive concepts. Most of the examples require written explanations, the use of multiple representations of functions, and technology for investigative purposes. Please see the following examples of supplementary student activities.

Name_____________________________

Lab: Instantaneous Rate of Change

Objective: Explore the instantaneous rate of change of a function

The diagram above shows a door with an automatic closer. At time t = 0 seconds someone pushes the door open. It swings open, slows down, stops, and then starts closing. It closes slowly until it hits a certain point, then it quickly closes completely. As the door is in motion, the number of degrees, d, it is from its closed position depends on time, t.

1. Sketch a reasonable graph of d as a function of t.

2. Suppose that d is a function of t defined as

(

)

t

t

t

d

-

=

2

200

)

(

.

Plot this graph on your calculator and sketch the results below. Use the indicated window.

X [0, 9.4]1 , Y[-50, 125]0

3. Make a table of values of d for each second from t = 0 through t = 10. Round answers to the

nearest 0.1 degrees.

t

0

1

2

3

4

5

6

7

8

9

10

d

4. At time, t =1 second, does the door appear to be opening or closing? Explain your answer.

5. At time, t = 3 seconds, does the door appear to be opening or closing? Explain your answer.

6. To find an estimate of the instantaneous rate at which the door is opening at time, t = 1 sec,

find the slope of the secant line from P(1, 100) to A(1.5, 106.1).

Rate of Change =

sec

deg

2

.

12

5

.

1

.

6

1

5

.

1

100

1

.

106

=

=

-

-

=

D

D

x

y

To get increasingly more accurate estimates, find the slope of the secant line as the x-coordinate of point A gets closer and closer to 1, the x-coordinate of point P. (Show your work.)

Coordinates of Point A

¾

¾

®

¬

PA

of

m

(1.4, _______)

(1.3, _______)

(1.2, _______)

(1.1, _______)

(1.01, 100.302)

(1.001, 100.0306)

(1.0001, 100.003068)

Geometrically, the instantaneous rate of change at time, t, is the slope of the tangent line to the curve at that time. Use the concept of limit to explain how you could find the exact value for the instantaneous rate of change at the given time, t.

[C – 4], [C – 5]

Name ______________________

Per_______________

Average Rate of Change of a Function and The Mean Value Theorem

1. Given the following data for a function f.

x

2

2.5

3

3.5

4

4.5

5

F(x)

12

9

8

6

5

0

-2

a. Find the average rate of change from x = 2 to 3.5 .

b. Find the equation of the corresponding secant line.

c. Find the average rate of change from x = 3 to 5.

d. Estimate the instantaneous rate of change of f at x = 3.

f. Use the equation of the tangent line to estimate f( 3.5).

2. Find the average rate of change of the function over the given interval.

Use the MVT Program to find the c on the given interval where the

instantaneous rate of change is equal to the average rate of change.

Include a sketch. Lastly, justify your answer by finding the value of c

analytically. If the MVT cannot be applied, CLEARLY STATE WHY.

a.

ú

û

ù

ê

ë

é

-

=

3

2

1

1

2

,

on

x

)

x

(

f

b.

[

]

6

1

2

1

,

on

x

)

x

(

f

-

+

=

c.

[

]

1

0

,

on

e

)

x

(

f

x

=

d.

[

]

3

1

,

on

x

Ln

)

x

(

f

=

e.

ú

û

ù

ê

ë

é

p

p

=

4

3

4

,

on

x

tan

)

x

(

f

f.

[

]

p

p

-

+

=

,

on

x

cos

)

x

(

f

3

g.

[

]

2

1

4

3

,

on

x

x

)

x

(

f

-

-

=

3. The Intermediate Value and Mean Value Theorems both deal with the

concept of “betweenness”. Both assert the existence of a number, c, that

is between a and b. However, the hypotheses and conclusions of the

theorems are quite different. In a well-constructed paragraph or two,

describe how the two theorems differ and how they are alike. Graphs

will help.

4. The rate at which water flows out of a pipe, in gallons per hour, is given

by a differentiable function R of time, t. The table shows the rate as

measured every 3 hours for a 24-hour period.

t

(hours)

R(t)

(gallons per hour)

0

9.6

3

10.4

6

10.8

9

11. 2

12

11.4

15

11.3

18

10.7

21

10.2

24

9.6

[C – 3], [C – 4], [C – 5]

Is there some time t, 0 < t < 24, such that R’(t) = 0? Justify.

Quotes from the AP Syllabus

· “Students should be able to communicate mathematics both orally and in well-written sentences. They should be able to explain solutions to problems.”

· “Students should be able to use technology to help solve problems, experiment, interpret results, and verify conclusions.”

Directions: Be aware of the above expectations as you solve each of the following. Keep your work extremely neat.

1. Consider the function

,

xe

)

x

(

f

kx

-

=

10

where k is a positive constant.

a. Using this window, sketch the graphs of f for these values of k = 1, 2, 3 .

X

[

]

1

5

0

,

Y

[

]

1

5

0

,

b. Find the x- and y- coordinates of the maximum point for the graph of f.

Express your answer in terms of k.

c. Use your answer to (b) to explain how the three graphs drawn in (a) are related

to each other.

d. Prove that the maximum point of f for different values of k all lie on the same

straight line, and give the equation for that line.

2. Let f be the function

x

sin

k

x

)

x

(

f

+

=

.

a. Sketch the graphs of f for k = .5, 1, 2, in the window: X

[

]

12

0

,

Y

[

]

12

0

,

.

b. Use the derivative of f to explain the differences in the three graphs drawn in

part (a).

c. Suppose that f(x) = 0 has EXACTLY THREE ROOTS. Find the roots.

[C – 4], [C – 5]

LAB: Introduction to Integration

Aims:

1. To introduce the integral as an accumulator of area.

2. To lead to the discovery of The Fundamental Theorem of the Calculus.

X

F(x) = ½ (base) (height)

0

1

2

3

4

x

1. F (x) =

ò

=

x

tdt

0

2

_______

t

2.F(x) =

=

+

ò

x

dt

t

0

3

2

1

________

t

3.F(x) =

ò

=

x

tdt

cos

1

_____________

y

= cos x X

[

]

2

2

4

p

p

-

,

Y

[

]

1

5

1

5

1

.

,

.

-

EMBED Equation.3 a. Use calc Menu 7 to develop table.

b.Copy the table into L1 and L2 and do a Scatter Plot.

c.Guess the function whose scatter plot you have & graph it in y2 using the thick

icon.

X

F(x)

0

.8

1.6

2.4

3.2

4.0

4.8

5.6

6.2

4. F(x) =

ò

=

x

dt

t

1

1

___________Y1 =

x

1

Zoom 4 & change Xmin to 0.

X

F(x)

.1

.5

1

1.5

2

3

4

In each case, how is the function under the integrand related to the accumulated area function? (Answer in complete sentence(s). ________________________________________

____________________________________________________________________________________________________________________________________________________________

[C – 3], [C – 4], [C – 5]

Name________________________ Per_____ INTEGRATION: A Family of Functions

Consider of family of functions of the form:

,

x

x

)

x

(

f

B

4

1

+

=

where

1

0

£

£

x

.

1. Show the family of plots for the parameter B = 0, 1, 2, 3, 4, 5. Use the following window:

EMBED Equation.3

[

]

1

0

1

0

.

,

X

[

]

1

0

1

0

.

,

Y

and turn on grid. (FORMAT menu)

00

1

.

y

B = 0

00

1

.

y

B = 1

00

1

.

y

B = 2

00

1

.

x

00

1

.

x

00

1

.

x

00

1

.

y

B = 3

00

1

.

y

B = 4

00

1

.

y

B = 5

00

1

.

x

00

1

.

x

00

1

.

x

2. By counting squares (each has an area of .01 sq units) and parts of a squares, give a two

decimal place approximation for the area under the curve between x = 0 and x = 1.

B

0

1

2

3

4

5

Estimated Area

fnInt (y1, x, 0, 1)

Definite Integral

3. Use the fnInt command (found in the MATH menu) of your calculator to find these same areas

correct to four decimal places. Record above in #2.

4. Write the 6 definite integrals representing the area under the curve between x = 0 and x = 1.

5. By looking at the graphs, state which B value(s) make LEFT RIEMANN SUMS larger than

the actual value. What does the slope of f(x) have to do with this? Explain.

6. By looking at the graphs, state which B value(s) make the TRAPEZOIDAL RULE

approximation LESS than the actual value. Use the concept of concavity to explain.

7. Without using anything other than the previous results and symmetry of EVEN and ODD

functions, find the values of each:

ò

-

1

1

)

(

dx

x

f

for B = 0, 1, 2, 3, 4, 5

B

0

1

2

3

4

5

Odd or Even

ò

-

1

1

)

(

dx

x

f

8. Sketch the graph of each of the six functions f(x) for B = 0, 1, 2, 3, 4, 5 . And visually check

the results of your findings in #7.

B = 0

00

.

1

y

B = 1

00

.

1

y

B = 2

00

.

1

y

B = 3

00

.

1

y

B = 4

00

.

1

y

B = 5

00

.

1

y

[C – 3], [C – 5]

AP® Calculus AB Course Outline

Unit 1: Precalculus Review (2-3 weeks)

A. Lines

1. Slope as rate of change

2. Parallel and perpendicular lines

3. Equations of lines

B. Functions and Graphs

1. Functions

2. Domain and range

3. Families of functions

4. Piecewise functions

5. Composition of functions

C. Exponential and Logarithmic Functions

1. Exponential growth and decay

2. Inverse functions

3. Logarithmic functions

4. Properties of logarithms

D. Trigonometric Functions

1. Graphs of basic trigonometric functions

a. Domain and range

b. Transformations

c. Inverse trigonometric functions

2. Applications

Unit 2: Limits and Continuity (3 weeks)

A. Rates of Change

B. Limits at a Point

1. Properties of limits

2. Two-sided limits

3. One-sided limits

C. Limits Involving Infinity

1. Asymptotic behavior

2. End behavior

3. Properties of limits

4. Visualizing limits

D. Continuity

1. Continuous functions

2. Discontinuous functions

a. Removable discontinuity

b. Jump discontinuity

c. Infinite discontinuity

E. Instantaneous Rates of Change

Unit 3: The Derivative (5 weeks)

A. Definition of the Derivative

B. Differentiability

1. Local linearity

2. Numeric derivatives using the calculator

3. Differentiability and continuity

C. Derivatives of Algebraic Functions

D. Derivative Rules when Combining Functions

E. Applications to Velocity and Acceleration

F. Derivatives of Trigonometric Functions

G. The Chain Rule

H. Implicit Derivatives

1. Differential method

2. y’ method

I. Derivatives of Inverse Trigonometric Functions

J. Derivatives of Logarithmic and Exponential Functions

Unit 4: Applications of the Derivative (4 weeks)

A. Extreme Values

1. Local (relative) extrema

2. Global (absolute) extrema

B. Using the Derivative

1. Mean Value Theorem

2. Rolle’s Theorem

3. Increasing and decreasing functions

C. 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

D. Optimization Problems

E. Linearization Models

F. Related Rates

Unit 5: The Definite Integral (3 weeks)

A. Approximating Areas

1. Riemann sums

2. Trapezoidal rule

3. Definite integrals

B. The Fundamental Theorem of Calculus (Part I)

C. Definite Integrals and Antiderivatives

1. The average value theorem

D. The Fundamental Theorem of Calculus (Part II)

Unit 6: Differential Equations and Mathematical

Modeling (3 - 4 weeks)

A. Antiderivatives

B. Integration using U – Substitution

C. Separable Differential Equations

1. Growth and decay

2. Slope fields

3. General differential equations

Unit 7: Applications of Definite Integrals (3 weeks)

A. Summing Rates of Change

B. Particle Motion

C. Areas in the Plane

D. Volumes

1. Volumes of solids with known cross-sections

2. Volumes of solids of revolution

a. Disk method

b. Shell method

This schedule leaves 4 – 6 weeks for Exam review. Students will complete calculator and non-calculator sample open response questions. During this time students will become familiar with the layout of the AP® Exam as well as the time constraints.

References and Materials

Primary Text

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. CALCULUS of a

Single Variable. Boston: Houghton Mifflin.

Supplementary Text

Finney, Ross L., Franklin D. Demana, Bert K. Waits, and Daniel Kennedy.

Calculus: Graphical, Numerical, Algebraic: AP Edition. Boston: Pearson

Prentice Hall

Online Resource

APEX Online Learning, Inc. AP Calculus AB Class Tools. AP Calculus AB Exam Review

Supplementary Worksheets and Lab Activities

Supplementary resources provided by Sister Alice Hess, I.H.M., Presenter at Lewes AP® Summer Institute, Lewes, Delaware.

3691215182124 hours

Gal/hr

12

9

6

3

Door

d degrees

t

d

d, degrees

t, seconds

x, seconds

P

A A

y, degrees

1

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