Course Overview
This
is an Advanced Placement Course and every student is expected to take
the AP Exam. Students are also preparing for their next mathematics
course. Expectations are clearly explained on the first day of school.
Students are informed that this course is a college level course that
is both demanding and rigorous. The students are told that we will work together to discover and develop an appreciation of calculus.
Course Assessments
Students
are regularly assessed on a cumulative basis through math labs,
homework, and past FreeResponse questions. Students are evaluated and
expected to understand calculus: Numerically, Algebraically,
Graphically, and Verbally (Rule of Four). All tests consist of general problems, multiplechoice questions, and past FreeResponse items. All assessments contain questions in which students are required to justify their solutions in wellwritten sentences. Students are allowed to use their graphing calculator on the last part of each test.
Calculus Journals
Students
are required to keep journals that are divided into three sections.
The first section is labeled “In Your Own Words” and is to be completed
at the end of every week. In this section, students are required to
explain previously taught concepts in their own words. Students may
also discuss real world applications of concepts learned in class in
this section. The next section is labeled “Cooperative Homework
Assignment”. Students are divided into to groups to collaboratively
complete one homework assignment each week. Students must record how
long the group met, who was in attendance, and what difficulties or
notable successes the group may have had during the session. Students
must report if they were any disagreements regarding any solutions and
explain alternative solutions. The last section is labeled “Error
Analysis”. Students are required to explain and correct their errors on
previous test/quizzes in wellwritten sentences.
Technology and Computer Software
Graphing
calculators are utilized in both teaching and learning. Both TI84
Plus and TI89 Titanium graphing calculators are used throughout the
course to help students interpret results and verify conclusions.
Students who are not able to purchase their own graphing calculator are
allowed to check out a calculator from their teacher. Additionally,
PowerPoint presentations downloaded from the Internet are used to aid
students’ comprehension of key calculus concepts, such as Reimann Sums.
Primary Textbook
Our primary textbook is Calculus of a Single Variable 2002, seventh edition, Larson, Hostetler, and Edwards, published by Houghton Mifflin: Boston, New York, ISBN 0618149163.
Course Planner: This course is planned by six weeks units. We have six units per year.
FIRST SIX WEEKS

1

Local
Linearity – we explore various functions and discover that by zooming
in, the function may appear to be a straight line. We can find the
slope of a line with these two points that are very close together.
Students work in pairs to find the slope of the segment connecting the
two points. Each pair is responsible for finding the slope at
several points on a given function. We explore the various
functions as each pair reports on their finding. We refer back to this
activity often in our later studies. We use this activity to review
concepts needed for calculus. Our students use the TI89 and some
also use the TI83/84. Calculators are used often to explore and
discover. We emphasize the four capabilities that students are
allowed to use on the exam – plot the graph of a function within an
arbitrary viewing window, find the zeros of functions, numerically
calculate the derivative of a function, and numerically calculate the
value of a definite integral.

2

Limits
– an intuitive understanding, using algebra, and estimating limits
from graphs and tables of values. We use the graphing calculator to
find limits on the graph screen and from the table.(TEST 1)

3

Continuity – definition of continuity, onesided limits, Intermediate Value Theorem.

4

Infinite Limits – Vertical Asymptotes, finding vertical asymptotes and determining infinite limits. (TEST 2)

5

The
Derivative and the Tangent Line Problem – students are reminded about
the opening activity, local linearity. We define the derivative of a
function and find some derivatives using this method. We want
students to recognize the definition of the derivative (limit of the
difference quotient) and the alternate form. This exploration allows us
to discover some of the rules for differentiation that we will
study. We compare this work to the slopes we found when we used local
linearity.

6

Differentiability
and Continuity – vertical tangent lines and points that have no
tangent line, Instantaneous rate of change, Graphical, Numerical, and
Analytical Differentiation. (TEST 3)

7

Rules
for Basic Differentiation: The Constant Rule, The Power Rule, The
Constant Multiple Rule, The Sum and Difference Rules, Sine and Cosine,
and Rates of Change. (SIX WEEKS TEST, AP style questions, both
multiple choice and free response)

Second Six Weeks

1

Rules
for Differentiation, continued: The Product Rule, The Quotient Rule,
Derivatives of Trig Functions, HigherOrder Derivatives.

2

Rules for Differentiation, continued: The Chain Rule

3

Derivatives of natural log functions and exponential functions. (TEST 1)

4

Implicit Differentiation

5

Related Rates (TEST 2)

6

Extrema, Relative Extrema, Critical Numbers

7

Rolle’s Theorem and Mean Value Theorem.

8

The First Derivative Test, Increasing and Decreasing Functions (SIX WEEKS TEST)

Third Six Weeks

1

Concavity,
Points of Inflection, The Second Derivative Test. This lesson
concludes with a matching game. Students match graphs of functions
with their derivatives. Also descriptions of the function and
derivative are matched with the graphs. Students verbalize what they
are looking for and how they know they have a match. This is a very
rich conversation filled with talk such as “The first derivative is
positive, so the function must be increasing over this interval.”

2

Derivative as a rate of change – position, velocity, acceleration, and the question of speed.

3

Connecting in tables and graphs.

4

Limit as x approaches infinity, Horizontal Asymptotes

5

Curve
Sketching without the graphing calculator. Many of our students will
attend universities that do not allow the graphing calculator in the
next course they take. We work on increasing our power to sketch
the functions without the calculator. We use the first and second
derivative to make accurate sketches. We emphasize the rational
functions and improve our ability to make a sketch so that we identify
intervals of increasing/decreasing and concave up/concave down
quickly. (TEST 1)

6

Optimization

7

Linear Approximations (TEST 2)

8

Antiderivatives and Indefinite Integration – general solutions and particular solutions. (SIX WEEKS TEST)

The
Semester Exam is given at the end of the Third Six Weeks. It is a
shorter version of an AP EXAM. Students have two hours for this exam.
Fourth Six weeks

1

Slope
fields – students explore several differential equations and draw the
slope fields by hand. They learn characteristics of various
differential equations. Students identify the slope field by the
differential equation that created it and by the function that it could
represent. We combine the graphical approach with the algebraic
approach, when possible, and compare our results.

2

Area and Definite Integrals – Riemann Sums, left, right, and midpoint

3

The Mean Value Theorem for Integrals and the Average Value of a Function (TEST 1)

4

The Fundamental Theorem of Calculus, I and II.

5

Integration
by Substitution – Pattern Recognition, Change of Variables for
Definite Integrals, Integration of Odd and Even Functions. (TEST 2)

6

Trapezoid Approach for finding Area

7

Total Distance Traveled (TEST 3)

8

Inverse Functions – Derivative of an Inverse Function.

9

Integration of Natural Log and Exponential Functions (SIX WEEKS TEST)

Fifth Six Weeks

1

Differential Equations: Growth and Decay Models (TEST 1)

2

Inverse Trigonometric Functions  Derivatives of Inverse Trigonometric Functions

3

Integrals Involving Inverse Trigonometric Functions. (TEST 2)

4

Area between curves.

5

Volumes
of solids – Disc, Washer, and Known Cross Sections – The first day of
this section is known as Candy Day. We anticipate it for days. As we
revolve the various functions around the xaxis, we see known candy
shapes. When we see a shape, everyone in the class gets a piece of
candy – kisses, peanut butter cups, eggs. This helps us visualize
the 3 dimensions and the circular cross sections we need for problem
solving. We also use playdoh as a base on poster board and we place
shapes of the knowncross sections in the playdoh so that they stand
up perpendicular to the appropriate axis. We also use Calculus in
Motion to help students find the volumes and powerpoints by Greg
Kelley. (SIX WEEKS TEST)

Sixth Six Weeks

1

Review
for the AP Exam – Our major assessments have been AP style. We
continue to work on Multiple Choice Questions and Free Response
Questions. We used released AP Tests.

2

Sequences and Series (After the exam)
