Welcome to Calculus, Gifted.
Minds are like parachutes. They only function when they are open.
Calculus AB
Syllabus
Below is an outline of the concepts we will cover as well as the sequence that these concepts are uncovered. The goal of the course is for the student to read, write, and manipulate the basic concepts of calculus in preparation for future, higher education courses of choice. This will be a class of discovery and exploration, so the amount of time suggested may vary. Each day consists of a 90 minute class period.
The recommended calculators include the TI-83, TI-83 Plus, and the TI-84 Plus. If the student does not own a calculator, one will be supplied. This tool will be used everyday to evaluate and reinforce concepts of calculus along with the Geometer’s Sketchpad software program shown on the projector. Calculators will be used on some of the testing opportunities.[C5]
Primary Textbook
Stewart, James. Calculus—Single Variable. 5th ed. Belmont, California: Brooks/Cole, a division of Thomson Learning, Inc., 2003
Classroom Activities
These activities help the learner get more deeply involved with the concepts, thereby reaching a higher level of learning with the increased ability to discover problems and work to solve them through calculus.
I. Students will be asked to develop real life situations that can be evaluated with graphs, derivatives, and integration. These creations will be analyzed by peers and the instructor. They will first be modeled after teacher examples, then later will be given as further inquiry on part of the student. [C4]
II. Geometer’s Sketchpad will be used often via overhead to plot a function, then estimate the slope of the tangent line at x values given, then plotting the slope values as a function of x with the program, and finally assessing understanding by allowing the program to actually plot the derivative. [C3] and [C5]
Calculus AB Course Outline [C2]
I. Pre-Assessment (1st Day)
The students will complete a worksheet showing mastery of basic principles including the following items. Students will show individual need of remediation which will be implemented during personal time.
A. Factoring Monomials
B. Binomial Factoring
C. Factoring Quadratic Equations
D. Reducing Rational Expressions
E. Using the Quadratic Formula
F. Dividing Synthetically
G. Special Factors
H. More Factoring, Trinomials and Trigonometric Functions
I. Rewrite Rational Expressions in Simplest Form
J. Applying the Conjugate
II. Functions and Models-Chapter 1 (3 Days)
The course begins with a review of the real line, set notation, interval notation, rational and irrational number identification, solving inequalities, solving absolute value equations, the Cartesian Plane, quadrants, abscissa, ordinate, Pythagorean theorem, mid point, distance, completing the square, equations of circles, piecewise functions, domain and range, slope, parallel and perpendicular lines, exponential growth and decay, inverse functions, logarithms and their properties, and representations of functions including equations, graphs, or verbally. Seven basic functions, y = x2, y = x3, y = x½, y = |x|, y = sin x, y = 2x, and y = log2x will be translated graphically and algebraically. Trigonometry and its inverse will be reviewed. Composite functions will be explored, as well as graphing functions with the calculator in appropriate rectangular viewing grids. Problem solving abilities are toned with real life applications and the translation of events into algebraic means of evaluation.
A. A Preview of Calculus (Intro)
B. Four Ways to Represent a Function (1.1)
C. Mathematical Models: A Catalog of Essential Functions (1.2)
D. New Functions from Old Functions (1.3)
E. Graphing Calculators and Computers (1.4)
F. Exponential, Logarithmic, and Inverse Functions (1.5, 1.6)
III. Limits and Rates of Change-Chapter 2 (5 Days)
The concept of a limit comes up in the methods of exhaustion as we try to find the area of a region, the slope of a tangent, the sum of infinite series, and real life applications that include velocity. Limits and the knowledge of is a basis for various branches of calculus and that is why it is appropriate to begin our study of calculus with the investigation of limits and their properties.
A. The Tangent and Velocity Problems (2.1)
B. The Limit of a Function (2.2)
C. Calculating Limits Using the Limit Laws (2.3)
D. The Precise Definition of a Limit (2.4)
E. Continuity and Limits at Infinity, Horizontal Asymptotes (2.5, 2.6)
F. Tangents, Velocities, and Other Rates of Change (2.7)
IV. Derivatives-Chapter 3 (7 Days)
Our study of differential calculus where how one quantity changes in relation to another quantity begins here. The derivative is an expansion of the velocities and slopes of tangent lines we found from the previous chapter. Now we will use the derivatives to solve problems involving rates of change as well as the approximation of functions.
A. Derivatives (and its interpretation as a rate of change) (2.8)
B. The Derivative as a Function (2.9)
C. Differentiation Formulas (3.1, 3.2)
D. Rates of Change in the Natural and Social Sciences (3.3)
E. Derivatives of Trigonometric Functions (3.4)
1. Special interest in proving the derivative of y = sin x is –cos x
F. The Chain Rule (3.5)
G. Implicit Differentiation (3.6)
H. Higher Derivatives (3.7)
I. Related Rates
J. Linear Approximations and Differentials
V. Applications of Differentiation-Chapter 4 (7 Days)
Now that we have investigated some derivative applications and our understanding level of the differentiation rules are better, we are able to pursue the applications of differentiation in greater depth. We will see how derivatives shape graphs of functions and how they help us find the maximum and minimum values of functions, which when applied to real life will help us find how to maximize profits in particular situations.
A. Maximum and Minimum Values
B. The Mean Value Theorem
C. How Derivatives Affect the Shape of a Graph
D. Limits at Infinity; Horizontal Asymptotes
E. Summary of Curve Sketching
F. Graphing with Calculus and Calculators
G. Optimization Problems
H. Applications to Business and Economics
I. Newton’s Method
J. Antiderivatives
VI. Integrals-Chapter 5 (5 Days)
Area and distance problems are used to formulate the idea of a definite integral, the basic idea of integral calculus. The Fundamental Theorem of Calculus relates the integral to the derivative and it greatly simplifies the solution of many real life problems.
A. Areas and Distances
B. The Definite Integral
C. The Fundamental Theorem of Calculus
D. Indefinite Integrals and the Net Change Theorem
E. The Substitution Rule
VII. Applications of Integration-Chapter 6 (5 Days)
Exploration into the applications of definite integrals and using integrals to compute areas between curves, volumes of solids, and the work done by varying forces will be enlightening and fun as we break down large quantities into smaller parts and approximating these pieces with the Riemann Sum. Then we will take the limit and express the quantity as an integral. To conclude, we will evaluate the integral using the Fundamental Theorem of Calculus, or the Midpoint Rule.
A. Areas between Curves
B. Volumes
C. Volumes by Cylindrical Shells
D. Work
E. Average Value of a Function
VIII. Inverse Functions-Chapter 7 (7 Days)
The functions in this chapter are common because they occur as pairs of inverse functions (especially the exponential function and its logarithmic inverse). One to one functions will be defined, as exponential and trigonometric functions occur frequently in models of nature and science as well as society which further determines the need for us to manipulate and find the derivatives and integrals for the function as well as its inverse.
A. Inverse Functions
B. Exponential Functions Their Derivatives and Their Integrals
C. Logarithmic Functions
D. Derivatives of Logarithmic Functions
E. Inverse Trigonometric Functions
F. Hyperbolic Functions
G. Indeterminate Forms and L’Hospital’s Rule
IX. Techniques and Further Application of Integration-
Chapter 8 & 9 (9 Days)
Since integration is not as straightforward as differentiation, we discover integration by parts, and methods special to trigonometric and rational functions, as well as develop further strategies of integration with Chapter 8. The next chapter, Chapter 9, is where we will extend our exploration of geometric applications by finding the length of a curve as well as the area of a surface.
A. Integration by Parts
B. Trigonometric Integrals
C. Trigonometric Substitution
D. Integration of Rational Functions by Partial Fractions
E. Strategy for Integration
F. Approximate Integration
G. Improper Integrals
H. Arc Length
I. Area of a Surface of Revolution
X. Differential Equations-Chapter 10 (7 Days)
Modeling real life situations by formulating mathematical models through intuitive reasoning about a certain physical law based on experimentation often shows the model taking the form of a differential equation. To help us predict future behavior based on current value changes, we examine those extrapolated equations.
A. Modeling with Differential Equations
B. Direction Fields and Euler’s Method
C. Separable Equations
D. Exponential Growth and Decay
E. The Logistic Equation
F. Linear Equations
G. Predator-Prey Systems
XI. Practice Released Testing Items for Calculus AB
(5 Days)
XII. Twelve 30 minute testing opportunities will be given
(4 Days)
XIII. The Final for Calculus AB will be given
See Contact Me on the navigational tool at the left for information on teacher expectations.
Supplies:
loose leaf paper
pencils
erasers
graph paper
markers or colored pencils
a scientific calculator will be provided (TIINSPIRE and TI84 Graphing Calculators :)
Students will be picked to supply one of the following: paper towels, kleenex, and hand sanitizer.
Tests are to be made up in the make up center three days after absence, or student will acquire a zero.