COURSE TITLE: AP Calculus AB
COURSE DESCRIPTION: Advanced Placement Calculus AB is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry and trigonometry. Basic differentiation and integration together with applications form the content. Upon completion of the course, the student is prepared to take the AB Advanced Placement test in Calculus which provides eligibility for one semester of credit in college calculus.
Prerequisite: Approval of Pre-calculus teacher.
COURSE REQUIREMENTS/REQUIRED MATERIALS:
1. Text: Calculus, Larson, Hostetler and Edward D.C.
Heath & Co. Sixth Edition.
2. Folder
3. Pencils and red pen
4. TI 83/84 Calculator
5. Scientific Calculator
COURSE OBJECTIVES/STUDENT OUTCOMES:
A. Students will gain understand of elementary functions.
Upon completing this goal the student will be able to:
•write the sum, product, quotient and composition of two functions
•write the functions in which an absolute value
function decomposes
•state the domain and range of typical functions
•form the inverse of a function
•calculate the limit of a function
•state the definition for a limit and do the necessary calculations
•state the definition of continuity
•calculate the values for which a function is discontinuous
B. Students will gain an understanding of differential calculus
Upon completing this goal the student will be able to:
•state the definition of the derivative
•form the derivative of elementary functions
•form the derivative of the sum, product, and quotient
of two functions
•form the derivative of a composite function
•form the derivative of the inverse of a function
•form higher derivatives
•form the equation for the tangent and normal to any point
on a curve
•do the necessary calculations for determining the maximum
and minimum point of a function and any points of inflection
•calculate the velocity and acceleration characteristics of
a particle moving along a straight line
•calculate the velocity and acceleration for motion along
a plane curve
•to apply the theory of related rates
C. Students will become familiar with integral calculus
Upon completing this goal the student will be able to:
•form the antiderivative of a function when possible
•form the distance and velocity equations when the
acceleration equation is given together with initial conditions
•solve differential equations of the type y=ky and
apply this information to growth and decay situations
•integrate integrals by substitution, by parts,
and by partial fractions
•state the definition and properties of the definite integral
•calculate the value of a definite integral
•use approximation techniques to evaluate a definite integral
•calculate the average value of a function on an interval
•determine the area between two curves
•calculate volumes of rotation
COURSE OUTLINE:
I. Elementary Functions
A. Properties of Functions
1. domain and range
2. sum, product, quotient, composition
3. absolute value
4. inverse
5. even, odd
6. periodicity
7. graphs, symmetry and asymptotes
B. Properties of particular functions
1. fundamental identities and addition formulas
for trig. functions
2. amplitude and periodicity of trig. functions
3. exponential and log functions and their inverses
C. Limits
1. statement of properties
2. definition for a limit
3. the number e expressed as a limit
4. functions which increase or decrease without bound
5. continuity
6. statements but not proofs of continuity theorems
II. Differential Calculus
A. The derivative
1. definition of the derivative
2. derivatives of elementary function
3. derivatives of sums, products, quotients
4. derivative of composite functions
5. derivative of implicitly defined functions
6. logarithmic differentiation
7. derivative of a rational power of a function
8. derivative of the inverse of a function
9. derivatives of higher order
10. Rolle's theorem: Mean value theorem
11. relationship between differentiation and continuity
12. Linear approximation of a function
13. L'Hospital's Rule
B. Application of the derivative
1. Slope of a curve; tangent and normal lines to a curve.
2. Curve sketching: increasing and decreasing functions;
relative and absolute maximum and minimum points;
concavity; points of inflection.
3. Extreme value problems
4. Velocity and acceleration of a particle moving along a line.
5. Velocity and acceleration for motion on a plane curve.
6. Average and instantaneous rates of change.
7. Related rates of change.
III. Integral Calculus
A. Antiderivatives
B. Applications of antiderivatives
1. Distance and velocity from acceleration with initial conditions.
2. Solutions of y' = ky and applications to growth and decay.
C. Techniques of integration
1. Basic integration formulas.
2. Integration by substitution.
3. Integration by parts.
4. Integration by partial fractions.
D. The definite integral
1. Concept of the definite integral as area.
2. Approximations to the definite integral using rectangles
or trapezoids.
3. Approximations: upper and lower sum, Trapezoidal rule.
4. Definition of the definite integral as a limit of a sum.
5. Recognition of limits of sums as definite integrals.
6. Properties of the definite integral.
7. The fundamental theorem of integral calculus.
8. Functions defined as integrals.
E. Applications of the integral
1. Average value of a function on an interval.
2. Area between two curves.
3. Volume of a solid of revolution.
4. Volumes of solids with known cross sections.
5. Interpretation of ln(x) as area under a graph: y = 1/x
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