EL SEGUNDO UNIFIED
EL SEGUNDO HIGH SCHOOL
COURSE OF STUDY
Course Title: A.P. Calculus & LMU Math 131
Department: Mathematics
Grade Levels: 11-12
COURSE DESCRIPTION
CALCULUS, A.P.
This course is taught with the same level of depth and rigor as entry level college and university calculus courses. These standards outline a complete college curriculum in one variable calculus. Calculus is a widely applied area of mathematics, and also involves a beautiful intrinsic theory. Students mastering this content will be exposed to both of these important aspects of the subject. The focus of this course is to prepare the students to pass the A.B. A.P. Calculus exam in the spring as well as be prepared for calculus in college.
Length: One Year
Prerequisite for Enrollment: 85% or better in Pre-Calculus Honors; 95% or better in Pre-Calculus. 80% or better on the Calculus Placement exam.
Students enrolled in AP Calculus must either take the A.P.
exam (the cost of this exam is approximately $80.00) or enroll in Math 131 at
LMU (the cost for this course is approximately $135). Students may choose to do both. LMU does not accept any waivers into this
course. Students must meet all criteria
for placement into Math 131.
Recommendation for Enrollment: Teacher recommendation, and a strong desire on the part of the student to succeed in this course.
Type of course: College preparatory (UC/CSU) and meets math graduation requirement.
COURSE OUTLINE & STANDARDS
Advanced Placement Curriculum
by the College Board
1. Functions, Graphs, and Limits
A. Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function. B. Limits of functions (including one-sided limits)
1. An intuitive understanding of the limiting process.
2. Calculating limits using algebra
3. Estimating limits from graphs or tables of data.
B. Asymptotic and unbounded behavior.
1. Understanding asymptotes in terms of graphical behavior.
2. Describing asymptotic behavior in terms of limits involving infinity.
3. Comparing relative magnitudes of functions and their rates of change.
(For example, contrasting exponential growth, polynomial growth, and logarithmic growth)
C. Continuity as a property of functions.
1. An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.
2. Understanding continuity in terms of limits.
3. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem.)
2. Derivatives
A. Concept of the derivative
1. Derivative presented graphically, numerically, and analytically.
2. Derivative interpreted as an instantaneous rate of change.
3. Derivative defined as the limit of the difference quotient
4. Relationship between differentiability and continuity
B. Derivative at a point.
1. Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are not tangents.
2. Tangent line to a curve at a point and local linear approximation.
3. Instantaneous rate of change as the limit of average rate of change.
4. Approximate rate of change from graphs and tables of values.
C. Derivative as a function.
1. Corresponding characteristics of graphs of f and f’.
2. Relationship between the increasing and decreasing behavior of f and the sign of f’.
3. The Mean Value Theorem and its geometric consequences.
4. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
D. Second derivatives.
1. Corresponding characteristics of the graphs of f, f’, f’’.
2. Relationship between the concavity of f and the sign of f’’.
3. Points of inflection as places where concavity changes.
E. Application of derivatives
1. Analysis of curves, including the notions of monotonicity and
concavity
2. Optimization, both absolute (global) and relative (local) extrema.
3. Modeling rates of change, including related rates problems.
4. Use of implicit differentiation to find the derivative of an inverse
function.
5. Interpretation of the derivative as a rate of change in varied applied
contexts, including velocity, speed, and acceleration.
F. Computation of derivatives
1. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
2. Basic rules for the derivative of sums, products, and quotients of functions.
3. Chain rule and implicit differentiation.
3. Integrals
A. Interpretations and properties of definite integrals.
1. Computation of Riemann sums using left, right, and midpoint evaluation points.
2. Definite integral as a limit of Riemann sums over equal subdivisions.
3. Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the definite integral.
4. Basic properties of definite integrals. (Examples include additivity and linearity.)
B. Applications of integrals: Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value fo a function, and the distance traveled by a particle along a line.
C. Fundamental Theorem of Calculus
1. Use of the Fundamental Theorem to evaluate definite integrals.
2. Use of the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so
defined.
D. Techniques of antidifferentiation
1. Antidifferentiation following directly from derivatives of basic functions.
2. Antiderivatives by substitution of variables (including change of limits for definite integrals)
E. Applications of antidifferentitation.
1. Finding specific antiderivatives using initial conditions, including applications to motion along a line.
2. Solving separable differential equations and using them in modeling. In particular, studying the equation y’=ky and exponential growth.
F. Numerical approximations to definite integrals.
1. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by table of values.
INSTRUCTIONAL METHODS
A. Lecture and Guided Practice
B. Class work and Homework
C. Individual work, and group work
D. Short term projects
E. Presentations
EVALUATION/GRADING OF STUDENT WORK
A. Tests
B. Semester Exam
C. Comprehensive final exam
D. Homework
F. Short term projects
G. Written essays
INSTRUCTIONAL MATERIALS
a. Text: Calculus by Leithold & Calculus by Stewart
b. TI-89 Graphing Calculator