Overview
A university level calculus course designed for mature, independent, and mathematically talented students. Topics include those typically taught in two semesters of college calculus: limits, derivatives, integrals, infinite series, elementary differential equations, and applications and modeling.
Objectives
The student will:
A. Functions, Graph, & Limits
1. Analyze graphs of functions with emphasis on the interplay between the geometric and analytic information and on the use of calculus to explain the observed local and global behavior of a function.
2. Calculate limits using algebra and estimate limits from graphs or tables of data.
3. Understand asymptotes in terms of graphical behavior and describe asymptotic behavior in terms of limits involving infinity.
4. Compare relative magnitudes of functions and their rates of change.
5. Understand continuity in terms of limits.
6. Understand graphs of continuous functions geometrically (Intermediate Value Theorem and Extreme Value Theorem).
7. Analyze planar curves given in parametric, polar, and vector form
B. Derivatives
1. Concept of the derivative.
a. Understand the concept of the derivative geometrically, numerically, and analytically.
b. Interpret the derivative as an instantaneous rate of change.
c. Define the derivative as the limit of the difference quotient.
d. Understand the relationship between differentiability and continuity.
2. Derivative at a point.
a. Find the slope of a curve at a point.
b. Find the tangent line to a curve at a point.
c. Find a linear approximation.
d. Find the instantaneous rate of change as the limit of average rate of change.
e. Approximate rate of change from graphs and tables of values.
3. Derivative as a function.
a. Identify corresponding characteristics of the graphs of f and f’.
b. Understand the relationship between increasing and decreasing behavior of f and the sign of f’.
c. Know the Mean Value Theorem and its geometric consequences.
d. Translate verbal descriptions into equations involving derivatives and vice versa.
4. Second derivatives.
a. Identify corresponding characteristics of the graphs of f, f’, and f”.
b. Understand the relationship between the concavity of f and the sign of f”.
c. Identify points of inflection as places where concavity changes.
5. Applications of derivatives.
a. Analyze curves, including the notions of monotonicity and concavity.
b. Analyze planar curves given in parametric, polar, and vector form, including velocity and acceleration vectors.
c. Solve optimization problems.
d. Model rates of change, including related rates problems.
e. Use implicit differentiation to find the derivative of an inverse function.
f. Interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
g. Interpret differential equations geometrically via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.
h. Solve differential equations numerically using Euler’s method.
i. Use L’Hospital’s Rule in determining convergence of improper integrals and series.
6. Computation of derivatives.
a. Find derivatives of basic functions, including x^n, exponential, trigonometric, and inverse trigonometric functions.
b. Use basic rules for the derivative of sums, products, and quotients of functions.
c. Find derivatives using the Chain Rule and implicit differentiation.
d. Find derivatives of parametric, polar, and vector functions.
C. Integrals
1. Riemann sums.
a. Understand the concept of a Riemann sum over equal subdivisions.
b. Calculate Riemann sums using left, right, and midpoint evaluation points.
2. Interpretations and properties of definite integrals.
a. Define the definite integral as a limit of Riemann sums.
b. Use the definite integral as the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
c. Use basic properties of definite integrals.
3. Applications of integrals.
a. Use appropriate integrals to model physical, social, or economic situations.
b. Use the integral as a rate of change to give accumulated change.
c. Use the method of setting up a Riemann sum and representing its limit as a definite integral.
d. Find the area of a region, including a region bounded by polar curves.
e. Find the volume of a solid with known cross sections.
f. Find the average value of a function.
g. Find the distance traveled by a particle along a line.
h. Find the length of a curve, including a curve given in parametric form.
4. Fundamental Theorem of Calculus.
a. Use the Fundamental Theorem to evaluate integrals.
b. Use the Fundamental Theorem to represent a particular antiderivative, including the analytical and graphical analysis of functions so defined.
5. Techniques of antidifferentiation.
a. Find anti-derivatives of basic functions.
b. Find anti-derivatives by substitution of variables, including change of limits for definite integrals.
c. Find anti-derivatives by parts and simple partial fractions (nonrepeating linear factors only).
d. Evaluate improper integrals as limits of definite integrals.
6. Applications of antidifferentiation.
a. Find specific anti-derivatives using initial conditions, including applications to motion along a line.
b. Solve separable differential equations and use them in modeling. In particular, the equation y’ = ky and exponent growth.
c. Solve logistic differential equations and use them in modeling.
7. Numerical approximations to definite integrals.
a. Use Riemann sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
b. Use the Trapezoidal rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
D. Polynomial Approximations and Series
1. Concept of series.
a. Recognize series as a sequence of partial sums.
b. Understand that convergence of a series is the limit of the sequences of partial sums.
c. Use technology to explore convergence and divergence of various series.
2. Series of constants.
a. Use series for decimal expansion.
b. Identify and use geometric series.
c. Identify and use harmonic series.
d. Identify and use alternating series with error bound.
e. Understand the relationship between terms of series as areas of rectangles and improper integrals, including the integral test and its use in testing the convergence of p-series.
f. Use the ratio test to determine convergence and divergence of a series.
g. Use the comparison test to determine convergence and divergence of a series.
3. Taylor series.
a. Use Taylor polynomials to approximate various functions, such as the sine function.
b. Find the general Taylor series centered at x = a.
c. Identify and use the Maclaurin series for the functions e^x, sin x , cos x and 1/(1 - x).
d. Manipulate Taylor series and use shortcuts to computing Taylor series, including differentiation, antidifferentiation, and the formation of new series from known series.
e. Use functions defined by power series and radius of convergence.
f. Calculate the Lagrange error bound for Taylor polynomials.
Methodology
New concepts are introduced through direct instruction and/or exploration. Students often work in small groups to extend understanding of concepts previously learned or to develop an intuitive understanding of new concepts prior to formal instruction. Homework assignments are given daily to reinforce learning.
Evaluation
Tests (60 %) – Tests are given at the end of each chapter. Each test is patterned after the AP exam and includes problems from previous AP exams. Each test consists of non-calculator and calculator sections, as well as multiple choice and free response sections.
Quizzes (30%) – Announced quizzes and take home quizzes are given periodically within each chapter.
Homework (10%) – All assignments are turned in at the end of each chapter and evaluated for completion and accuracy.
Resources
Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
Finney, Ross L., et al. Assessment Resources. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
Finney, Ross L., et al. Solutions Manual. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
Finney, Ross L., et al. Texas Instruments Technology Resource Manual. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
Finney, Ross L., et al. Student Practice Workbook. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
Finney, Ross L., et al. Teacher’s AP Correlations and Preparation Guide. Calculus: Graphical, Numerical, Algebraic. Boston: Pearson Education, Inc., 2007.
TI-83 Plus or TI-84 Plus Calculator
Overhead Calculator Tools
Advanced Placement Program Course Description: Calculus. Princeton: College Board, 2007.
Rogawski, Jon, Calculus: Early Transcendentals. New York: W.H. Freeman and Company, 2008.
Kelley, W. Michael. Arco’s Master the AP Calculus AB & BC Tests. Lawrenceville: Peterson’s, 2002.
Ostebee, Arnold, and Paul Zorn. Calculus: From Graphical, Numerical, and Symbolic Points of View. Fort Worth: Harcourt Brace, 2002.
Larson,Ron, et al. Calculus. Boston: Houghton Mifflin Company, 2002.
Hughes-Hallett, Deborah, and Andrew M. Gleason. Calculus. New York: Wiley, 1994.