Approach: I use a multirepresentational approach, "The Rule of Five", to teach the concepts of calculus. I present topics graphically, numerically, analytically, verbally, and in writing, and emphasize the connections between them. The students learn to work with functions presented in each of the five ways and understand the connections between them. I emphasize learning the "language of calculus" and insist that my students communicate using the correct terminology in writing and orally when they ask questions and discuss their problem solving strategies.
Technology: Each student has at least a TI83+ or TI84+. I use Geo-sketchpad, PowerPoint, Podcasts and applets from websites for teaching purposes. The students are taught four basic calculator capabilities and use technology regularly to solve problems, experiment, interpret results and support conclusions.
Major Assignments and Assessments: Students have homework for each topic and must show their work. When we discuss the homework problems in class, the students write their Solutions on the board and explain their work. We discuss the concepts involved and alternate problemsolving methods. They have several quizzes and a test in each unit, and a final exam at the end of the semester. The exam and tests are made up of both multiple choice and free response questions from previous AP exams. AP problem sets are assigned as homework and as projects several times each semester. Students are allowed to work with a partner and also get help from me. Students are assigned weekly journal problems and or questions. They are required to write about concepts, procedures and their solutions.
Textbook:
Single Variable Calculus: Early Transcendentals Rogawkski, W.H Freeman and Company, 2008
Other Instructional Materials:
Calculus, Graphical, Numerical, Algebraic, Finney, Demanna, Waits, Kennedy, Pearson Prentice Hall, 2007
Calculus Concepts and Applications Foerster, Key Curriculum Press, 2005
Calculus: Anton, Bivens, and Davis, Anton Textbooks, 2002
Concepts and Calculators in Calculus, Best, Carter, and Crabtree, Venture Publishing, 1997
Calculus of a Single Variable Larson, Edwards, and Hostetler, Houghton Mifflin 2006
AP Calculus AP Test Prep Series Finney, Demanna, Waits, Kennedy, Pearson Prentice Hall, 2007
Syllabus
Unit 1 Functions, Graphs and Limits
Topics:
- Review of functions
- Limits
- Continuity
- Intermediate Value Theorem
Applications and Skills:
- Use the TI84 to graph functions in an arbitrary viewing windows and find zeros of functions
- Estimate limits from tables and graphs
- Find limits algebraically
- Find onesided limits
- Find nonexistent limits
- Apply properties of limits
- Describe a function's asymptotic behavior in terms of limits involving infinity
- Determine continuity at a point and on an interval
- Identify types of discontinuity
- Apply the Intermediate Value Theorem
Text:
Single Variable Calculus: Early Transcendentals Rogawkski: Chapter 1 and 2
Resources:
- Calculus: by Larson (Chapter 1)
- Calculus, Graphical, Numerical, Algebraic, Finney
- Calculus Concepts and Applications Foerster
Unit 2 Derivatives
Topics:
- Concept of The Derivative
- Computing Derivatives
- Derivative at Point
- Derivative as Function
Second Derivative
Applications and Skills:
- Approximate average rate of change and instantaneous rate of change (including average velocity and instantaneous velocity) from graphs and tables
- Use a difference quotient to approximate a derivative at a point & Compute the value of a derivative at a point using the definition of f '(a), the limit of a difference quotient
- Estimate the derivative from graphs and tables
- Apply the rules for derivatives of polynomial, exponential and trigonometric functions
- Apply the rules for derivatives of sums, products, and quotients of functions
- Apply the Chain Rule
- Determine if a function is continuous and differentiable at a point, and the relationship between continuity and differentiability
- Solve particle motion problems involving position, velocity, acceleration, speed, and total distance traveled
- Use the TI84 to compute the derivative numerically
- Find a derivative of a function using the definition of f '(x)
- Find equations of tangent lines and normal lines
- Recognize graphs that have points at which there are vertical tangents or no tangents
- Find a linear approximation of f at a point and use it to approximate f at another value
- Interpret the meaning of a derivative as an instantaneous rate of change
- Translate equations involving derivatives into verbal descriptions (in sentence form) and vice versa
- Interpret the second derivative as a rate of change of a rate of change
- Compare relative magnitudes of functions and their rates of change
- Apply the relationships between differentiability and continuity
Text:
Single Variable Calculus: Early Transcendentals Rogawkski: Chapter 3
Resources:
Unit 3 Applications of Derivatives
Topics:
- Extreme Values
- Graphing
- Second Derivatives
- Optimization
- Related Rates
- Linearization
Applications and Skills:
- Use the sign of f ' to determine the increasing, decreasing behavior of f and to determine if f is monotonic
- Use the sign of f " to determine the concavity of f
- Find derivatives of logarithmic and inverse trigonometric functions
- Use implicit differentiation to find and apply rates of change of implicitly defined functions
- Use implicit differentiation to find the derivative of an inverse function
- Solve related rates problems and explain the solution using a sentence and units
- Determine if a function satisfies the conditions of the Extreme Value Theorem
- Find critical points of a function
- Find extreme values of functions (local and global) and where they occur
- Apply First and Second Derivative Tests to determine local extrema
- Analyze graphs of f ' and f " as aids in sketching a graph of f
- Use a graph or sign chart of f ' as an aid in determining increasing and decreasing behavior of f.
- Use a graph or sign chart of f " as aids in determining concavity of f and inflection points
- Relate inflection points of f, the local extrema of f ', and the change in sign of f "
- Apply Rolle's Theorem and the Mean Value Theorem
- Apply L'Hospital's Rule
- Use derivatives to analyze families of functions
- Solve optimization problems
- Find a linear approximation of f at a point and use it to approximate f at another value
Text: Single Variable Calculus: Early Transcendentals Rogawkski: Chapter 4
Resources:
Unit 4 Definite Integrals
Topics:
Applications and Skills:
- Compute Riemann sums using left, right, and midpoint evaluation points
- Use Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically and by tables of values.
- Given a table or graph of the rate of change of a quantity, approximate the change in the quantity
- Approximate distance traveled using velocity tables and graphs
- Solve application problems by setting up an approximating Riemann sum and representing its limit as a definite integral
- Define the definite integral as a limit of Riemann sums
- Identify the relationship between graphs of f and f ' and sketch the graph of f given f '
- Use the FTC to compute values of f, given f '
- Find antiderivatives that follow directly from derivatives of basic functions
Determine Antiderivatives by substitution of variables (including change of limits
definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
- Calculate Improper integrals (as limits of definite integrals).
- Apply the properties of antiderivatives
- Use the Fundamental Theorem of Calculus to evaluate definite integrals and to find area
- Find specific antiderivatives using initial conditions
- Find distance and velocity from acceleration with initial conditions
- Use the FTC to represent a particular antiderivative and analytically and graphically analyze the functions so defined
- Apply the Second Fundamental Theorem of Calculus
- Use the definite integral to compute area
- Use the TI84 to compute definite integrals numerically
- Use the definite integral of the rate of change of a quantity over an interval to find the change in the quantity over the interval
- Use sentences to explain the meaning of a definite integral and give units
- Use the definite integral to compute distance traveled by a particle along a line Find the average value of a function using the definite integral
Text: Variable Calculus: Early Transcendentals Rogawkski: Chapter 5
Resources:
- Calculus by Larson
- Supplementary problems from Calculus by Anton
- Students use a program on the Nspire (TI84) to find Reimann sums and the trapezoidal sums and the corresponding graphs for large values of n to help them visualize and understand the limit definition of the definite integral
- Students fill a region below a function's graph with dried beans, then smooth out the beans to form a rectangle whose height approximates the average value of the function
- Supplementary problems from Calculus by Best
- Calculus, Graphical, Numerical, Algebraic, Finney
- Calculus Concepts and Applications Foerster
Unit 5 Applications of Integrals
Topics:
- Techniques for finding Antiderivatives
- Initial Condition Problems
- Differential Equations
- Integral as net change over time
- Areas of regions between curves
- Volumes of solids with known cross-sections
- Volumes of solids of revolution including disc and washer method
Applications and Skills:
- Use integrals to calculate net change and total accumulation
- Find areas of regions between curves using horizontal or vertical representative rectangles
- Use definite integrals to find volumes of solids with known cross sections
- Use definite integrals to find volumes of solids of revolution using discs and washers
- Use Riemann sums to approximate volume given a table of values
- Find antiderivatives by Chain Rule for Integration and by u substitution of variables
- Find integrals that lead to inverse trig functions
- Find definite integrals by substitution including change of limits
- Write a differential equations that model a given statement
- Solve logistic differential equations and use them in modeling.
Use slope fields to interpret differential equations geometrically and relate slope
fields and solution curves for differential equations
- Find Numerical solution of differential equations using Euler's method.
Apply L'Hospital's Rule, including its use in determining limits and convergence of
improper integrals
Text: Variable Calculus: Early Transcendentals Rogawkski: Chapter 6, 7, 8 and 9
Calculus, Graphical, Numerical, Algebraic, Finney: Chapter 6
Resources:
Unit 6 Polynomial Approximations and Series
Topics:
- Concept of Series
- Series of Constants
- Taylor Series
Application and Skills:
- Manipulate Taylor series to create new series
- Determine Convergence and divergence of series
- Utilize Taylor polynomial approximation with graphical demonstration of convergence
- Solve problems for Maclaurin series and the general Taylor series centered at x = a.
- Understand Maclaurin series for the functions ex , sin x, cos x, and 1/(1-x).
- Manipulate Taylor series and shortcuts to compute Taylor series, including substitution, differentiation, antidifferentiation and the formation of new series from known series.
- Utilize Functions defined by power series.
- Calculate the Radius and interval of convergence of power series.
- Understand Lagrange error bound for Taylor polynomials.
- Use Technology to explore convergence and divergence.
- Understand decimal expansion using series
- Understand Geometric series with applications.
- Understand The harmonic series.
- Understand Alternating series with error bound.
- Understand that Terms of series as areas of rectangles and their relationship to improper
- integrals, including the integral test and its use in testing the convergence
- of p-series.
- Understand the ratio test for convergence and divergence.
- Compare series to test for convergence or divergence.
- Apply L'Hospital's Rule, including its use in determining limits and convergence of improper integrals
Text: Variable Calculus: Early Transcendentals Rogawkski: Chapter 8.4 and 10
Resouces:
- Calculus, Graphical, Numerical, Algebraic, Finney
- Calculus Concepts and Applications Foerster
Unit 7 Parametric, polar and vector functions.
Topics:
- Compute of derivatives of Parametric, polar and vector functions
- Analyzing planar curves
- Applications of derivatives in parametric form, polar form and vector form,
- including velocity and acceleration
Applications and skills
- Calculate derivatives of parametric, polar and vector functions
- Find arc length of parametric curves
- Differentiate and integrate vector valued functions
- Calculate slopes, lengths and areas inside polar curves
Text: Variable Calculus: Early Transcendentals Rogawkski: Chapter 11
Resources:
Unit 8 Review for AP Exam
