AP Calculus ABObjectives

This course will cover the topics as outlined in the AP Calculus Course Description for Calculus AB, including integration by parts. Students will study four major concepts throughout the year: limits, derivatives, indefinite integrals, and definite integrals. Each of these concepts and their connections will be studied graphically, numerically, analytically, and verbally. Technology will be used throughout the course to emphasize and reinforce an appreciation of calculus as a coherent body of knowledge.

The expectations and rigor of this course are high. Students will be held to a college level academic standard. They will be required to communicate their understanding of the topics covered, using proper vocabulary and terms, both written and verbally. The main objective of this course is to give students the confidence, knowledge, and skills necessary to be successful in future mathematics courses.

Course Outline

Primary TextbookHughes-Hallett, Gleason, McCallum, et al. Calculus Single Variable, 4th ed, NJ: John Wiley & Sons, Inc., 2005.

Chapter 1: A Library of Functions (2 weeks)• Precalculus Review & Evaluation• Continuity

o Continuity on an interval – Intermediate Value Theoremo Continuity at a point

• Limitso Definition of a limito Properties of limitso Limits to infinity

Chapter 2: Key Concept – The Derivative (3 weeks)• Definition of the derivative• Derivative at a point• Slope of a curve at a point• Tangent line to a curve• Derivative function

o Graphically – relating the graphs of f and f′o Numerically – estimate values of f′ using a tableo Formula - constants, linear & power rule for differentiationo Verbal interpretations of the derivative - Unit Analysis

• Second derivative as a rate of change

• Differentiability and continuity1.

Chapter 3: Shortcuts to Differentiation (4weeks)• Rules for differentiation

o Constant multipleso Sums & differenceso Power rule (revisited)o Product ruleo Quotient ruleo Chain rule

• Derivatives of exponential, logarithmic, trigonometric, & inverse trigonometric functions

• Implicit differentiation• Local linearity – tangent line approximation• Mean Value Theorem

Chapter 4: Using the Derivative (4 weeks)• Using the derivative to find:

o Critical pointso Local maxima & minima

First-Derivative Test Second-Derivative Test

o Inflection points and concavityo Global maxima & minima – The Extreme Value Theorm

• Optimization and modeling• Related rates• L’Hopital’s rule and dominance

Chapter 5: The Definite Integral (3 weeks)• Riemann sums• Fundamental Theorem of Calculus (given F(x))• Interpretations of the definite integral – area, total change from a rate of change, and

average value of a function• Estimate definite integral from graph, table, or formula

Chapter 6: Constructing Antiderivatives (3 weeks)• Fundamental Theorem of Calculus• Constructing antiderivatives

o Numericallyo Graphicallyo Analytically – Properties of antiderivatives

Indefinite integral

Definite integral• Differential equations• Second Fundamental Theorem of Calculus

2.Chapter 7: Integration (2 weeks)• Techniques of integration

o Substitution methodo Integration by parts

• Approximating definite integrals numericallyo Midpoint ruleo Trapezoid rule

Chapter 8: Using the Definite Integral (3 weeks)• Areas and volumes

o Volumes of solids with known cross sectionso Volumes of solids of revolution

Disk method Shell method

• Density and center of mass

Chapter 11: Differential Equations (3 weeks)• Solve first order differential equations

o Graphically – Slope fieldso Numerically – Euler’s method (if time allows)o Analytically – Separation of variables

• Modeling growth and decay – Newton’s Law of Heating and Cooling

Student Evaluations