AP ® Calculus AB Syllabus Course Overview Calculus is mathematics that deals with the dynamic. It is a study of how to describe, quantify, and qualify changes on a very large scale and on a very small scale. It employs both careful precision and inspired creativity. My hope is that you will not only develop a proficiency in the subject, but an appreciation for it as well. The following is a brief outline of the goals of the course. •
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Students will develop a conceptual understanding of and connect the two major concepts of Calculus (of a single variable): differentiation and integration. o Differentiation: a linear operation on functions, a rate of change, and the slope of tangent lines of a curve at different points o Integration: a linear operation on functions, a summation of change, and the limit of Riemann sums o The Fundamental Theorem of Calculus, connecting differentiation and integration Problem solving o Develop proficiency in the techniques of Calculus and be able to find analytic solutions by pencil and paper. o Find numerical solutions using technology (graphing calculators or other software). o Read and analyze a problem, model the problem mathematically, and solve the problem correctly. o Evaluate the validity of arguments and solutions. Math literacy: o Accurately communicate and discuss mathematics verbally and in writing Represent and view functions analytically, numerically, and graphically. Do well on the AP® Calculus AB Exam.
Course Planner Unit 0: Functions [~ 1 week] Text Ref.
Topics
N/A
The graphing calculator
1.1
Concepts of functions
1.1
The 4 representations of concepts (graphing calculator usage)
1.2
Linear functions
1.2
Polynomial & rational functions (roots & poles)
1.2
Exponential & logarithmic functions
1.2
Trigonometric functions
1.3
Transformations of functions
1.3 & 7.1
Compositions & inverses Page 1 of 4
1.4
The graphing calculator
1.1
Piecewise defined functions
Unit 1: Limits & Continuity [1 ~ 2 weeks] Text Ref.
Topics
2.1
The tangent problem and instantaneous velocity
2.2
The limit of a function & one-sided limits (graphs & table of values)
2.2 & 4.4
Infinite limits & asymptotes
2.3
Evaluating limits & the laws of limits
2.3 & 3.4
The Squeeze Theorem & the special limits: lim sin(x) & lim cos(x) - 1
2.5
Continuity
2.5
The Intermediate Value Theorem
x→ 0
x
x→ 0
x
Unit 2: The Derivative [3 ~ 4 weeks] Text Ref.
Topics
3.1
Motivation for the derivative: the slope of the tangent line & instantaneous velocity
3.1
The definition of a derivative (finding derivatives using the definition)
3.1
The derivative as the slope of the tangent line at a point (the limit of the slopes of secant lines) & as an instantaneous rate of change (the limit of the average rates of change)
3.1
Approximations of the derivative using a table of values and difference quotients
3.2
The derivative as a function & as a linear operator; higher order derivatives
3.2
Differentiability & continuity
3.3
Derivative of constants & power functions; the sum and difference rules
3.3
The product & quotient rules of derivatives
3.4
Derivatives of trigonometric functions
3.5
The Chain Rule
3.6
Implicit differentiation
7.2
Derivatives of exponential functions
7.3 & 7.4
Logarithmic functions & their derivatives
7.6
The derivatives of inverse trigonometric functions
Unit 3: Applications of Differentiation [4 ~ 5 weeks] Text Ref.
Topics
3.7
Application of the derivative as the instantaneous rate of change to problem solving
3.8
Related rates Page 2 of 4
3.9
Differentials, local linearity and linear approximations
4.1
The Extreme Value Theorem: local & global extrema
4.1
Critical numbers & the maximum and minimum problems
4.2
The Mean Value Theorem
4.3
The derivative and the graphs of functions • Increasing/decreasing intervals & the first derivative o The derivative as a rate of change of a function • The first derivative test for local max/min • Concavity & the second derivative • Inflection points • The second derivative test for local max/min
4.5
Curve sketching using derivatives and limits (asymptotes)
4.6
Graphing using Calculus concepts and the graphing calculator
4.7
Application of the derivative: optimization
7.8
Indeterminate forms & l’Hospital’s Rule
Unit 4: The Integral [3 ~ 4 weeks] Text Ref.
Topics
4.9
The antiderivative
5.1
Motivation for the integral: finding the area under a curve between two bounds & finding the distance traveled by an object
5.2 & 8.7
The approximations of definite integral • Riemann sums: left-hand, right-hand and midpoint rules • The trapezoidal rule
5.2
The definite integral as the limit of Riemann sums
5.2
The properties of definite integrals & comparison properties of definite integrals
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite integrals
5.2 & 5.4
The integral as a linear operator on functions
5.4
The definite integral as a net change from a rate of change function
5.5
The substitution method of finding integrals
8.2
Integrals involving trigonometric functions
Unit 5: Applications of Integration [3 ~ 4 weeks] Text Ref.
Topics
6.1
Areas of a bounded region
6.2 & 6.3
Volumes of solids: solids of revolution and cross section problems
6.4
Work problems
6.5
Average value (Mean Value Theorem for integrals)
Page 3 of 4
Unit 6: Differential Equations [~ 2 weeks] Text Ref.
Topics
7.5
Differential equations for exponential growth and decay
10.3
Differential equations: modeling problems & solving separable differential equations
10.2
Graphing the solution curves of a differential equations using slope fields
Unit 7: Review for the AP® Exam [4 ~ 5 weeks] Text Ref.
Topics Released Free Response Questions Multiple Choice Exam Practice (Calculator) Multiple Choice Exam Practice (Non-Calculator) Practice AP Exam
Unit 8: Additional Topics [after the AP® Exam] Text Ref.
Topics
4.8
Newton’s method for finding roots [Optional]
8.1
Integration by parts
8.3
Trigonometric substitution
8.4
Partial fractions
6.3
The cylindrical shell method of finding volumes of solids of revolution
2.4
The definition of limits using ε and δ [introduce topic, but do not dwell on the topic]
Primary Textbook James Stewart. Calculus: Single Variable, 6th ed.
Required Supplies • •
TI 89 graphing calculator 2 spiral notebooks or composition books
Student Evaluation • •
Unit exams, final exam AP® free-response questions and multiple choice practice exams
My Contact Information • • • •
Steve Lu (Room 209) Conference Period: 5 Office Hour: Mondays 3:30-4:30 Email:
[email protected]
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