AP Calculus AB
Syllabus
Lindsay High School
2015 to 2016
AP Calculus AB
The previous mathematics courses your have studied dealt with finite solutions to a given problem or problems. Calculus
deals more with continuous mathematics and it deals primarily with the rates of change (called a derivative) associated with
graphs (notice I did not specifically say functions), and the inverse of the derivative (called the anti-derivative, if it exists).
Derivatives are the tangential slope of a graph and the anti-derivative is the accumulation of area under a graph.
The Limit is what makes calculus “work.” It is used to define the derivative and the anti-derivative. It is the baseline that
mathematicians also return to when trying to determine “hard” solutions to particular problems.
Your set perspective of independent and dependent variables will be generalized. For a given problem, it is sometimes better
if “y” is the independent variable and “x” is the dependent variable. In some cases, they will both be independent variables.
Your algebra skills need to be second nature in this class.
You will learn new ways to apply the algebra skills you honed in Precalculus. This course is not about algebra. The algebra
is often used to get at the calculus presented in practice problems assigned during this course of study. To get good at
calculus and its many sub areas, you will need to work problems. The number of problems will depend on your ability to
learn the lessons being taught by the problems.
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AP Calculus AB
The following areas will constitute the contents of this AP Calculus AB course.
Review
Limits
Derivates
Applications of Derivatives
Integrals
Applications of Integrals
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Review
Functions
Inverse Functions
Trigonometric Functions
Solving Trigonometric Functions
Exponential and Logarithmic Functions
Common Graphs
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Limits
Tangent Lines and Rates of Change
The Limit
One-sided Limits
Limit Properties
Computing Limits
Limits Involving Infinity
Continuity
The Definition of the Limit
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Derivatives
Applications of Derivatives
The Definition of the Derivative
Critical Points
Interpretation of the Derivative
Minimum and Maximum Values
Differential Formulas
Finding Absolute Extrema
Product and Quotient Rules
The Shape of the Graph
Chain Rule
Part I
Derivatives of Trigonometric Functions
Part II
Derivatives of Exponential and Logarithmic Functions The Mean Value Theorem (MVT)
Derivatives of Inverse Trigonometric Functions
Optimization Problems
Derivatives of Hyperbolic Trigonometric Functions
L’Hospital’s Rule and Indeterminate Forms
Implicit Differentiation
Linear Approximations
Related Rates
Differentials
Higher Order Derivatives
Newton’s Method
Logarithmic Differentiation
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Integrals
Indefinite Integrals
Computing Indefinite Integrals
Substitution Rule for Indefinite Integrals
More Substitution Rules
Area Problem
Definition of the Definite Integral
Computing Definite Integrals
Substitution Rules for Definite Integrals
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Applications of Integrals
Average Function Value
Area Between Two Curves
Volumes of Solids of Revolution (Disk Method)
Work
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Review
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Review
Existence Theorems
Functions
domain (independent variable, pre-image)
range (dependent variable, image)
Evaluation, Function
Graphs
Intercepts
x-intercepts
roots
zeros
factors
y-intercepts
Symmetry
Solutions (Points of Intersection)
Elementary Functions
Algebraic (polynomial, radical, rational)
degree of polynomial
polynomial coefficients
leading coefficient
constant term
Trigonometric
Sine
Cosine
Tangent
Exponential and Logarithmic
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Review
Functions
Even
Odd
Slope (Rise over Run)
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Composite
Function
Absolute Value
Properties
Inverse Functions
Trigonometric Functions
Solving Trigonometric Functions
Exponential and Logarithmic Functions
Definition of the Natural Logarithmic
Function ( integral definition)
Common Graphs
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AP Calculus AB
Limits
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Limits
Tangent Lines and Rates of Change
Secant Line
Difference Formula
Area Problem
The Limit
open interval
closed interval
Bounded and Unbounded Behavior
Linear Behavior of a non-linear equation
,  definition of a limit
One-sided Limits
Limit from the left
Limit from the right
Existence of a limit
Limit Properties
Basic Limits
Scalar Multiple
Sum and Difference
Product and Quotient
Radical
Composite
Trigonometric
Power
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Limits
Computing Limits
Functions that Agree in all but one point
Dividing Out Technique
Rationalizing Technique (numerator and denominator)
The Squeeze Theorem
Two Special Trigonometric Limits
Continuity
open interval
closed interval
Definition
Discontinuity
removable
non-removable
Properties Of Continuity
Scalar Multiple
Sum and Difference
Product and Quotient
Composite
Intermediate Value Theorem (IVT) (an existence Theorem)
The Definition of the Limit
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Limits
Limits Involving Infinity
Definition of Limits at Infinity
Vertical Asymptotes
Horizontal Asymptotes
Limits at Infinity
Properties of Infinite Limits
Sum and Difference
Product and Quotient
Applied Minimum and Maximum Problems
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Derivatives
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Derivatives
Slope of a Secant Line
Difference Equation (Rise over Run)
Definition of Tangent Line with Slope m
The Definition of the Derivative
Definition of Differentiable (open interval)
Differentiability and Continuity Relationship
Differentiability  Continuity
Interpretation of the Derivative
Differential Formulas
Constant Rule
Power Rule
Sum and Difference Rules
Product and Quotient Rules
Sine and Cosine Rules
Position Function (ballistics, position, velocity, acceleration)
Derivatives of Trigonometric Functions
Tangent and Cotangent
Secant and Cosecant
Chain Rule (inner and outer derivative)
The General Power Rule
Higher Order Derivatives
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Derivatives
Derivatives of Exponential and Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
Derivatives of Hyperbolic Trigonometric Functions
Implicit Differentiation
Logarithmic Differentiation
Related Rates
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Applications of Derivatives
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Applications of Derivatives
Critical Points
Definition of Extrema
The Extreme Value Theorem
Minimum and Maximum Values
Definition of a Critical Number
Relative Extrema Relationship to Critical Numbers
Finding Absolute Extrema
Definition of Increasing and Decreasing Functions
First Derivative Test
The Shape of the Graph
Definition of Concavity
Test for Concavity
Definition of Point of Inflection
Points of Inflection
Second Derivative Test
Part I
Part II
The Mean Value Theorem (MVT)
Rolle’s Theorem (existence theorem)
Optimization Problems
L’Hospital’s Rule and Indeterminate Forms
Linear Approximations
Differentials
Error Propagation
Differential Formulas
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Applications of Derivatives
Newton’s Method
Approximating the Zero of a Function
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Anti-Derivatives (Integrals)
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Integrals
Indefinite Integrals (Anti-derivative)
Definition
Constant of Integration
Indefinite Integral  Anti-derivative
Slope Fields
Particular Solution
Initial Condition
Computing Indefinite Integrals
Sigma Notation
Summation Formulas
Upper and Lower Sums
Inscribed and Circumscribed
Limits of Lower and Upper Sums
Definition of the Area of a Region in the Plane
Definition of a Riemann Sum
Definition of Definite Integral
Continuity implies Integrability
The Definite as the Area of a Region
Definition of Two Special Integrals
Additive Interval Property
Properties of Definite Integrals
Preservation Of Inequality
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Integrals
The Fundamental Theorem Of Calculus (FTC)
Mean Value Theorem for Integrals
Definition of the Average Value of a Function in an Interval
The Second Fundamental Theorem of Calculus
Substitution Rule for Indefinite Integrals
General Power Rule for Integration
Change of Variables for Definite Integrals
Integration of Even and Odd Functions
Computing Definite Integrals
The Trapezoidal Rule
Error in the Trapezoidal Rule
Natural Logarithmic Functions (Integral perspective)
Definition of the Natural Logarithm
Properties of the Natural Logarithm
Definition of e
Derivative of the Natural Logarithmic Function
Derivative Involving Absolute Value
Log Rule for Integration
Substitution Rules for Definite Integrals
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Integrals
Trigonometric Functions
Basic Integrals
Sine and Cosine
Secant and Cosecant
Tangent and Cotangent
Inverse Functions
Definition
Reflective Property of Inverse Functions
Existence of an Inverse Function
Continuity and Differentiability of Inverse Functions
The Derivative of an Inverse Function
Trigonometric Functions
Definition of Inverse Trigonometric Functions
Properties of Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Natural Exponential Function
Definition
Operations with Exponential Functions
Properties
Derivative of the Natural Exponential Function
Integration Rules for Exponential Functions
Definition of Exponential Functions to Base a
Definition of Logarithmic Function to Base a (Change of Base)
Properties of Inverse Functions (base a)
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Applications of Integration
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Applications of Integrals
Average Function Value
Area Between Two Curves
Volumes of Solids of Revolution (Disk Method)
Work
Definition of Work Done by a Constant Force
Definition of Work Done by a Variable Force
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