ap calculus ab syllabus lindsay high school 2015 to 2016
TRANSCRIPT
AP Calculus ABSyllabus
Lindsay High School2015 to 2016
AP Calculus AB
30 May 2010 AP Calculus Course 2
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.
AP Calculus AB
30 May 2010 AP Calculus Course 3
Review
Limits
Derivates
Applications of Derivatives
Integrals
Applications of Integrals
The following areas will constitute the contents of this AP Calculus AB course.
AP Calculus AB
30 May 2010 AP Calculus Course 4
Limits Tangent Lines and Rates of ChangeThe LimitOne-sided LimitsLimit PropertiesComputing LimitsLimits Involving InfinityContinuityThe Definition of the Limit
Review FunctionsInverse FunctionsTrigonometric FunctionsSolving Trigonometric FunctionsExponential and Logarithmic FunctionsCommon Graphs
AP Calculus AB
30 May 2010 AP Calculus Course 5
Derivatives The Definition of the DerivativeInterpretation of the DerivativeDifferential FormulasProduct and Quotient RulesChain RuleDerivatives of Trigonometric FunctionsDerivatives of Exponential and Logarithmic FunctionsDerivatives of Inverse Trigonometric FunctionsDerivatives of Hyperbolic Trigonometric FunctionsImplicit DifferentiationRelated RatesHigher Order DerivativesLogarithmic Differentiation
Applications of DerivativesCritical PointsMinimum and Maximum ValuesFinding Absolute ExtremaThe Shape of the Graph
Part IPart II
The Mean Value Theorem (MVT)Optimization ProblemsL’Hospital’s Rule and Indeterminate FormsLinear ApproximationsDifferentialsNewton’s Method
AP Calculus AB
30 May 2010 AP Calculus Course 6
Integrals Indefinite IntegralsComputing Indefinite IntegralsSubstitution Rule for Indefinite IntegralsMore Substitution RulesArea ProblemDefinition of the Definite IntegralComputing Definite IntegralsSubstitution Rules for Definite Integrals
Applications of Integrals Average Function ValueArea Between Two CurvesVolumes of Solids of Revolution (Disk Method)Work
AP Calculus AB
30 May 2010 AP Calculus Course 7
Review
AP Calculus AB
30 May 2010 AP Calculus Course 8
Review Existence TheoremsFunctions
domain (independent variable, pre-image)range (dependent variable, image)Evaluation, FunctionGraphs
Interceptsx-intercepts
rootszerosfactors
y-interceptsSymmetrySolutions (Points of Intersection)
Elementary FunctionsAlgebraic (polynomial, radical, rational)
degree of polynomialpolynomial coefficientsleading coefficientconstant term
TrigonometricSineCosineTangent
Exponential and Logarithmic
Review Functions
EvenOddSlope (Rise over Run)
Composite FunctionAbsolute Value
PropertiesInverse FunctionsTrigonometric FunctionsSolving Trigonometric FunctionsExponential and Logarithmic Functions
Definition of the Natural Logarithmic Function ( integral definition)
Common Graphs
AP Calculus AB
30 May 2010 AP Calculus Course 9
Limits
AP Calculus AB
30 May 2010 AP Calculus Course 10
Limits Tangent Lines and Rates of Change
Secant LineDifference FormulaArea Problem
The Limitopen intervalclosed intervalBounded and Unbounded BehaviorLinear Behavior of a non-linear equation, definition of a limit
One-sided LimitsLimit from the leftLimit from the rightExistence of a limit
Limit PropertiesBasic LimitsScalar MultipleSum and DifferenceProduct and QuotientRadicalCompositeTrigonometricPower
AP Calculus AB
30 May 2010 AP Calculus Course 11
Limits Computing Limits
Functions that Agree in all but one pointDividing Out TechniqueRationalizing Technique (numerator and denominator)The Squeeze TheoremTwo Special Trigonometric Limits
Continuityopen intervalclosed intervalDefinitionDiscontinuity
removablenon-removable
Properties Of ContinuityScalar MultipleSum and DifferenceProduct and QuotientComposite
Intermediate Value Theorem (IVT) (an existence Theorem)The Definition of the Limit
AP Calculus AB
30 May 2010 AP Calculus Course 12
Limits Limits Involving Infinity
Definition of Limits at InfinityVertical AsymptotesHorizontal Asymptotes Limits at InfinityProperties of Infinite Limits
Sum and DifferenceProduct and Quotient
Applied Minimum and Maximum Problems
AP Calculus AB
30 May 2010 AP Calculus Course 13
Derivatives
AP Calculus AB
30 May 2010 AP Calculus Course 14
Derivatives Slope of a Secant Line
Difference Equation (Rise over Run)Definition of Tangent Line with Slope m
The Definition of the DerivativeDefinition of Differentiable (open interval)
Differentiability and Continuity Relationship Differentiability Continuity
Interpretation of the DerivativeDifferential Formulas
Constant RulePower RuleSum and Difference RulesProduct and Quotient RulesSine and Cosine RulesPosition Function (ballistics, position, velocity, acceleration)
Derivatives of Trigonometric FunctionsTangent and CotangentSecant and Cosecant
Chain Rule (inner and outer derivative)The General Power RuleHigher Order Derivatives
AP Calculus AB
30 May 2010 AP Calculus Course 15
Derivatives Derivatives of Exponential and Logarithmic FunctionsDerivatives of Inverse Trigonometric FunctionsDerivatives of Hyperbolic Trigonometric FunctionsImplicit DifferentiationLogarithmic Differentiation Related Rates
AP Calculus AB
30 May 2010 AP Calculus Course 16
Applications of Derivatives
AP Calculus AB
30 May 2010 AP Calculus Course 17
Applications of DerivativesCritical Points
Definition of ExtremaThe Extreme Value TheoremMinimum and Maximum ValuesDefinition of a Critical NumberRelative Extrema Relationship to Critical Numbers
Finding Absolute ExtremaDefinition of Increasing and Decreasing FunctionsFirst Derivative Test
The Shape of the GraphDefinition of ConcavityTest for ConcavityDefinition of Point of InflectionPoints of InflectionSecond Derivative TestPart IPart II
The Mean Value Theorem (MVT)Rolle’s Theorem (existence theorem)
Optimization ProblemsL’Hospital’s Rule and Indeterminate FormsLinear ApproximationsDifferentials
Error PropagationDifferential Formulas
Applications of DerivativesNewton’s Method
Approximating the Zero of a Function
AP Calculus AB
30 May 2010 AP Calculus Course 18
Anti-Derivatives (Integrals)
AP Calculus AB
30 May 2010 AP Calculus Course 19
Integrals Indefinite Integrals (Anti-derivative)
DefinitionConstant of IntegrationIndefinite Integral Anti-derivativeSlope Fields
Particular SolutionInitial Condition
Computing Indefinite IntegralsSigma NotationSummation FormulasUpper and Lower SumsInscribed and Circumscribed
Limits of Lower and Upper SumsDefinition of the Area of a Region in the PlaneDefinition of a Riemann Sum
Definition of Definite IntegralContinuity implies IntegrabilityThe Definite as the Area of a RegionDefinition of Two Special IntegralsAdditive Interval PropertyProperties of Definite IntegralsPreservation Of Inequality
AP Calculus AB
30 May 2010 AP Calculus Course 20
Integrals The Fundamental Theorem Of Calculus (FTC)
Mean Value Theorem for IntegralsDefinition of the Average Value of a Function in an Interval
The Second Fundamental Theorem of CalculusSubstitution Rule for Indefinite Integrals
General Power Rule for IntegrationChange of Variables for Definite IntegralsIntegration of Even and Odd Functions
Computing Definite IntegralsThe Trapezoidal Rule
Error in the Trapezoidal RuleNatural Logarithmic Functions (Integral perspective)
Definition of the Natural LogarithmProperties of the Natural LogarithmDefinition of eDerivative of the Natural Logarithmic FunctionDerivative Involving Absolute ValueLog Rule for Integration
Substitution Rules for Definite Integrals
AP Calculus AB
30 May 2010 AP Calculus Course 21
Integrals Trigonometric Functions
Basic IntegralsSine and CosineSecant and CosecantTangent and Cotangent
Inverse FunctionsDefinitionReflective Property of Inverse FunctionsExistence of an Inverse FunctionContinuity and Differentiability of Inverse FunctionsThe Derivative of an Inverse FunctionTrigonometric Functions
Definition of Inverse Trigonometric FunctionsProperties of Inverse Trigonometric FunctionsDerivatives of Inverse Trigonometric Functions
Natural Exponential FunctionDefinitionOperations with Exponential FunctionsProperties Derivative of the Natural Exponential FunctionIntegration Rules for Exponential FunctionsDefinition of Exponential Functions to Base aDefinition of Logarithmic Function to Base a (Change of Base)Properties of Inverse Functions (base a)
AP Calculus AB
30 May 2010 AP Calculus Course 22
Applications of Integration
AP Calculus AB
30 May 2010 AP Calculus Course 23
Applications of Integrals Average Function ValueArea Between Two CurvesVolumes of Solids of Revolution (Disk Method)Work
Definition of Work Done by a Constant ForceDefinition of Work Done by a Variable Force