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

1
(No Transcript)
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.
3
The following areas will constitute the contents
of this AP Calculus AB course.
Review Limits Derivates Applications of
Derivatives Integrals Applications of Integrals
4
Limits Tangent Lines and Rates of Change The
Limit One-sided Limits Limit Properties Computi
ng Limits Limits Involving Infinity Continuity
The Definition of the Limit
Review Functions Inverse Functions Trigonometr
ic Functions Solving Trigonometric
Functions Exponential and Logarithmic
Functions Common Graphs
5
Derivatives The Definition of the
Derivative Interpretation of the
Derivative Differential Formulas Product and
Quotient Rules Chain Rule Derivatives of
Trigonometric Functions Derivatives of
Exponential and Logarithmic Functions Derivatives
of Inverse Trigonometric Functions Derivatives
of Hyperbolic Trigonometric Functions Implicit
Differentiation Related Rates Higher Order
Derivatives Logarithmic Differentiation
Applications of Derivatives Critical
Points Minimum and Maximum Values Finding
Absolute Extrema The Shape of the Graph Part
I Part II The Mean Value Theorem
(MVT) Optimization Problems LHospitals Rule
and Indeterminate Forms Linear
Approximations Differentials Newtons Method
6
Integrals Indefinite Integrals Computing
Indefinite Integrals Substitution Rule for
Indefinite Integrals More Substitution
Rules Area Problem Definition of the Definite
Integral Computing Definite Integrals Substituti
on Rules for Definite Integrals
Applications of Integrals Average Function
Value Area Between Two Curves Volumes of Solids
of Revolution (Disk Method) Work
7
Review
8
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 Exp
onential and Logarithmic
Review Functions Even Odd Slope (Rise
over Run) 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
9
Limits
10
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
11
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 Disconti
nuity 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
12
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
13
Derivatives
14
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
15
Derivatives Derivatives of Exponential and
Logarithmic Functions Derivatives of Inverse
Trigonometric Functions Derivatives of
Hyperbolic Trigonometric Functions Implicit
Differentiation Logarithmic Differentiation
Related Rates
16
Applications of Derivatives
17
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) Rolles Theorem (existence
theorem) Optimization Problems LHospitals
Rule and Indeterminate Forms Linear
Approximations Differentials Error
Propagation Differential Formulas
Applications of Derivatives Newtons
Method Approximating the Zero of a Function
18
Anti-Derivatives (Integrals)
19
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 Propertie
s of Definite Integrals Preservation Of
Inequality
20
Integrals The Fundamental Theorem Of Calculus
(FTC) Mean Value Theorem for Integrals Definit
ion 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
21
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)
22
Applications of Integration
23
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