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. 30 May 2010 AP Calculus Course

Applications of Derivatives Integrals Applications of Integrals The following areas will constitute the contents of this AP Calculus AB course. Review Limits Derivates Applications of Derivatives Integrals Applications of Integrals 30 May 2010 AP Calculus Course

Trigonometric Functions Solving Trigonometric Functions Review Functions Inverse Functions Trigonometric Functions Solving Trigonometric Functions Exponential and Logarithmic Functions Common Graphs 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 30 May 2010 AP Calculus Course

The Definition of the Derivative Interpretation of the Derivative 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 L’Hospital’s Rule and Indeterminate Forms Linear Approximations Differentials Newton’s Method 30 May 2010 AP Calculus Course

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 Applications of Integrals Average Function Value Area Between Two Curves Volumes of Solids of Revolution (Disk Method) Work 30 May 2010 AP Calculus Course

Review 30 May 2010 AP Calculus Course

domain (independent variable, pre-image) 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 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 30 May 2010 AP Calculus Course

Limits 30 May 2010 AP Calculus Course

Tangent Lines and Rates of Change Secant Line Difference Formula 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 30 May 2010 AP Calculus Course

Functions that Agree in all but one point Dividing Out Technique 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 30 May 2010 AP Calculus Course

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 30 May 2010 AP Calculus Course

Derivatives 30 May 2010 AP Calculus Course

Difference Equation (Rise over Run) 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 30 May 2010 AP Calculus Course

Derivatives of Exponential and Logarithmic Functions Derivatives of Inverse Trigonometric Functions Derivatives of Hyperbolic Trigonometric Functions Implicit Differentiation Logarithmic Differentiation Related Rates 30 May 2010 AP Calculus Course

Applications of Derivatives 30 May 2010 AP Calculus Course

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 Applications of Derivatives Newton’s Method Approximating the Zero of a Function 30 May 2010 AP Calculus Course

Anti-Derivatives (Integrals) 30 May 2010 AP Calculus Course

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 30 May 2010 AP Calculus Course

The Fundamental Theorem Of Calculus (FTC) 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 30 May 2010 AP Calculus Course

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 Definition of Inverse Trigonometric Functions Properties of Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions Natural Exponential Function 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) 30 May 2010 AP Calculus Course

Applications of Integration 30 May 2010 AP Calculus Course

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 30 May 2010 AP Calculus Course