Students in this course will walk in the footsteps of Newton and Leibnitz.
An interactive course framework combines with the exciting on-line course delivery to make calculus an adventure. The course includes a study of limits, continuity, differentiation, integration, differential equations, and the applications of derivatives and integrals, parametric and polar equations, and infinite sequences and series.
An Advanced Placement (AP) course in calculus consists of a full high school year of work that is comparable to calculus courses in colleges and universities. It is expected that students who take an AP course in calculus will seek college credit, college placement, or both, from institutions of higher learning.
Most colleges and universities offer a sequence of several courses in calculus, and entering students are placed within this sequence according to the extent of their preparation, as measured by the results of an AP examination or other criteria.
Topics and Concepts
Segment One
Module 01 – Limits and Continuity
- Using Limits to Analyze Instantaneous Change
- Estimating Limit Values from Graphs and Tables
- Determining Limits Using Algebraic Properties and Manipulation
- Selecting Procedures for Determining Limits
- Squeeze Theorem and Representations of Limits
- Determining Continuity and Exploring Discontinuity
- Connecting Limits, Infinity, and Asymptotes
- The Intermediate Value Theorem (IVT)
Module 02 – Differentiation: Definition and Fundamental Properties
- Average and Instantaneous Rates of Change and the Derivative Definition
- Determining Differentiability and Estimating Derivatives
- Derivative Rules: Constant, Sum, Difference, Constant Multiple, and Power
- The Product Rule and the Quotient Rule
- Derivatives of Trigonometric Functions
- Derivatives of Exponential and Logarithmic Functions
Module 03 – Differentiation: Composite, Implicit, and Inverse Functions
- The Chain Rule
- Implicit Differentiation
- Differentiating Inverse Functions
- Differentiating Inverse Trigonometric Functions
- Selecting Procedures for Calculating Derivatives
- Calculating Higher-Order Derivatives
Module 04 – Contextual Applications of Differentiation
- Interpreting and Applying the Derivative in Motion
- Rates of Change in Applied Contexts Other Than Motion
- Related Rates
- Approximating Values of a Function Using Local Linearity and Linearization
- L’Hospital’s Rule
Module 05 – Analytical Applications of Differentiation
- Mean Value and Extreme Value Theorems
- Determining Function Behavior and the First Derivative Test
- Using the Candidates Test to Determine Absolute Extrema
- Determining Concavity of Functions and the Second Derivative Test
- Connecting Graphs of Functions and Their Derivatives
- Optimization Problems
- Exploring Behaviors of Implicit Relations
Segment Two
Module 06 – Integration and Accumulation of Change
- Exploring Accumulations of Change
- Riemann Sums and the Definite Integral
- Accumulation Functions Involving Area and the Fundamental Theorem of Calculus
- Applying Properties of Definite Integrals
- Finding Antiderivatives and Indefinite Integrals
- Integrating Using Substitution
- Integrating Using Integration by Parts
- Integrating Using Linear Partial Fractions
- Evaluating Improper Integrals
- Integrating Functions Using Long Division and Completing the Square
- Selecting Techniques for Antidifferentiation
Module 07 – Differential Equations
- Solutions of Differential Equations
- Sketching and Reasoning Using Slope Fields
- Approximating Solutions Using Euler’s Method
- Finding Solutions Using Separation of Variables
- Exponential Models with Differential Equations
- Logistic Models with Differential Equations
Module 08 – Applications of Integration
- Average Value and Connecting Position, Velocity, and Acceleration Using Integrals
- Using Accumulation Functions and Definite Integrals in Applied Contexts
- Finding the Area Between Curves
- Finding the Area Between Curves That Intersect at More Than Two Points
- Volumes with Discs
- Volumes with Washers
- Volumes with Cross Sections
- The Arc Length of a Smooth, Planar Curve and Distance Traveled
Module 09 – Parametric, Polar, and Vector-Valued Equations
- Differentiating Parametric Equations and Finding Arc Length
- Differentiating and Integrating Vector-Valued Functions
- Solving Motion Problems Using Parametric and Vector-Valued Functions
- Defining Polar Coordinates and Differentiating in Polar Form
- Finding Area Bounded by Polar Curves
Module 10 – Infinite Sequences and Series
- Convergent and Divergent Infinite Series and Geometric Series
- Integral Test for Convergence, Harmonic Series, and p-Series
- Comparison Tests for Convergence
- Additional Tests to Determine Convergence
- Alternating Series and Their Error Bound
- Taylor Polynomial Approximations of Functions and Evaluating Error
- Radius and Interval of Convergence of Power Series
- Finding Taylor or Maclaurin Series for a Function
- Representing Functions as Power Series
