relationships, manipulation
sufficiency, necessity, existence, for-all
Definition and properties
Basics and extrema
Summation, extreme values
Complex numbers and Manipulations
Proof by induction
Change in Tandem, Rates of Change (linear, quadratic, polynomial functions)
Polynomial Functions: Complex Zeros, End Behavior
Rational Functions: End Behavior, Zeros, Vertical Asymptotes, Holes
Equivalent Representations of Polynomial and Rational Expressions
Transformations of Functions
Function Model Selection, Assumption Articulation, Construction, Application
Change in Arithmetic and Geometric Sequences
Change in Linear and Exponential Functions
Exponential Functions: Manipulation, Context and Data Modeling
Competing Function Model Validation
Composition, Inverse Functions
Logarithmic Expressions, Inverses of Exponential Functions
Logarithmic Functions: Manipulation, Context and Data Modeling
Exponential and Logarithmic Equations and Inequalities
Semi-log Plots
Periodic Phenomena, sine, cosine, and tangent
Sine and Cosine: Function Values, Function Graphs
Sinusoidal Functions: Transformations, Context and Data Modeling 3 2
The Tangent Function, Inverse Trigonometric Functions
Trigonometric Equations and Inequalities
The secant, cosecant, and cotangent functions
Equivalent Representations of Trigonometric Functions
Polar Coordinates, Polar Function Graphs, Rates of Change in Polar Functions
Parametric Functions: Modeling Planar Motion, Rates of Change
Parametrically Defined Circles and Lines, Conic Sections
Implicitly Defined Functions: Parametrization
Vectors, Vector-Valued Functions
Matrices, The Inverse and Determinant, Linear Transformations
Matrices as Functions, Matrices Modeling contexts
Coordinate geometry, Circular measure
Graphs, cos/sin theorem, triangular shape, area
Linear arithmetics, products, angle
Structural features, volumes and surface areas
Parallel / perpendicular proof of line, planes; normals, angle and distance
Parametrically Defined Circles and Lines, relationships, arc and tangent, extrema
Conic Sections, eccentricity, geometric properties, arc foci problems
Permutations and combinations
Binomial Theorem
Discrete and Continuous random variables, Linear combinations of random variables
One-Variable data: representation, description, summary statistics, graph, comparison
Two-Variable data: representation, statistics, correlation
Mean and Standard Deviation
Random and Non-random patterns, Classical model,
Estimating Probabilities using Simulation
Mutually Exclusiveness, Independence, Union of Events
Conditional Probability
Probability generating functions
The Binomial Distribution and n-Bernoulli
The Geometric Distribution
The normal distribution
The Poisson distribution
Central Limit Theorem
Biased and Unbiased Point Estimates
Sampling and estimation
Experimental Design
Inference using normal and t-distributions
Categorical and Quantitative Data: confidence interval, testing and p-Values, errors
Slopes, Regressions, Least Squares, Residuals, Linearity / Non-linearity
Chi-Square, Goodness of Fit Test, Test for Homogeneity or Independence
Hypothesis tests
2 | -tests
Non-parametric tests
Introducing Calculus: Can Change Occur at an Instant? Defining Limits
Estimating Limit Values from Graphs, Tables
Determining Limits: Using Algebraic Properties of Limits, Algebraic Manipulation, the Squeeze Theorem, Selecting Procedures for Determining Limits
Connecting Multiple Representations of Limits
Exploring Types of Discontinuities, Defining Continuity at a Point Confirming Continuity over an Interval, Removing Discontinuities
Connecting Infinite Limits and Vertical Asymptotes , Connecting Limits at Infinity and Horizontal Asymptotes
Working with the Intermediate Value Theorem
Defining Average and Instantaneous Rates of Change at a Point
Defining, Estimating the Derivative of a Function Derivatives (at a Point)
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
Applying the Power Rule
Derivative Rules: Constant, Sum, Difference, and Constant Multiple, The Product Rule, The Quotient Rule
Derivatives of cos x, sin x, ex, and ln x, Tangent, Cotangent, Secant, and/or Cosecant Functions
Monotonicity, extrema, solving inequalities
The Chain Rule, Implicit Differentiation
Differentiating Inverse Functions, Inverse Trigonometric Functions
Selecting Procedures for Calculating Derivatives, Calculating HigherOrder Derivatives
Interpreting the Meaning of the Derivative in Context
Straight-Line Motion: Connecting Position, Velocity, and Acceleration
Rates of Change in Applied Contexts Other Than Motion
Introduction to Related Rates, Solving Related Rates Problems
Approximating Values of a Function Using Local Linearity and Linearization
Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms
Mean Value Theorem, Extreme Value Theorem
Global Versus Local Extrema, and Critical Points
Determining Intervals on Which a Function Is Increasing or Decreasing
Using the First Derivative Test to Determine Relative (Local) Extrema
Using the Candidates Test to Determine Absolute (Global) Extrema
Determining Concavity of Functions over Their Domains
Using the Second Derivative Test to Determine Extrema
Sketching Graphs of Functions and Their Derivatives
Connecting a Function, Its First Derivative, and Its Second Derivative
Introduction to Optimization Problems Solving Optimization Problems
Exploring Behaviors of Implicit Relations
Exploring Accumulations of Change, Approximating Areas with Riemann Sums
Riemann Sums, Summation Notation, and Definite Integral Notation
The Fundamental Theorem of Calculus and Accumulation Functions
Interpreting the Behavior of Accumulation Functions Involving Area
Applying Properties of Definite Integrals
The Fundamental Theorem of Calculus and Definite Integrals
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Integrating Using Substitution, Using Long Division and Completing the Square
Integrating Using Integration by Parts, Using Linear Partial Fractions
Evaluating Improper Integrals, Selecting Techniques for Antidifferentiation
Modeling Situations with Differential Equations, Verifying Solutions
Sketching, Reasoning Using Slope Fields
Approximating Solutions Using Euler’s Method
Finding General Solutions Using Separation of Variables
Finding Particular Solutions Using Initial Conditions and Separation of Variables
Exponential, Logistic Models with Differential Equations
Finding the Average Value of a Function on an Interval
Connecting Position, Velocity, and Acceleration of Functions Using Integrals
Using Accumulation Functions and Definite Integrals in Applied Contexts
Finding the Area Between Curves Expressed as Functions of x, y, and Curves That Intersect at More Than Two Points
Volumes with Cross Sections: Squares and Rectangles, Triangles and Semicircles
Volume with Disc Method: Revolving Around the x-, y-Axis or Other Axes
Volume with Washer Method: Revolving 4 Around the x-, y-Axis or Other Axes
The Arc Length of a Smooth, Planar Curve and Distance Traveled
Defining and Differentiating Parametric Equations
Second Derivatives of Parametric Equations
Finding Arc Lengths of Curves Given by Parametric Equations
Defining and Differentiating VectorValued Functions
Integrating Vector Valued Functions
Solving Motion Problems Using Parametric and VectorValued Functions
Defining Polar Coordinates and Differentiating in Polar Form
Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
Finding the Area of the Region Bounded by Two Polar Curves
Infinite Sequences and Series
Defining Convergent and Divergent Infinite Series
Working with Geometric Series
The nth Term Test for Divergence, Integral Test for Convergence
Harmonic Series and p-Series
Comparison, Alternating Series Test, Ratio Test for Convergence
Determining Absolute or Conditional Convergence
Alternating Series Error Bound
Finding Taylor Polynomial Approximations of Functions
Lagrange Error Bound
Radius and Interval of Convergence of Power Series
Finding Taylor or Maclaurin Series for a Function
Representing Functions as Power Series