Course Description

AP Calculus: An Interactive Intelligent Textbook

Course Overview

This course provides comprehensive coverage of differential and integral calculus, aligned with the College Board's Advanced Placement (AP) Calculus curriculum. The course is designed to prepare students for both the AP Calculus AB and AP Calculus BC examinations while building deep conceptual understanding through interactive MicroSimulations and real-world applications.

Calculus represents one of humanity's greatest intellectual achievements—a mathematical framework that describes change, motion, and accumulation. From predicting planetary orbits to optimizing machine learning algorithms, calculus provides the foundational language for understanding our dynamic world.

Target Audience

High school students preparing for AP Calculus AB or BC examinations
College students seeking supplementary resources for Calculus I and II
Self-learners interested in building strong calculus foundations
Educators seeking interactive teaching resources

Prerequisites

Students should have successfully completed:

Algebra II - Polynomial, rational, exponential, and logarithmic functions
Precalculus or Trigonometry - Trigonometric functions, identities, and equations
Functions and Graphs - Domain, range, composition, and transformations

Recommended mathematical maturity includes comfort with:

Algebraic manipulation and equation solving
Function notation and multiple representations
Basic proof reading and logical reasoning
Graphing calculator operations

Course Structure

Unit 1: Limits and Continuity

Topics Covered

Introducing Calculus: Can Change Occur at an Instant?
Average vs. instantaneous rate of change
The tangent line problem
Historical context: Newton and Leibniz
Defining Limits and Using Limit Notation
Intuitive definition of limits
One-sided limits
Limit notation and interpretation
Estimating Limit Values from Graphs and Tables
Graphical analysis of limits
Numerical estimation techniques
When limits fail to exist
Determining Limits Using Algebraic Properties
Sum, difference, product, and quotient rules
Limits of polynomial and rational functions
Direct substitution property
Determining Limits Using Algebraic Manipulation
Factoring and cancellation
Rationalization techniques
Complex fraction simplification
Determining Limits Using the Squeeze Theorem
Statement and proof of the Squeeze Theorem
Applications to trigonometric limits
The fundamental limit: lim(x→0) sin(x)/x = 1
Exploring Types of Discontinuities
Removable discontinuities
Jump discontinuities
Infinite discontinuities
Defining Continuity at a Point
Three conditions for continuity
One-sided continuity
Continuity of composite functions
Confirming Continuity over an Interval
Continuity on open and closed intervals
Properties of continuous functions
Continuity of elementary functions
Removing Discontinuities
Defining functions to remove discontinuities
Continuous extensions
Piecewise function continuity
Connecting Infinite Limits and Vertical Asymptotes
Definition of infinite limits
Identifying vertical asymptotes
Behavior near vertical asymptotes
Connecting Limits at Infinity and Horizontal Asymptotes
Limits as x approaches infinity
Identifying horizontal asymptotes
End behavior of functions
Working with the Intermediate Value Theorem
Statement of the IVT
Existence of roots
Applications to real-world problems

Unit 2: Differentiation: Definition and Fundamental Properties

Topics Covered

Defining Average and Instantaneous Rates of Change
Difference quotients
Instantaneous rate as limit of average rate
Physical interpretations
Defining the Derivative of a Function
Derivative as limit definition
Derivative at a point vs. derivative function
Notation: f'(x), dy/dx, Df
Estimating Derivatives of a Function at a Point
Graphical estimation using tangent lines
Numerical estimation using difference quotients
Symmetric difference quotient
Connecting Differentiability and Continuity
Differentiability implies continuity
Continuity does not imply differentiability
Points of non-differentiability: corners, cusps, vertical tangents
Applying the Power Rule
Derivation of the power rule
Extension to negative and fractional exponents
Derivatives of polynomial functions
Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Linearity of the derivative
Combining rules for complex functions
Efficiency in differentiation
Derivatives of cos(x), sin(x), eˣ, and ln(x)
Derivation using limit definition
Pattern recognition
Natural exponential and logarithm relationship
The Product Rule
Statement and proof
Applications to products of functions
Extension to more than two factors
The Quotient Rule
Statement and proof
Applications to rational functions
Connection to product rule
Finding the Derivatives of Tangent, Cotangent, Secant, and Cosecant
Using quotient rule with sine and cosine
Memorization strategies
Domain considerations

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Topics Covered

The Chain Rule
Statement and intuitive explanation
Leibniz notation interpretation
Nested applications
Implicit Differentiation
Differentiating relations
Finding dy/dx from implicit equations
Higher-order implicit derivatives
Differentiating Inverse Functions
Inverse function theorem
Derivative of f⁻¹(x)
Graphical interpretation
Differentiating Inverse Trigonometric Functions
Derivatives of arcsin, arccos, arctan
Derivatives of arcsec, arccsc, arccot
Domain and range considerations
Selecting Procedures for Calculating Derivatives
Strategic approach to complex derivatives
Logarithmic differentiation
Combining multiple techniques
Calculating Higher-Order Derivatives
Second and higher derivatives
Notation for higher derivatives
Physical interpretation: acceleration, jerk

Unit 4: Contextual Applications of Differentiation

Topics Covered

Interpreting the Meaning of the Derivative in Context
Units of the derivative
Rate of change interpretation
Marginal analysis in economics
Straight-Line Motion: Connecting Position, Velocity, and Acceleration
Position function s(t)
Velocity as derivative of position
Acceleration as derivative of velocity
Speed vs. velocity
Rates of Change in Applied Contexts Other Than Motion
Population growth rates
Temperature change
Economic applications
Introduction to Related Rates
Setting up related rates problems
Implicit differentiation in context
Drawing diagrams and identifying relationships
Solving Related Rates Problems
Systematic problem-solving approach
Common related rates scenarios
Ladder, balloon, and shadow problems
Approximating Values of a Function Using Local Linearity
Tangent line approximation
Linear approximation formula
Error estimation
Using L'Hospital's Rule for Indeterminate Forms
Indeterminate forms: 0/0 and ∞/∞
Statement of L'Hospital's Rule
Repeated application
Other indeterminate forms: 0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰

Unit 5: Analytical Applications of Differentiation

Topics Covered

Using the Mean Value Theorem
Statement of MVT
Rolle's Theorem as special case
Applications and interpretations
Extreme Value Theorem, Global vs. Local Extrema
Statement of EVT
Finding global extrema on closed intervals
Critical points and endpoints
Determining Intervals of Increase and Decrease
First Derivative Test for monotonicity
Sign charts
Connection to function behavior
Using the First Derivative Test to Find Local Extrema
Identifying local maxima and minima
Sign change analysis
Critical points where f'(x) = 0 or DNE
Using the Candidates Test to Find Absolute Extrema
Systematic approach for closed intervals
Comparing critical point and endpoint values
Global optimization
Determining Concavity of Functions
Second derivative and concavity
Concave up vs. concave down
Inflection points
Using the Second Derivative Test to Find Extrema
Local extrema classification
When the test is inconclusive
Efficiency considerations
Sketching Graphs of Functions and Their Derivatives
Connecting f, f', and f'' graphs
Determining one graph from another
Comprehensive curve sketching
Connecting a Function, Its First Derivative, and Its Second Derivative
Summary of relationships
Analyzing all three simultaneously
Real-world interpretations
Introduction to Optimization Problems
Setting up optimization problems
Primary and constraint equations
Practical constraints
Solving Optimization Problems
Systematic problem-solving approach
Common optimization scenarios
Verifying solutions
Exploring Behaviors of Implicit Relations
Critical points of implicit curves
Tangent and normal lines
Second derivatives implicitly

Unit 6: Integration and Accumulation of Change

Topics Covered

Exploring Accumulations of Change
Accumulation functions
Area interpretation
Distance from velocity
Approximating Areas with Riemann Sums
Left, right, and midpoint Riemann sums
Trapezoidal approximation
Over and underestimates
Riemann Sums, Summation Notation, and Definite Integral Notation
Sigma notation review
Limit of Riemann sums
Definite integral definition
The Fundamental Theorem of Calculus and Accumulation Functions
FTC Part 1: Derivative of accumulation function
Accumulation functions defined by integrals
Chain rule applications
Interpreting the Behavior of Accumulation Functions
When accumulation functions increase/decrease
Extreme values of accumulation functions
Concavity of accumulation functions
Applying Properties of Definite Integrals
Additivity over intervals
Reversing limits
Integral of sum and constant multiple
The Fundamental Theorem of Calculus and Definite Integrals
FTC Part 2: Evaluation theorem
Computing definite integrals
Net change interpretation
Finding Antiderivatives and Indefinite Integrals
Basic antiderivative rules
Power rule for integration
Trigonometric integrals
Exponential and logarithmic integrals
Integrating Using Substitution
u-substitution technique
Recognizing patterns
Changing bounds for definite integrals
Integrating Functions Using Long Division and Completing the Square
Polynomial long division
Completing the square for quadratics
Resulting standard forms
Selecting Techniques for Antidifferentiation (BC only)
Strategic approach to integration
Recognizing appropriate techniques
Combining methods

Unit 7: Differential Equations

Topics Covered

Modeling Situations with Differential Equations
What is a differential equation?
Setting up DEs from word problems
Initial conditions
Verifying Solutions for Differential Equations
Substitution to verify
General vs. particular solutions
Initial value problems
Sketching Slope Fields
Creating slope fields
Interpreting slope fields
Isoclines
Reasoning Using Slope Fields
Qualitative behavior of solutions
Equilibrium solutions
Stability analysis
Approximating Solutions Using Euler's Method (BC only)
Euler's method algorithm
Step size considerations
Error accumulation
Finding General Solutions Using Separation of Variables
Separable differential equations
Integration technique
Constant of integration
Finding Particular Solutions
Applying initial conditions
Verifying particular solutions
Domain considerations
Exponential Models with Differential Equations
Natural growth and decay
Half-life and doubling time
Newton's Law of Cooling
Logistic Models with Differential Equations (BC only)
Logistic differential equation
Carrying capacity
Logistic growth formula

Unit 8: Applications of Integration

Topics Covered

Finding the Average Value of a Function
Average value formula
Mean Value Theorem for Integrals
Interpretations
Connecting Position, Velocity, and Acceleration Using Integrals
Integrating acceleration to find velocity
Integrating velocity to find position
Displacement vs. total distance
Using Accumulation Functions and Definite Integrals in Applied Contexts
Net change problems
Rate in, rate out problems
Interpreting integrals in context
Finding the Area Between Curves Expressed as Functions of x
Setting up area integrals
Determining bounds of integration
Multiple regions
Finding the Area Between Curves Expressed as Functions of y
Integrating with respect to y
Choosing the appropriate variable
Horizontal slices
Finding the Area Between Curves That Intersect at More Than Two Points
Finding intersection points
Breaking into subintervals
Absolute value considerations
Volumes with Cross Sections: Squares and Rectangles
Cross-sectional area approach
Setting up volume integrals
Visualizing 3D solids
Volumes with Cross Sections: Triangles and Semicircles
Various cross-sectional shapes
Area formulas in terms of x or y
Applications
Volume with Disc Method: Revolving Around the x- or y-Axis
Disc method derivation
Revolving around coordinate axes
Setting up integrals
Volume with Disc Method: Revolving Around Other Axes
Horizontal and vertical lines
Adjusting the radius formula
Visualizing the solid
Volume with Washer Method: Revolving Around the x- or y-Axis
Washer method for hollow solids
Inner and outer radius
Setting up integrals
Volume with Washer Method: Revolving Around Other Axes
Generalizing washer method
Careful radius identification
Complex regions
The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC only)
Arc length formula derivation
Computing arc length integrals
Distance traveled by a particle

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only)

Topics Covered

Defining and Differentiating Parametric Equations
Parametric representation of curves
Eliminating the parameter
Derivatives: dy/dx from parametric form
Second Derivatives of Parametric Equations
Finding d²y/dx² parametrically
Concavity of parametric curves
Applications
Finding Arc Lengths of Curves Given by Parametric Equations
Parametric arc length formula
Speed along a parametric curve
Applications
Defining and Differentiating Vector-Valued Functions
Vector functions and components
Derivatives of vector functions
Tangent vectors
Integrating Vector-Valued Functions
Antiderivatives of vector functions
Definite integrals of vectors
Position from velocity vectors
Solving Motion Problems Using Parametric and Vector-Valued Functions
Position, velocity, acceleration vectors
Speed as magnitude of velocity
Projectile motion
Defining Polar Coordinates and Differentiating in Polar Form
Polar coordinate system
Converting between rectangular and polar
Derivatives of polar curves
Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
Polar area formula
Setting up polar area integrals
Common polar curves
Finding the Area of the Region Bounded by Two Polar Curves
Intersection of polar curves
Area between polar curves
Careful bound determination

Unit 10: Infinite Sequences and Series (BC only)

Topics Covered

Defining Convergent and Divergent Infinite Series
Sequences vs. series
Partial sums
Convergence and divergence definitions
Working with Geometric Series
Geometric series formula
Convergence condition
Sum of convergent geometric series
The nth Term Test for Divergence
Statement of the test
What the test cannot determine
Applications
Integral Test for Convergence
Statement and conditions
Connection to improper integrals
p-series
Harmonic Series and p-Series
Divergence of harmonic series
Convergence of p-series for p > 1
Comparison applications
Comparison Tests for Convergence
Direct Comparison Test
Limit Comparison Test
Choosing comparison series
Alternating Series Test for Convergence
Alternating series definition
Leibniz's theorem
Alternating series error bound
Ratio Test for Convergence
Statement of ratio test
When the test is inconclusive
Applications to factorial and exponential series
Determining Absolute or Conditional Convergence
Absolute vs. conditional convergence
Rearrangement theorems
Classification of series
Alternating Series Error Bound
Error estimation for alternating series
Number of terms for desired accuracy
Applications
Finding Taylor Polynomial Approximations of Functions
Taylor polynomials centered at a
Maclaurin polynomials (a = 0)
Pattern recognition
Lagrange Error Bound
Remainder in Taylor's theorem
Error estimation
Applications to approximation
Radius and Interval of Convergence of Power Series
Power series definition
Finding radius of convergence
Testing endpoints
Representing Functions as Power Series
Geometric series manipulations
Differentiation and integration of power series
New series from known series
Taylor Series and Maclaurin Series
Taylor series definition
Common Maclaurin series
Recognition and manipulation

Learning Objectives by Bloom's Taxonomy Level

The 2001 revision of Bloom's Taxonomy organizes cognitive skills into six hierarchical levels. The following learning objectives are organized by these levels, progressing from foundational knowledge to higher-order thinking skills.

Level 1: Remember

Retrieve relevant knowledge from long-term memory

Upon successful completion of this course, students will be able to:

Limits and Continuity

LO 1.1 Recall the formal definition of a limit using epsilon-delta notation
LO 1.2 State the three conditions required for continuity at a point
LO 1.3 List the types of discontinuities (removable, jump, infinite)
LO 1.4 Recite the Squeeze Theorem and its conditions
LO 1.5 State the Intermediate Value Theorem

Differentiation

LO 1.6 Recall the limit definition of the derivative
LO 1.7 List the basic derivative rules (power, product, quotient, chain)
LO 1.8 State the derivatives of all six trigonometric functions
LO 1.9 Recall the derivatives of eˣ, ln(x), and inverse trigonometric functions
LO 1.10 State the conditions for differentiability at a point

Integration

LO 1.11 State both parts of the Fundamental Theorem of Calculus
LO 1.12 List the basic antiderivative formulas
LO 1.13 Recall the formulas for Riemann sums (left, right, midpoint, trapezoidal)
LO 1.14 State the formula for average value of a function

Applications

LO 1.15 Recall the formulas for area between curves
LO 1.16 State the disc and washer method formulas for volumes of revolution
LO 1.17 Recall the Mean Value Theorem and Rolle's Theorem
LO 1.18 State L'Hospital's Rule and its conditions

Series (BC)

LO 1.19 List the common convergence tests (nth term, integral, comparison, ratio, alternating series)
LO 1.20 Recall the Maclaurin series for eˣ, sin(x), cos(x), 1/(1-x), and ln(1+x)
LO 1.21 State the formula for the sum of a convergent geometric series
LO 1.22 Recall the Lagrange error bound formula

Parametric and Polar (BC)

LO 1.23 State the formula for dy/dx in parametric form
LO 1.24 Recall the formula for arc length of parametric curves
LO 1.25 State the formula for area in polar coordinates

Level 2: Understand

Construct meaning from instructional messages

Upon successful completion of this course, students will be able to:

Limits and Continuity

LO 2.1 Explain the intuitive meaning of a limit in terms of function behavior
LO 2.2 Interpret one-sided limits graphically and numerically
LO 2.3 Describe the relationship between limits and asymptotes
LO 2.4 Explain why continuity requires limits to match function values
LO 2.5 Illustrate the Intermediate Value Theorem with graphical examples

Differentiation

LO 2.6 Explain the derivative as instantaneous rate of change
LO 2.7 Interpret the derivative graphically as the slope of the tangent line
LO 2.8 Describe the relationship between differentiability and continuity
LO 2.9 Explain why corners, cusps, and vertical tangents preclude differentiability
LO 2.10 Interpret higher-order derivatives in context (velocity, acceleration, jerk)

Integration

LO 2.11 Explain the definite integral as a limit of Riemann sums
LO 2.12 Interpret the definite integral as net signed area under a curve
LO 2.13 Describe the relationship between differentiation and integration via FTC
LO 2.14 Explain why the constant of integration is necessary for indefinite integrals
LO 2.15 Interpret accumulation functions graphically and contextually

Applications

LO 2.16 Explain the meaning of derivative units in real-world contexts
LO 2.17 Interpret critical points and their relationship to extrema
LO 2.18 Describe how the first and second derivatives determine function shape
LO 2.19 Explain the geometric meaning of the Mean Value Theorem
LO 2.20 Interpret optimization solutions in context

Differential Equations

LO 2.21 Explain slope fields as visual representations of differential equations
LO 2.22 Describe the difference between general and particular solutions
LO 2.23 Interpret exponential growth/decay models in real-world contexts
LO 2.24 Explain carrying capacity in logistic growth models (BC)

Series (BC)

LO 2.25 Explain the difference between sequences and series
LO 2.26 Interpret convergence as the limit of partial sums
LO 2.27 Describe absolute vs. conditional convergence
LO 2.28 Explain the meaning of radius and interval of convergence
LO 2.29 Interpret Taylor polynomials as local approximations to functions

Parametric and Polar (BC)

LO 2.30 Explain how parametric equations describe motion in the plane
LO 2.31 Interpret vector-valued functions geometrically
LO 2.32 Describe the polar coordinate system and its relationship to rectangular coordinates

Level 3: Apply

Carry out or use a procedure in a given situation

Upon successful completion of this course, students will be able to:

Limits and Continuity

LO 3.1 Calculate limits using algebraic techniques (factoring, rationalization, simplification)
LO 3.2 Apply the Squeeze Theorem to evaluate trigonometric limits
LO 3.3 Determine continuity of piecewise functions at boundary points
LO 3.4 Use the Intermediate Value Theorem to prove existence of roots
LO 3.5 Find horizontal and vertical asymptotes using limits

Differentiation

LO 3.6 Compute derivatives using the power, product, quotient, and chain rules
LO 3.7 Apply implicit differentiation to find dy/dx from implicit equations
LO 3.8 Calculate derivatives of inverse functions using the inverse function theorem
LO 3.9 Use logarithmic differentiation for complex products and quotients
LO 3.10 Find higher-order derivatives of various function types

Integration

LO 3.11 Evaluate definite integrals using the Fundamental Theorem of Calculus
LO 3.12 Apply u-substitution to compute indefinite and definite integrals
LO 3.13 Use long division and completing the square to rewrite integrands
LO 3.14 Calculate Riemann sum approximations from graphs and tables
LO 3.15 Apply integration by parts to evaluate integrals (BC)

Applications

LO 3.16 Solve related rates problems using implicit differentiation
LO 3.17 Use linear approximation to estimate function values
LO 3.18 Apply L'Hospital's Rule to evaluate indeterminate limits
LO 3.19 Calculate average value of a function over an interval
LO 3.20 Find area between curves using definite integrals
LO 3.21 Compute volumes of solids using disc, washer, and cross-section methods
LO 3.22 Calculate arc length of curves (BC)

Differential Equations

LO 3.23 Verify solutions to differential equations by substitution
LO 3.24 Solve separable differential equations
LO 3.25 Apply initial conditions to find particular solutions
LO 3.26 Use Euler's method to approximate solutions numerically (BC)

Series (BC)

LO 3.27 Apply convergence tests to determine series behavior
LO 3.28 Find Taylor and Maclaurin series for functions
LO 3.29 Determine radius and interval of convergence using the ratio test
LO 3.30 Use power series operations (differentiation, integration) to find new series
LO 3.31 Apply Lagrange error bound to estimate approximation accuracy

Parametric and Polar (BC)

LO 3.32 Calculate derivatives of parametric and vector-valued functions
LO 3.33 Find area enclosed by polar curves
LO 3.34 Compute arc length of parametric curves
LO 3.35 Solve motion problems using vector functions

Level 4: Analyze

Break material into constituent parts and determine relationships

Upon successful completion of this course, students will be able to:

Limits and Continuity

LO 4.1 Distinguish between different types of discontinuities and their causes
LO 4.2 Analyze function behavior to determine where limits exist or fail to exist
LO 4.3 Compare and contrast one-sided limits and two-sided limits
LO 4.4 Investigate the relationship between limits, continuity, and differentiability

Differentiation

LO 4.5 Analyze the relationship between a function and its derivative graphically
LO 4.6 Determine intervals of increase/decrease from derivative sign analysis
LO 4.7 Investigate concavity using second derivative analysis
LO 4.8 Classify critical points using first and second derivative tests
LO 4.9 Examine the connections among f, f', and f'' simultaneously

Integration

LO 4.10 Analyze accumulation functions to determine their behavior (increasing/decreasing, concavity)
LO 4.11 Compare different integration techniques for efficiency
LO 4.12 Distinguish between displacement and total distance traveled
LO 4.13 Investigate the relationship between rate functions and accumulation

Applications

LO 4.14 Analyze word problems to identify relevant calculus concepts and relationships
LO 4.15 Break down optimization problems into constraints and objective functions
LO 4.16 Examine related rates scenarios to identify changing quantities and their relationships
LO 4.17 Analyze curve sketching by integrating multiple derivative properties
LO 4.18 Investigate the relationship between position, velocity, and acceleration functions

Differential Equations

LO 4.19 Analyze slope fields to predict solution behavior without solving
LO 4.20 Distinguish between stable and unstable equilibrium solutions
LO 4.21 Compare exponential and logistic growth models (BC)
LO 4.22 Examine how initial conditions affect solution trajectories

Series (BC)

LO 4.23 Analyze series to determine appropriate convergence tests
LO 4.24 Compare convergence rates of different series
LO 4.25 Investigate the relationship between a function and its Taylor series
LO 4.26 Examine how error bounds depend on the number of terms and center point

Parametric and Polar (BC)

LO 4.27 Analyze parametric curves to identify key features (direction, speed, turning points)
LO 4.28 Compare rectangular, parametric, and polar representations of curves
LO 4.29 Investigate motion along curves using velocity and acceleration vectors

Level 5: Evaluate

Make judgments based on criteria and standards

Upon successful completion of this course, students will be able to:

Problem-Solving Strategies

LO 5.1 Assess which differentiation technique is most efficient for a given function
LO 5.2 Judge which integration method is most appropriate for a given integral
LO 5.3 Evaluate the reasonableness of calculated answers using estimation and context
LO 5.4 Critique solution methods for efficiency, elegance, and correctness

Approximations and Error

LO 5.5 Evaluate the accuracy of Riemann sum approximations
LO 5.6 Assess the quality of linear approximations based on function curvature
LO 5.7 Judge the precision of Taylor polynomial approximations using error bounds
LO 5.8 Evaluate Euler's method approximations and their limitations (BC)

Applications and Modeling

LO 5.9 Assess whether a mathematical model appropriately describes a real-world situation
LO 5.10 Evaluate optimization solutions for practical feasibility
LO 5.11 Judge the validity of related rates solutions in context
LO 5.12 Critique the assumptions underlying differential equation models

Series Analysis (BC)

LO 5.13 Evaluate which convergence test provides the most conclusive result
LO 5.14 Assess the trade-off between accuracy and computational complexity in series approximations
LO 5.15 Judge whether absolute or conditional convergence affects series manipulations

Mathematical Reasoning

LO 5.16 Evaluate the logical validity of calculus-based arguments
LO 5.17 Assess whether conditions for theorems (IVT, MVT, FTC) are satisfied
LO 5.18 Judge the appropriateness of mathematical tools for given problems
LO 5.19 Critique graphical and numerical evidence for mathematical claims

Level 6: Create

Put elements together to form a coherent whole; reorganize into new patterns

Upon successful completion of this course, students will be able to:

Problem Construction

LO 6.1 Design optimization problems that model real-world scenarios
LO 6.2 Construct related rates problems from physical situations
LO 6.3 Formulate differential equations to model growth, decay, and other phenomena
LO 6.4 Create word problems that require specific calculus techniques to solve

Mathematical Communication

LO 6.5 Compose clear, rigorous mathematical arguments using calculus concepts
LO 6.6 Develop complete solutions with appropriate justification at each step
LO 6.7 Create visual representations (graphs, diagrams) to illustrate calculus concepts
LO 6.8 Produce coherent explanations of calculus ideas for various audiences

Synthesis and Extension

LO 6.9 Combine multiple calculus techniques to solve novel problems
LO 6.10 Develop new integration techniques by modifying known methods
LO 6.11 Construct Taylor series for functions not directly covered in the course
LO 6.12 Design numerical methods to approximate solutions when exact methods fail

Applications and Modeling

LO 6.13 Build mathematical models for phenomena in physics, biology, economics, and engineering
LO 6.14 Create parametric and polar representations for curves with specified properties (BC)
LO 6.15 Develop solution strategies for unfamiliar differential equations
LO 6.16 Construct volume calculations for solids with unusual cross-sections

Investigation and Discovery

LO 6.17 Formulate conjectures about calculus relationships and test them systematically
LO 6.18 Design experiments using technology to explore calculus concepts
LO 6.19 Create connections between calculus and other areas of mathematics
LO 6.20 Develop original problems that extend course concepts in meaningful ways

Assessment Alignment

AP Calculus AB Coverage

This course fully addresses all topics and skills tested on the AP Calculus AB examination:

Units 1-8 provide comprehensive coverage of AB content
Learning objectives at all Bloom's levels align with AP mathematical practices
MicroSimulations reinforce conceptual understanding emphasized on the AP exam

AP Calculus BC Coverage

Students pursuing BC credit will additionally complete:

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Unit 10: Infinite Sequences and Series
BC-specific topics within Units 6, 7, and 8

College Credit Equivalency

AP Calculus AB: Equivalent to Calculus I (typically 3-4 semester hours)
AP Calculus BC: Equivalent to Calculus I and II (typically 6-8 semester hours)

Course Features

Interactive MicroSimulations

Over 150 interactive MicroSimulations bring calculus concepts to life:

Visualize limits approaching values dynamically
Explore derivatives through tangent line animations
See Riemann sums converge to definite integrals
Manipulate slope fields and watch solution curves form
Experiment with Taylor polynomial approximations
Interact with polar and parametric curves in real-time

Multiple Representations

Every concept is presented through multiple representations:

Graphical: Interactive visualizations and dynamic graphs
Numerical: Tables, computations, and approximations
Algebraic: Symbolic manipulation and formula derivation
Verbal: Clear explanations and real-world contexts

Real-World Applications

Connections to authentic applications include:

Physics: Motion, work, fluid pressure
Engineering: Optimization, rates of change
Biology: Population dynamics, drug concentration
Economics: Marginal analysis, consumer surplus
Data Science: Curve fitting, numerical methods

Conclusion

This AP Calculus course combines rigorous mathematical content with innovative interactive pedagogy. By engaging with concepts at all levels of Bloom's Taxonomy—from foundational recall to creative synthesis—students develop both the procedural fluency and conceptual understanding necessary for success on AP examinations and in future STEM coursework.

The integration of MicroSimulations transforms abstract calculus concepts into tangible, explorable ideas. Students don't just learn calculus; they experience it as a living, dynamic discipline that describes the patterns of change in our world.