Calculus FlexBook® for Middle School

Calculus FlexBook® for High School

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Levels are CK-12's student achievement levels.

Basic
Students matched to this level have a partial mastery of prerequisite knowledge and skills fundamental for proficient work.

At Grade (Proficient)
Students matched to this level have demonstrated competency over challenging subject matter, including subject matter knowledge, application of such knowledge to real-world situations, and analytical skills appropriate to subject matter.

Advanced
Students matched to this level are ready for material that requires superior performance and mastery.

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- Limit
- Concept of Limit
- Evaluate Limits Graphically and Vertical Asymptotes
- Direct Substitution
- Properties of Limits
- One-Sided Limit Type
- Infinite Limit Type
- Basic Trigonometric Limits
- Limits of Polynomial and Rational Functions
- Limits Involving Radical Functions
- Continuity
- Continuity at a Point, Continuity Test, Types of Discontinuity
- Intermediate Value Theorem, Existence of Solutions

- Derivatives and Applications of Derivatives
- Definition of Derivative
- Slope of Tangent Line
- Average and Instantaneous Rates of Change
- Differentiability Implies Continuity
- Constant, Identity, and Power Rules
- Sum and Difference Differentiation Rules
- Products and Quotients Differentiation Rules
- Motion, Velocity, Speed, Acceleration and Jerk
- Derivatives of Trigonometric Functions
- Derivative of Sine
- Differentiation Rules and Multi-Step Differentiation
- Chain Rule
- Implicit Differentiation
- Higher Order Derivatives-Acceleration and Jerk
- Related Rates
- Derivatives of Exponential Functions
- Differentials and Approximations
- Tangent Line Approximation
- Approximation Errors
- Maxima and Minima, Mean Value Theorem, Rolle's Theorem
- Absolute versus Local Extrema
- First Derivative Test
- Concavity and Inflection
- Second Derivative Test
- Limits at Infinity
- Absolute Extrema and Optimization
- Newton's Method
- Analyzing the Graphs of Functions
- Graphing Functions Using Calculus

- Indefinite Integral and Integral Formulas
- Antiderivative
- Trigonometric Rules
- Area Computations
- Area Sums
- Area as Limit
- Probability and Probability Density Functions
- Definite Integrals and Finite Sum Estimation
- Properties of Definite Integrals
- Riemann Sum
- Using Finite Sums to Distance
- Fundamental Theorem of Calculus, Integration Techniques, Numerical Integration
- Mean Value Theorem
- Endpoint Approximations
- Trapezoidal and Midpoint Approximations
- Simpson's Rule
- Setting Up Integration Problems, Volumes, Volumes of Solids of Revolution
- Area Between Curves
- Cross Section Method
- Volumes by Disks
- Volumes by Washers
- Method of Cylindrical Shells
- Length of a Plane Curve
- Basic Formula of Areas of Surfaces of Revolution
- Moments and Centers of Mass, Work and Force, Fluid Forces
- Work by a Variable Force, Work by a Constant Force

- Exponential Function, a to the x power
- Natural Exponent, e to the x power
- Derivatives and Integrals of Exponential Functions
- Inverse Functions
- Definition of Inverse Functions
- Reflective Property
- Existence (One-to-One, Monotonic)
- Finding Inverse Functions
- Continuity and Differentiability
- Derivative Rule for Inverses
- Natural Log
- Analysis of Logarithmic Graphs
- Logarithmic Differentiation
- Logarithmic Integration
- Other Exponential and Logarithmic Functions
- Indeterminate Forms
- Growth and Decay
- Exponential Change
- Compound Interest
- Compound Interest per Year
- Compound Interest per Period
- Continuous Interest
- APR and APY
- Annuities
- Annuities for Loans
- Relative Rates of Growth and Decay
- Properties of Inverse Trigonometric Functions
- Derivatives of Six Inverse Trigonometric Properties
- Analyzing Graphs of Inverse Trigonometric Functions
- Integration of Inverse Trigonometric Functions by Substitution
- Hyperbolic Functions
- Hyperbolic Functions (Definitions, Evaluation, Properties, Identities)
- Derivatives and Integrals of Hyperbolic Functions
- Inverse Hyperbolic Functions (Definitions, Evaluation, Properites)
- Derivatives and Integrals of Inverse Hyperbolic Functions

- Trigonometric Integration
- Strategies for Fitting Integrands to Basic Forms
- Integration by Parts
- Rule for Integration by Parts
- Guidelines for Integration by Parts
- Recursive Integration by Parts
- Trigonometric Integrals
- Powers of Sines and Cosines
- Powers of Secants and Tangents
- Products of Sines and Cosines of Different Angles
- Definite Integrals of Even and Odd Functions
- Trigonometric Substitution
- Partial Fractions
- Partial Fraction Decomposition
- General Method
- Convergence and Divergence of Integrals
- Improper Integrals and Infinite Discontinuities
- Domination Test
- Limit Comparison Test
- General and Particular Solutions
- Euler's Numerical Methods (Improved Euler, Runge-Kutta)
- Slope Fields and Isoclines
- Differential Equations and Integration
- Solving Separable First-Order Differential Equations
- Exponential and Logistic Growth

- Sequences in Calculus
- Definition of a Sequence
- Limit of a Sequence
- Convergence and Divergence of Sequences
- L'Hopital's Rule
- Rules, Sandwich/Squeeze
- Picard's Method
- Infinite Series
- Infinite Polynomials
- Sequence of Partial Sums
- Geometric Series
- Alternating Series
- Convergence Tests
- nth-Term Test
- Ratio Test
- Root Test
- Integral Test
- Limit Comparison Test, Simplified Limit Comparison Test
- Alternating Series Test
- Alternating Series Remainder
- Absolute and Conditional Convergence
- Rearrangement
- Summary and Comparison of Procedures for Determining Convergence
- Power Series and Convergence
- Term-By-Term Integration of Power Series
- Series Multiplication of Power Series
- Taylor and Maclaurin Series
- Taylor and Maclaurin Polynomials
- Convergence of Taylor and Maclaurin Polynomials
- Maclaurin Series Truncation Error
- Calculations with Series
- Binomial Series for Powers
- Choosing Centers
- Evaluating Non-Elementary Integrals
- Finding and Identifying Maclaurin Series

- Properties of Ellipses
- Properties of Hyperbolas
- Quadratic Equations and the Conic
- Rotation to Eliminate XY-Term
- Parametric Equations and Plane Curves
- Parametric Equations
- Eliminating the Parameter (Domain, Curve Sketching)
- Finding Parametric Equations (Parabola, Cycloids)
- Evaluating Parametric Equations
- Parametric-Inverses
- Parametric Equations and Calculus
- Differentiation and Parametric Form
- Parametric Formula for Length of a Curve
- Parametric Formulas for Area of a Surface of Revolution
- Parametric Formula for Area in the Plane
- Polar Axis and Polar Coordinates
- Special Polar Equations and Graphs
- Polar Equations and Conics
- Polar Equations of Conics in Calculus
- Kepler's Laws
- Polar Equations and Calculus
- Polar Formula for Area in the Plane
- Polar Formula for Length of a Curve
- Polar Formulas for Area of a Surface of Revolution
- Polar Formulas for Mass, Moments, and Centers

- Vectors in the Plane
- Sum, Difference, Product of Vector and Scalar
- Dot Product (Scalar Product, Inner Product, Tail to Tail)
- Angle Between Vectors
- Vector-Valued Functions in the Plane
- Derivative of a Vector-Valued Function
- Properties of the Derivative
- Indefinite Integral of a Vector-Valued Function
- Position, Velocity, and Acceleration Along a Plane Curve (Differentiation)
- Acceleration, Velocity, and Position Along a Plane Curve (Integration)
- Arc Length of a Plane Curve
- Curvature Formulas (Circle of Curvature)
- Tangential and Normal Components of Acceleration

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