YEAR 5: Calculus
Pass the AP Calculus exam with ease. Differential and integral calculus. Applications to physical problems. Numerical methods for solutions of equations and interpolation. Theory of ordinary differential equations.
Course Outline
Part I

What is a Derivative?

Motion, Distance, and Velocity.

Derivative of a Function. Simple Examples of Calculating Derivatives.

Examples of Calculating Derivatives.

Properties of Derivatives. Approximating the Values of a Function by Means of a Derivative.

Tangent to a Curve.

Increase and Decrease of Functions. Maxima and Minima.

The Second Derivative of a Function. Convexity and Concavity of a Curve. Points of Inflection.

Derivatives and Tangents in Polar Coordinates.


What is an Integral?

Determining Distance from Rate of Motion. The Area Bounded by a Curve.

The Definite Integral.

Relationship Between the Integral and the Derivative.

The Indefinite Integral.

Properties of Integrals.

Examples and Applications.

Part II

Calculation of Derivatives.

Derivative of a Sum and of a Product of Functions.

Composite Functions. Derivative of the Fraction of Two Functions.

Inverse Function. Parametric Representation of a Function.

Power Function. Polynomials.

Exponential Function. Logarithms.

Trigonometric Functions. Inverse Trigonometric Functions.

Partial Derivatives. Functions of Several Variables.

Derivative of an Implicit Function.

L'Hopital's Rule.


Approximation.

Taylor's Formula. Taylor's Series.

Rate of Convergence. EulerAbel Method.

Application of Differentials to Approximation.

Constructing Graphs of Functions Using Derivatives.

Finite Increments Method to Approximate Roots of Equations.

Lagrange's Interpolation Formula.

Finite Differences. Newton's Interpolation Formula. Numerical Differentiation.

Approximate Solutions of Algebraic and Transcendental Equations.

Method of Chords. Newton's Method.

BudanFourier. LobachevskyGraeffe.



Integration Techniques.

Elementary Integrals.

General Properties of Integrals.

Change of Variables.

Integration by Parts.

Trigonometric Substitution.

Integration by Partial Fractions.

Part III

Applications to Problems of Physics.

Velocity. Acceleration.

Pendulum. Elastic Vibration of a Spring.

Newton's Laws. Kepler's Laws.

Specific Heat Capacity of an Object. Thermal Expansion.

Center of Mass. Moments of Inertia.


Ordinary Differential Equations.

First Order Equations. Ricatti Equation. Orthogonal Trajectories.

Higher Order Linear Equations and Systems of Differential Equations.

Bessel Equation. Bessel Functions.

Euler's Equations. Euler's Method.

Series Solutions.

RungeKutta Method. Adams Method. Milne Method.

Laplace Transformation. Heaviside Operational Calculus.

Stability Theory. Lyapunov Functions. Autonomous Systems.

Asymptotic Behavior. Perturbation Method.
