YEAR 5: Calculus

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.

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. Euler-Abel 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.

  • Budan-Fourier.  Lobachevsky-Graeffe.

Integration Techniques.

• Elementary Integrals.

• General Properties of Integrals.

• Change of Variables.

• Integration by Parts.

• Trigonometric Substitution.

• Integration by Partial Fractions.

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.

• Runge-Kutta Method. Adams Method. Milne Method.

• Laplace Transformation. Heaviside Operational Calculus.

• Stability Theory. Lyapunov Functions. Autonomous Systems.

• Asymptotic Behavior.  Perturbation Method.