Newton’s Law of Cooling and Orthogonal Trajectories

Salient features of this module are:

• Pre-requisites to polar curves
• Graphical visualization
• Concept testing
• Unravelling examiners mind
• Model question paper

Course Contents: 

Pre-requisites to Maxima & Minima

• Sign of Trigonometric Identities
  • Sign of Trigonometric Identities – Introduction
  • Sign of Trigonometric Identities – Quad I
  • Sign of Trigonometric Identities – Quad II
  • Sign of Trigonometric Identities – Quad III
  • Sign of Trigonometric Identities – Quad IV
  • Sign of Trigonometric Identities – Summary
  • Sign of Trigonometric Identities – 1
  • Sign of Trigonometric Identities – 2
• General solution for Trigonometric  ratios
  • General solution for Trigonometric ratios (Example 1)
  • General solution for Trigonometric ratios (Example 2)
  • General solution for Trigonometric ratios (Example 3)
  • General solution for Trigonometric ratios (Example 4)
• Concept of derivate – Slope
  • Concept of Derivative – Slope – Straight line
  • Concept of Derivative – Slope – Curve
• Differentiation of Basic functions
  • Introduction – Chain and Quotient Rules
  • Derivative of tan x
  • Derivative of sec x
  • Derivative of cosec x
  • Derivative of cot x
  • Derivative of [y = e^x * x^n]
  • Derivative of [y = tan^(-1) x
  • Derivative of [y = x/(ax+b)] – Quotient rule
  • Derivative of [y = x/(ax+b)] – Chain rule
  • Derivative of [x^2 + y^2 = k]
  • Derivative of [y = 1/sqrt(ax^2 +b)]
  • Derivative of [y = a^x (log (1+x^2))]
  • Derivative of [y = a^sinx (1+sin^2 (x)) sec x]
  • Graphical understanding of trigonometric derivatives
• Application of Derivative
  • Finding nature of the curve
  • Interpreting curvature of a curve
  • Linearization of a curve
  • L-Hospital Rule
  • Use of second derivative (curvature)
  • Maxima & Minima – 1
  • Maxima & Minima – 2
• Angle between Curves & Radius of curvature
  • Essential trigonometric identities 1
  • Essential trigonometric identities 2
  • Essential trigonometric identities 3
  • Essential trigonometric identities 4
  • Essential trigonometric identities 5

PRE-REQUISITES – ELEMENTS OF INTEGRAL CALCULUS

• Pre-requisites – Elements of Integral Calculus
  • Understanding Integration
  • Concept of Integration – Elementary Example
  • Introducing the Learning matrix
  • Why is Integration an anti-differentiation process?
  • Useful Rules and Symbolisms of Integration – (Rule 1)
  • Useful Rules and Symbolisms of Integration – (Rule 2)
  • Introducing Standard Integrals
  • Deriving Integrals of Basic function
  • Deriving Specific Integrals – 1
  • Deriving Specific Integrals – 2
  • Deriving Specific Integrals – 3
  • Deriving Specific Integrals – 4
  • Deriving Specific Integrals – 5
  • Useful Rules of Definite Integrals – (Rule 1)
  • Useful Rules of Definite Integrals – (Rule 2)
  • Useful Rules of Definite Integrals – (Rule 3)
  • Useful Rules of Definite Integrals – (Rule 4)

NEWTON’S LAW OF COOLING & ORTHOGONAL TRAJECTORIES

• Application od ODE’s
  • Newton’s Law of Cooling – Conceptual Understanding
  • Newton’s Law of Cooling – Problem 1 – Cooling
  • Newton’s Law of Cooling – Problem 2 – Heating
  • Orthogonal Tragectories – Conceptual Understanding
  • Orthogonal Tragectories – Problem 1
  • Orthogonal Tragectories – Problem 2
  • Orthogonal Tragectories – Problem 3
  • Orthogonal Tragectories – Problem 4