Unit 1

• Hyperbolic functions
• Higher order derivatives
• Leibnitz rule and its applications to problems of type:
  • \(\pmb{e^{ax} + b \sin x}\)
  • \(\pmb{e^{ax} + b \cos x}\)
  • \(\pmb{(ax + b)^n \sin x}\)
  • \(\pmb{(ax + b)^n \cos x}\)
• Concavity and inflection points
• Envelopes
• Asymptotes
• Curve tracing in Cartesian coordinates
• Tracing in polar coordinates of standard curves
• L’Hospital’s rule
• Applications in business, economics, and life sciences

Unit 2

• Reduction formulae
• Derivations and illustrations of reduction formulae of the type:
  • \(\pmb{\int \sin^n x \, dx}\)
  • \(\pmb{\int \cos^n x \, dx}\)
  • \(\pmb{\int \tan^n x \, dx}\)
  • \(\pmb{\int \sec^n x \, dx}\)
  • \(\pmb{\int (\log x)^n \, dx}\)
  • \(\pmb{\int \sin^n x \sin mx \, dx}\)
• Parametric equations
• Parameterizing a curve
• Arc length of a curve
• Arc length of parametric curves
• Area under a curve
• Area and volume of surface of revolution
• Techniques of sketching conics

Unit 3

• Reflection properties of conics
• Rotation of axes and second degree equations
• Classification of conics using the discriminant
• Polar equations of conics
• Spheres
• Cylindrical surfaces
• Central conicoids, paraboloids
• Plane sections of conicoids
• Generating lines
• Classification of quadrics
• Illustrations of graphing standard quadric surfaces like:
  • Cone
  • Ellipsoid

Unit 4

• Differential equations and mathematical models
• General, particular, explicit, implicit, and singular solutions of a differential equation
• Exact differential equations and integrating factors
• Separable equations and equations reducible to this form
• Linear equation and Bernoulli equations
• Special integrating factors and transformations

Graphical Demonstration (Teaching Aid)

• Plotting of graphs of functions:
  • \(\pmb{e^{ax} + b}\)
  • \(\pmb{\log(ax + b)}\)
  • \(\pmb{\frac{1}{ax + b}}\)
  • \(\pmb{\sin(ax + b)}\)
  • \(\pmb{\cos(ax + b)}\)
  • \(\pmb{|ax + b|}\)
• Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph, and comparing them
• Sketching parametric curves (e.g., trochoid, cycloid, epicycloids, hypocycloid)
• Obtaining surface of revolution of curves
• Tracing of conics in Cartesian coordinates/polar coordinates
• Sketching:
  • Ellipsoid
  • Hyperboloid of one and two sheets
  • Elliptic cone
  • Elliptic paraboloid
  • Hyperbolic paraboloid