Chapter 2 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 10
Type: Notes

TITLESOLUTION
Exercise 2.1View
Exercise 2.2 View
Exercise 2.3 View
Exercise 2.4 View
Exercise 2.5 View
Exercise 2.6 View
Exercise 2.7 View
Exercise 2.8 View
Exercise 2.9 View
Exercise 2.10 View

In this chapter we will read these topics:

  • Name the mathematicians who invented different calculus.
  • Define dependent and independent varaibles.
  • Define the derivative of a function.
  • What is differentiation?
  • Differentiation by definition.
  • The first principle or AB-Initio Method.
  • Write different notations used for derivatives by different mathematicians.
  • Find derivative when n is a positive integer.
  • Find derivative when n is a negative integer.
  • Find derivative when n is zero.
  • Find from the definition, the differential coefficient of the given variable.
  • Find from the 1st principle.
  • Theorems on differentiation
  • State and prove the theorem of derivatives.
  • State and prove the difference theorem of the derivative.
  • State and prove the product theorem of derivatives.
  • State and prove the quotient theorem of derivatives.
  • State and prove the reciprocal rule of the derivative.
  • The Chain Rule.
  • A derivative of Parametric equations.
  • Define Parameter.
  • Define Parametric equations.
  • Define explicit and implicit functions.
  • A derivative of Implicit Relations.
  • Differentiation of trigonometric functions.
  • Prove by the ab-initio method.
  • Find derivatives from the first principle.
  • Derivation of inverse trigonometric functions.
  • Differentiation of exponential functions.
  • Define the exponential function.
  • Define general exponential function.
  • Define natural exponential function.
  • Differentiation of logarithmic functions.
  • Define the general logarithm function.
  • Define common logarithm function.
  • Define natural logarithm function.
  • Differentiation of hyperbolic functions.
  • Define hyperbolic functions.
  • A derivative of inverse hyperbolic functions.
  • Higher derivatives.
  • First derivative
  • Second derivative
  • Third derivative.
  • Fourth derivative.
  • Nth derivative.
  • Higher derivative.
  • Series expansions of functions
  • Define Maclaurin’s series expansion of a function.
  • State and prove Maclaurin’s series expansion of a function.
  • Find Maclaurin’s series for sinx.
  • State and prove Taylor’s series expansion of a function.
  • Geometrical meaning of derivative.
  • State the geometrical interpretation of derivative.
  • Define the tangent line.
  • Prove that geometrically derivative of a function gives the slope of a tangent line.
  • Discuss the tangent line to the graph of function |x| at x=0.
  • Define increasing function.
  • Define a decreasing function.
  • Define a constant function.
  • Define an increasing function in terms of derivatives.
  • Define decreasing function in terms of derivatives.
  • Define constant function in terms of derivatives.
  • Define stationary point.
  • Define critical value and critical point.
  • Define turning point of a function.
  • Define point of inflection.
  • Define relative minima of a function.
  • Define relative maxima of a function.
  • What is the relative extrema of a function.
  • Define first derivative test.
  • Applications of maxima and minima.
  • Find two positives integers whose sum is 9 and the product of one with the square of the other will be maximum.
  • What are dimensions of a box of a square base having largest volume if the sum of one side of the base and height is 12cm.
  • The perimeter of a triangle is 20cm. If one side is of the length 8cm, what are length of the other sides for the mum area of the triangle?
  • FInd the two positive integers whose sum is 30 and their product will be maximum.

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