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

Is there actually a way to “unlock math” for ourselves and learn the subject quickly? It is a universal and undebatable knowledge that mathematics is a darn difficult subject.

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Exercise 2.1 | View |

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.