Fsc 2nd Year Math Notes

• Math Notes 2nd Year Fsc 1.5
• Math City Fsc
• Fsc 2nd Year Math Notes
• Fsc Online 2nd Year Maths Notes

Notes (Solutions) of Unit 02: Differentiation, Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. You can view online or download PDF. To view PDF, you must have PDF Reader installed on your system and it can be downloaded from Software section.

Here are few online resource, which are very helpful to find derivative.

In this online lecture,Muhammad Naveed Jaffar explains FSC part 2 Mathematics Chapter 3 Integration.The topic being discussed is Topic Exercise 3.1 Question. Level: F.Sc (Pre-Engg)Class: 2nd YearBoard: Punjab Textbook BoardSubject: MathematicsTotal Exercises: 10Type: Notes TITLE SOLUTION 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. I sometimes be busy in preparing notes in pdf. All the students of 2nd year fsc pre engineering and ICS part 2 students should use these notes for MCQs preparation for math paper. 2nd year maths keybook pdf download. F.Sc and ICS part 2 math MCQs Notes pdf download. The latest notes of mathematics for 2nd year are given below in pdf. Class 12 Mathematics Notes are free and will always remain free. We will keep adding updated notes, past papers, guess papers and other materials with time. We will also introduce a mobile app for viewing all the notes on mobile. Make sure to comment down your experience regarding our website.

Contents & summary

• • Average Rate of Change
• Finding $f'(x)$ from Definition of Derivative
  • Exercise 2.1
• • Exercise 2.2
• • Exercise 2.3
• Derivatives of Inverse Function
• Derivative of a Function given in form of parametric Equations
• • Exercise 2.4
• Derivatives of Inverse Trigonometric Functions
• Derivative of Exponential Functions
• Logarithmic Differentiation
• Derivative of the Inverse Hyperbolic Function
• Successive Differentiation ( or Derivatives)
• Series Expansions of Function
• • Exercise 2.8
• Increasing and Decreasing Function
• Critical Values and Critical Points
  • Exercise 2.10

Which method is better

In this chapter many questions can be solved in much easier way. Actually in every exercise some formula/method is introduced to solve the question. In examination it is not necessary to do the same method as given in exercise. Here is one example:

We have to find the derivative of $frac{x+1}{x-1}$ with respect to $x$.

Method 1

$$begin{aligned}frac{d}{dx}left(frac{x+1}{x-1}right) &= frac{(x-1)frac{d}{dx}(x+1)-(x+1)frac{d}{dx}(x-1)}{(x-1)^2}&= frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}&= frac{x-1-x-1}{(x-1)^2}&= frac{-2}{(x-1)^2}end{aligned}$$

Method 2

By converting improper to proper fraction $$frac{x+1}{x-1}= 1+frac{2}{x-1}=1+2(x-1)^{-1}$$Now$$begin{aligned}frac{d}{dx}left(frac{x+1}{x-1}right) &=frac{d}{dx}left(1+2(x-1)^{-1}right)&= 0-2(x-1)^{-2}(1)&= frac{-2}{(x-1)^2}end{aligned}$$

This was a simple example but try it to find the derivative of $frac{x^2+1}{x^2-1}$.

Solutions

• Exercise 2.1 | View Online | Download PDF (156KB)

• Exercise 2.2 | View Online | Download PDF (130KB)

• Exercise 2.3 | View Online | Download PDF (209KB)

• Exercise 2.4 | View Online | Download PDF (175KB)

• Exercise 2.5 | View Online | Download PDF (239KB)

• Exercise 2.6 | View Online | Download PDF (236KB)

• Exercise 2.7 | View Online | Download PDF (194KB)

• Exercise 2.8 | View Online | Download PDF (147KB)

• Exercise 2.9 | View Online | Download PDF (200KB)

• Exercise 2.10 | View Online | Download PDF (180KB)

The following notes was written and sent by Mr. Amir Shehzad.

• Unit 02: Differentiation | Download PDF (1.53 MB)

Here are previous and next chapters

Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.

Math Notes 2nd Year Fsc 1.5

Unit 01: Functions and Limits

Objectives

After reading this unit the students will be able to:

• identify the domain and range of a functions through graphs.
• draw the graph of modulus function and identify its domain and range.
• recognize the composition of a function and then to find out the composition of two functions.
• describe the inverse of a function and then to find out the inverse of composition of two functions.
• recognize the algebraic and transcendental functions as well as the concepts of explicit, implicit and parametric functions.
• display graphically the explicit, implicit and parametric functions as well as the compound functions.
• introduce the limit of a function with respect to real number intervals on the real number line, the open and closed intervals and its location on a real number line.
• explain the meaning of x tends to zero, x tends to a and x tends to infinity
• define the limit of a sequence when the limit of a sequence with nth term is given.
• define the limit of a function and the statement of theorems on limits of sum, difference, product and quotient of functions.
• evaluate the limits of a function in case of some special functions.
• evaluate the limit of algebraic, exponential and trigonometric functions.
• introduce the continuous and discontinuous functions.
• recognize the left hand and right hand limits through examples.
• define the continuity of a function at a point and in an interval.
• test the continuity and discontinuity of a function a point and in an interval.

Download

Unit 02: Differentiation

Objectives

After reading this unit the students will be able to:

• distinguish between independent and dependent variables.
• estimate the change in the dependent variable, when the independent variable is incremented or decremented.
• define the derivative of a function as an instantaneous rate of change of variable with respect to another variable.
• define derivative or differential coefficient of a function.
• differentiate $y=x^n$ and $y=(ax+b)^n$ by first principle rule.
• introduce the theorems of differentiation, such as the derivative of a constant function, the derivative of any constant multiple of a function, the derivative of a sum or difference of two functions, the derivative of the product of two functions and the derivative of a quotient of two functions.
• apply theorems of differentiation in solving problems.
• introduce chain rule of differentiation in different situations.
• introduce implicit differentiation of a function.
• introduce differentiation of trigonometric and inverse trigonometric functions.
• introduce differentiation of exponential and logarithmic functions.
• introduce the differentiation of hyperbolic and inverse hyperbolic functions.

Download

Unit 03: Higher Order Derivative and Application

Objectives

After reading this unit the students will be able to:

Download

Unit 04: Differentiation of Vector Functions

Objectives

After reading this unit the students will be able to:

Download

Unit 05: Integration

Objectives

After reading this unit the students will be able to:

Download

Unit 09: Integration

Math City Fsc

• • Average Rate of Change
• Finding $f'(x)$ from Definition of Derivative
  • Exercise 2.1
• • Exercise 2.2
• • Exercise 2.3
• Derivatives of Inverse Function
• Derivative of a Function given in form of parametric Equations
• • Exercise 2.4
• Derivatives of Inverse Trigonometric Functions
• Derivative of Exponential Functions
• Logarithmic Differentiation
• Derivative of the Inverse Hyperbolic Function
• Successive Differentiation ( or Derivatives)
• Series Expansions of Function
• • Exercise 2.8
• Increasing and Decreasing Function
• Critical Values and Critical Points
  • Exercise 2.10

Which method is better

In this chapter many questions can be solved in much easier way. Actually in every exercise some formula/method is introduced to solve the question. In examination it is not necessary to do the same method as given in exercise. Here is one example:

We have to find the derivative of $frac{x+1}{x-1}$ with respect to $x$.

Method 1

$$begin{aligned}frac{d}{dx}left(frac{x+1}{x-1}right) &= frac{(x-1)frac{d}{dx}(x+1)-(x+1)frac{d}{dx}(x-1)}{(x-1)^2}&= frac{(x-1)(1)-(x+1)(1)}{(x-1)^2}&= frac{x-1-x-1}{(x-1)^2}&= frac{-2}{(x-1)^2}end{aligned}$$

Method 2

By converting improper to proper fraction $$frac{x+1}{x-1}= 1+frac{2}{x-1}=1+2(x-1)^{-1}$$Now$$begin{aligned}frac{d}{dx}left(frac{x+1}{x-1}right) &=frac{d}{dx}left(1+2(x-1)^{-1}right)&= 0-2(x-1)^{-2}(1)&= frac{-2}{(x-1)^2}end{aligned}$$

This was a simple example but try it to find the derivative of $frac{x^2+1}{x^2-1}$.

Solutions

• Exercise 2.1 | View Online | Download PDF (156KB)

• Exercise 2.2 | View Online | Download PDF (130KB)

• Exercise 2.3 | View Online | Download PDF (209KB)

• Exercise 2.4 | View Online | Download PDF (175KB)

• Exercise 2.5 | View Online | Download PDF (239KB)

• Exercise 2.6 | View Online | Download PDF (236KB)

• Exercise 2.7 | View Online | Download PDF (194KB)

• Exercise 2.8 | View Online | Download PDF (147KB)

• Exercise 2.9 | View Online | Download PDF (200KB)

• Exercise 2.10 | View Online | Download PDF (180KB)

The following notes was written and sent by Mr. Amir Shehzad.

• Unit 02: Differentiation | Download PDF (1.53 MB)

Here are previous and next chapters

Notes of FSc Part 2 of “A Textbook of Mathematics For Class XII” published by Khyber Pakhtunkhwa Textbook Board, Peshawar. We are posting the notes chapter-wise. These notes are shared as open educational resources. This page will be continuously updated.

Math Notes 2nd Year Fsc 1.5

Unit 01: Functions and Limits

Objectives

After reading this unit the students will be able to:

• identify the domain and range of a functions through graphs.
• draw the graph of modulus function and identify its domain and range.
• recognize the composition of a function and then to find out the composition of two functions.
• describe the inverse of a function and then to find out the inverse of composition of two functions.
• recognize the algebraic and transcendental functions as well as the concepts of explicit, implicit and parametric functions.
• display graphically the explicit, implicit and parametric functions as well as the compound functions.
• introduce the limit of a function with respect to real number intervals on the real number line, the open and closed intervals and its location on a real number line.
• explain the meaning of x tends to zero, x tends to a and x tends to infinity
• define the limit of a sequence when the limit of a sequence with nth term is given.
• define the limit of a function and the statement of theorems on limits of sum, difference, product and quotient of functions.
• evaluate the limits of a function in case of some special functions.
• evaluate the limit of algebraic, exponential and trigonometric functions.
• introduce the continuous and discontinuous functions.
• recognize the left hand and right hand limits through examples.
• define the continuity of a function at a point and in an interval.
• test the continuity and discontinuity of a function a point and in an interval.

Download

Unit 02: Differentiation

Objectives

After reading this unit the students will be able to:

• distinguish between independent and dependent variables.
• estimate the change in the dependent variable, when the independent variable is incremented or decremented.
• define the derivative of a function as an instantaneous rate of change of variable with respect to another variable.
• define derivative or differential coefficient of a function.
• differentiate $y=x^n$ and $y=(ax+b)^n$ by first principle rule.
• introduce the theorems of differentiation, such as the derivative of a constant function, the derivative of any constant multiple of a function, the derivative of a sum or difference of two functions, the derivative of the product of two functions and the derivative of a quotient of two functions.
• apply theorems of differentiation in solving problems.
• introduce chain rule of differentiation in different situations.
• introduce implicit differentiation of a function.
• introduce differentiation of trigonometric and inverse trigonometric functions.
• introduce differentiation of exponential and logarithmic functions.
• introduce the differentiation of hyperbolic and inverse hyperbolic functions.

Download

Unit 03: Higher Order Derivative and Application

Objectives

After reading this unit the students will be able to:

Download

Unit 04: Differentiation of Vector Functions

Objectives

After reading this unit the students will be able to:

Download

Unit 05: Integration

Objectives

After reading this unit the students will be able to:

Download

Unit 09: Integration

Math City Fsc

Objectives

This unit tells us, how to:

Fsc 2nd Year Math Notes

• define the differential equation, its order, degree, general and particular solutions, and its identification as linear and nonlinear ordinary differential equations.
• demonstrate the concept in forming a differential equation.
• solve the first order linear and nonlinear ordinary differential equations by separable variable form, and homogenuous form and then how to reduce differential equations in standard form of homogenuous.
• solve the real life problems related to differential equations.
• define the orthogonal trajectories and then how to show the orthogonal trajectories of the two families of curve.

Download

Fsc Online 2nd Year Maths Notes