Fsc 2nd Year Math Notes

FSc Part 2 Solutions Calculus and Analytic Geometry, MATHEMATICS 12 Notes (Solutions) of Calculus and Analytic Geometry, MATHEMATICS 12 (Mathematics FSc Part 2 or HSSC-II), Punjab Text Book Board Lahore. There are seven units in this book and we have work hard to make easy and suitable solutions for students and teachers so that it help them learn things quickly and easily. Dear Students, We have brought to you FSc Part 2 (12th Class) Math Notes/Solutions for you today. These notes have been prepared with very hardwork.

Fsc 2nd Year Math Notes
2nd Year Chemistry Notes

Notes (Solutions) of Unit 01:Functions and Limits, 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.

Contents & summary

- Concept of Function
- Notation and Values of a Function
- Graph of Functions Defined Piece-Wise
- Algebraic Function
- Inverse Trigonometric Functions
- Exponential Function
- Hyperbolic Function
- Explicit Function
- Odd Function
Composition of Function and Inverse of a Function
- Inverse of a Function
- Exercise 1.2
- Meaning of the phrase “x approaches Zero”
- Concept of Limit of a Function
- Theorems on Limits of Function
- $lim_{xto a}frac{x^n-a^n}{x-a} = na^{n-1}$, where n is an integer and a>0
- $lim_{xto0}frac{sqrt{x+a} - sqrt{a}}{x} = frac{1}{2sqrt{a}}$
- Methods for Evaluating the limits at Infinity
- $lim_{xto0}frac{a^x-1}{x} = {log_e}^a$
- If $theta$ is measured in radian, then $lim_{thetato 0}frac{sintheta}{theta} = 1$
Continuous and Discontinuous Function
- Criterion for Existence of Limit of a Function
- Exercise 1.4
- Graph of the Exponential Function $f(x) = a^x$
- Graph of Common Logarithmic Function $f(x) = log x$
- Graph of natural logarithmic Function $f(x) = ln x$
- Graph of parametric Equations
- Graphical Solution of the Equations

Did you know?
Consider two functions $f(x)=x+3$ and $displaystyle g(x)=frac{x^2-9}{x-3}$. Is $f=g$? (see answer here)

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Exercise 1.2 | View online | Download PDF

$Fsc 2nd year math notes$

Exercise 1.4 | View online | Download PDF

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

Unit 01: Function & Limits (Complete notes with MCQs) | Download PDF (0.7 MB)

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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.

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 $frac{x^2+1}{x^2-1}$.

Finding a derivative is not a difficult thing. It has few rules and formulas. Instead of learning sums try to understand the rules for differentiation. Also remember derivative of important function. To remember the derivative just take a print of Important Derivatives & Integrals. At the time of solving question, use this page to see formulas. This method will help to put these formulas in your mind without any trouble.

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)

Contents & summary

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Contents & summary

Which method is better

Method 1

Method 2

Solutions

Fsc 2nd Year Math Notes

2nd Year Chemistry Notes

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