[PDF] LIMITS AND DERIVATIVES - NCERT PDF keep213.pdf

18 avr 2018 · To evaluate the limits of trigonometric functions, we shall make use of the following 13 1 2 Derivatives Suppose f is a real valued function, then Example 6 Find the derivative of f(x) = ax + b, where a and b are non-zero

1 4 Differentiation Using Limits inclusion of real-world applications, detailed art pieces, and illustrative limits, continuity, derivatives, and the Chain Rule

[PDF] Chapter 5: Applications of Derivatives

Optimization is the application of calculus-based graphical analysis to particular physical I am trying to tell a story about how to take apart a function using the is not real, the middle limit you should have thought about in the earlier

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the real world imposes some constraint on the values that x may have Why? Because to compute the derivative at 1 we must compute the limit lim ∆x→0

[PDF] Applications of differentiation - Australian Mathematical Sciences

Applications of differentiation – A guide for teachers (Years 11–12) Principal author: The derivative of the function can be used to determine when a local maximum or local minimum occurs The domain of f is all real numbers The first two

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functions; applications to business and economics, such as maximum- minimum problems, marginal analysis, Lesson 2: One-Sided Limits and Continuity: The Derivative Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Introduction f x , is a polynomial and hence continuous for all real numbers

[PDF] Applications of Derivatives - Higher Education Pearson

Overview One of the most important applications of the derivative is its use as a tool for finding the This means that the right-hand and left-hand limits both exist at x = c and equal ƒ(c) second result is not important to the real-life situation

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A real-valued mathematical expression, such as the quadratic function in exhibit 1 1, The concept of limit is now all-pervasive in calculus and its applications

[PDF] LIMITS AND DERIVATIVES - NCERT

[PDF] Exercises and Problems in Calculus - Portland State University

1 août 2013 · MORE APPLICATIONS OF THE DERIVATIVE 239 interesting “real world” problems require, in general, way too much background to fit comfortably limits 4 1 1 Definition Suppose that f is a real valued function of a real

13.1 Overview

13.1.1 Limits of a function

Let f be a function defined in a domain which we take to be an interval, say, I. We shall study the concept of limit of f at a point 'a' in I.

We say -lim ( )

x af x → is the expected value of f at x = a given the values of f near to the left of a. This value is called the left hand limit of f at a.

We say lim ( )

x af x+→ is the expected value of f at x = a given the values of f near to the right of a. This value is called the right hand limit of f at a. If the right and left hand limits coincide, we call the common value as the limit of f at x = a and denote it by lim ( )x af x→.

Some properties of limits

Let f and g be two functions such that both

lim ( )x af x → and lim ( )x ag x→ exist. Then (i) lim[ ( ) ( )] lim ( ) lim ( ) x ax a x af x g x f x g x →→→+ = +(ii) lim[ ( ) ( )] lim ( ) lim ( ) x ax a x af x g x f x g x →→→- = -(iii)For every real number α lim ( )( ) lim ( ) x ax af x f x→→α = α(iv) lim[ ( ) ( )] [lim ( ) lim ( )] x ax a x af x g x f x g x lim ( )( )lim ( ) lim ( ) x a x a x af xf x g x g x →=, provided g (x) ≠ 0Chapter 13

LIMITS AND DERIVATIVES

226 EXEMPLAR PROBLEMS - MATHEMATICS

Limits of polynomials and rational functions

If f is a polynomial function, then lim ( )

x af x → exists and is given by lim ( ) ( )x af x f a→=An Important limit An important limit which is very useful and used in the sequel is given below:

1limn nn

x ax anax a- →-=-Remark The above expression remains valid for any rational number provided ' a' is positive.

Limits of trigonometric functions

To evaluate the limits of trigonometric functions, we shall make use of t he following limits which are given below: (i) 0 sinlim xx x→ = 1(ii)0limcos 1 xx →=(iii)

0limsin 0

xx →=13.1.2 Derivatives Suppose f is a real valued function, then f ′(x) = 0 ( ) ( )lim hf x h f x h→ + -... (1) is called the derivative of f at x, provided the limit on the R.H.S. of (1) exists. Algebra of derivative of functions Since the very definition of derivatives involve limits in a rather direct fashion, we expect the rules of derivatives to follow closely that of limits as given below: Let f and g be two functions such that their derivatives are defined in a common domain. Then : (i)Derivative of the sum of two function is the sum of the derivatives of t he functions.

[ ]( ) ( )df x g xdx+ =( ) ( )d df x g xdx dx+(ii)Derivative of the difference of two functions is the difference of the d

erivatives of the functions. [ ]( ) ( )df x g xdx- =( ) ( )d df x g xdx dx-

LIMITS AND DERIVATIVES 227

(iii)Derivative of the product of two functions is given by the following product rule.[ ]( ) ( )df x g xdx? =( ) ( ) ( ) ( )ddf x g x f x g xdxdx ? + ? This is referred to as Leibnitz Rule for the product of two functions. (iv)Derivative of quotient of two functions is given by the following quotient rule (wherever the denominator is non-zero). d f x dx g x =()2 ( )d d f x g x f x g xdxdx g x ? - ? 13.2 Solved Examples

Short Answer Type

Example 1 Evaluate

3 22

1 2(2 3)lim23 2xx

xx x x→- - -- + Solution We have 3 22

1 2(2 3)lim23 2xx

xx x x→- - -- + =21 2(2 3)lim2 ( 1) ( 2xx x x x x 2 ( 1) 2(2 3)lim( 1)( 2)xx x x x x x 2

25 6lim( 1)( 2)xx x

x x x 2 ( 2) ( 3)lim( 1)( 2)xx x x x x - - [x - 2 ≠ 0] 2

3 1lim( 1) 2xx

x x

228 EXEMPLAR PROBLEMS - MATHEMATICS

Example 2

Evaluate 0

2 2lim

xx x →+ -Solution Put y = 2 + x so that when x → 0, y → 2. Then 0

2 2lim

xx x 1 1 2 2 2

2lim2yy

111221 1 1(2) 22 22 2

--= ? =Example 3 Find the positive integer n so that

33lim 1083n n

xx x

Solution We have

3 3lim3 n n xx x - =n(3)n - 1

Therefore,n(3)n - 1 =108 = 4 (27) = 4(3)4 - 1

Comparing, we getn =4

Example 4 Evaluate

2 lim (sec tan ) xx xπ→-Solution Put y =

2xπ-. Then y → 0 as x → 2

π. Therefore

2 lim (sec tan ) xx xπ→- =

0lim[sec( ) tan ( )]2 2yyy

0lim (cosec cot )yy y→-=

1 coslimsin sin

yy y y→ 0

1 coslimsinyy

y→-

LIMITS AND DERIVATIVES 229

= 2 02sin 2lim

2sin cos2 2

y y y y→

21 cossince, sin2 2

sin 2sin cos

2 2y y

y yy-

02lim tan2y

y → = 0

Example 5 Evaluate

0 sin (2 ) sin(2 )lim xx x x→ + - -Solution (i) We have 0 sin (2 ) sin(2 )lim xx x x→ 0 (2 2 ) (2 2 )2cossin22limxx x x x x→ 0

2cos 2sinlim

xx x→=2 cos 2

0sinlim 2cos 2

xx x →= 0sinas lim 1 xx x = Example 6 Find the derivative of f(x) = ax + b, where a and b are non-zero constants, by first principle.

Solution By definition,

f′(x) = 0 ( ) ( )lim hf x h f x h→ 0 ( ) ( )lim ha x h b ax b h→ 0lim h bh h→ = b

Example 7

Find the derivative of f(x) = ax2 + bx + c, where a, b and c are none-zero constant, by first principle.

Solution By definition,

f′(x) = 0 ( ) ( )lim hf x h f x h→

230 EXEMPLAR PROBLEMS - MATHEMATICS

=22

0( ) ( )lim

ha x h b x h c ax bx c h→ 2

02limhbh ah axh

h→+ + =0limh→ ah + 2ax + b = b + 2ax Example 8 Find the derivative of f(x) = x3, by first principle.

Solution By definition,

f′(x) = 0 ( ) ( )lim hf x h f x h→ 3 3

0( )limh

x h x h→ 3 33

03 ( )lim

h x h xh x h x h→+ + + -=

0limh→(h2 + 3x (x + h)) = 3x2

Example 9 Find the derivative of f(x) =

1 x by first principle.

Solution By definition,

f′(x) = 0 ( ) ( )lim hf x h f x h→ 0

1 1 1lim

hh x h x→

0lim( )hhh x h x→

+ = 2 1- x. Example 10 Find the derivative of f(x) = sin x, by first principle.

Solution By definition,

f′(x) = 0 ( ) ( )lim hf x h f x h→

LIMITS AND DERIVATIVES 231

=0 sin ( ) sinlim hx h x h→ 02

2cos sin2 2lim

22h
x h h h→+ 00 sin(2 )2lim cos lim2 2 hh h x h h→→+?=cos x.1 = cos x Example 11 Find the derivative of f(x) = xn, where n is positive integer, by first principle.quotesdbs_dbs8.pdfusesText_14

[PDF] [PDF] LIMITS AND DERIVATIVES - NCERT

13.1 Overview

13.1.1 Limits of a function

We say -lim ( )

We say lim ( )

Some properties of limits

Let f and g be two functions such that both

LIMITS AND DERIVATIVES

226 EXEMPLAR PROBLEMS - MATHEMATICS

Limits of polynomials and rational functions

If f is a polynomial function, then lim ( )

1limn nn

Limits of trigonometric functions

0limsin 0

LIMITS AND DERIVATIVES 227

Short Answer Type

Example 1 Evaluate

1 2(2 3)lim23 2xx

1 2(2 3)lim23 2xx

25 6lim( 1)( 2)xx x

3 1lim( 1) 2xx

228 EXEMPLAR PROBLEMS - MATHEMATICS

Example 2

Evaluate 0

2 2lim

2 2lim

2lim2yy

111221 1 1(2) 22 22 2

33lim 1083n n

Solution We have

Therefore,n(3)n - 1 =108 = 4 (27) = 4(3)4 - 1

Comparing, we getn =4

Example 4 Evaluate

2xπ-. Then y → 0 as x → 2

π. Therefore

0lim[sec( ) tan ( )]2 2yyy

0lim (cosec cot )yy y→-=

1 coslimsin sin

1 coslimsinyy

LIMITS AND DERIVATIVES 229

2sin cos2 2

21 cossince, sin2 2

2 2y y

02lim tan2y

Example 5 Evaluate

2cos 2sinlim

0sinlim 2cos 2

Solution By definition,

Example 7

Solution By definition,

230 EXEMPLAR PROBLEMS - MATHEMATICS

0( ) ( )lim

02limhbh ah axh

Solution By definition,

0( )limh

03 ( )lim

0limh→(h2 + 3x (x + h)) = 3x2

Example 9 Find the derivative of f(x) =

Solution By definition,

1 1 1lim

0lim( )hhh x h x→

Solution By definition,

LIMITS AND DERIVATIVES 231

2cos sin2 2lim