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 Karnataka Class 12 Commerce Maths Continuity And Differentiability Complete Notes

Karnataka Class 12 Commerce Maths Continuity And Differentiability : Here we provides you Karnataka Class 12 Commerce Maths Continuity And Differentiability Notes in PDF Format. Karnataka Class 12 Commerce Maths Continuity And Differentiability topic covers those are under following given :

 1) Continuity: Definition, continuity of a function at a point and on a domain. Examples and problems, Algebra of continuous functions, problems , continuity of composite function and problems

 2)  Differentiability: Definition, Theorem connecting differentiability and continuity with a counter example. Defining logarithm and mentioning its properties , Concepts of exponential, logarithmic functions, Derivative of ex , log x from first principles, Derivative of composite functions using chain rule, problems. Derivatives of inverse trigonometric functions, problems. Derivative of implicit function and problems. Logarithmic differentiation and problems . Derivative of functions expressed in parametric forms and problems. Second order derivatives and problems Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric Interpretations and problems

Download here Karnataka Class 12 Commerce Maths Complete Notes In PDF Format 

Karnataka Class 12 Commerce Maths Continuity And Differentiability Complete Notes

Karnataka Class 12 Commerce Maths Continuity And Differentiability : Definition of continuity for English Language Learners. : the quality of something that does not stop or change as time passes : a continuous quality. : something that is the same or similar in two or more things and provides a connection between them. Continuity is ‘the state of being continuous’ and continuous means ‘without any interruption or disturbance’. For example, the price of a commodity and its demand are inversely proportional. The graph of demand curve of a commodity is a continuous curve without any breaks or gaps.

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Continuity, in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say x—is associated with a value of a dependent variable—say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question “How close?” is asked, difficulties arise.

Karnataka Class 12 Commerce Maths Continuity And Differentiability : For close x-values, the distance between the y-values can be large even if the function has no sudden jumps. For example, if y = 1,000x, then two values of x that differ by 0.01 will have corresponding y-values differing by 10. On the other hand, for any point x, points can be selected close enough to it so that the y-values of this function will be as close as desired, simply by choosing the x-values to be closer than 0.001 times the desired closeness of the y-values. Thus, continuity is defined precisely by saying that a function f(x) is continuous at a point x0 of its domain if and only if, for any degree of closeness ε desired for the y-values, there is a distance δ for the x-values (in the above example equal to 0.001ε) such that for any x of the domain within the distance δ from x0, f(x) will be within the distance ε from f(x0). In contrast, the function that equals 0 for x less than or equal to 1 and that equals 2 for x larger than 1 is not continuous at the point x = 1, because the difference between the value of the function at 1 and at any point ever so slightly greater than 1 is never less than 2.

A function is said to be continuous if and only if it is continuous at every point of its domain. A function is said to be continuous on an interval, or subset of its domain, if and only if it is continuous at each point of the interval. The sum, difference, and product of continuous functions with the same domain are also continuous, as is the quotient, except at points at which the denominator is zero. Continuity can also be defined in terms of limits by saying that f(x) is continuous at x0 of its domain if and only if, for values of x in its domain,

 Function.

A more abstract definition of continuity can be given in terms of sets, as is done in topology, by saying that for any open set of y-values, the corresponding set of x-values is also open. (A set is “open” if each of its elements has a “neighborhood,” or region enclosing it, that lies entirely within the set.) Continuous functions are the most basic and widely studied class of functions in mathematical analysis, as well as the most commonly occurring ones in physical situations.

CONTINUITY OF FUNCTIONS OF ONE VARIABLE

The following problems involve the CONTINUITY OF A FUNCTION OF ONE VARIABLE. Function y = f(x) is continuous at point x=a if the following three conditions are satisfied :

i.) f(a) is defined ,

ii.) $ \displaystyle{ \lim_{ x \to a } \ f(x) } $ exists (i.e., is finite) ,

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and

iii.) $ \displaystyle{ \lim_{ x \to a } \ f(x) } = f(a) $ .

Function f is said to be continuous on an interval I if f is continuous at each point x in I. Here is a list of some well-known facts related to continuity :

1. The SUM of continuous functions is continuous.

2. The DIFFERENCE of continuous functions is continuous.

3. The PRODUCT of continuous functions is continuous.

4. The QUOTIENT of continuous functions is continuous at all points x where the DENOMINATOR IS NOT ZERO.

5. The FUNCTIONAL COMPOSITION of continuous functions is continuous at all points x where the composition is properly defined.

6. Any polynomial is continuous for all values of x.

7. Function ex and trigonometry functions $ \sin x $ and $ \cos x $ are continuous for all values of x.

Karnataka Class 12 Commerce Maths Continuity And Differentiability : Most problems that follow are average. A few are somewhat challenging. All limits are determined WITHOUT the use of L’Hopital’s Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the above step-by-step definition of continuity at a point and the well-known facts, and by giving careful consideration to the indeterminate form $ \displaystyle{ ^{^{^{^{''}}}}{0 \over 0}^{''} }$during the computation of limits. Knowledge of one-sided limits will be required. For a review of limits and indeterminate forms click here.

Questions with answers on the continuity of functions with emphasis on piecewise functions.

Example 1: For what values of x are each of the following functions discontinuous?

functions of question 1

Karnataka Class 12 Commerce Maths Continuity And Differentiability :

Solution to Example 1

a) For x = 0, the denominator of function f(x) is equal to 0 and f(x) is not defined and does not have a limit at x = 0. Therefore function f(x) is discontinuous at x = 0.

b) For x = 2 the denominator of function g(x) is equal to 0 and function g(x) not defined at x = 2 and it has no limit. Function g(x) is not continuous at x = 2.

c) The denominator of function h(x) can be factored as follows: x2 -1 = (x – 1)(x + 1). The denominator is equal to 0 for x = 1 and x = -1 values for which the function is undefined and has no limits. Function h is discontinuous at x = 1 and x = -1.

d) tan(x) is undefined for all values of x such that x = π/2 + k π , where k is any integer (k = 0, -1, 1, -2, 2,…) and is therefore discontinuous for these same values of x.

e) The denominator of function j(x) is equal to 0 for x such that cos(x) – 1 = 0 or x = k (2 π), where k is any integer and therefore this function is undefined and therefore discontinuous for all these same values of x.

f) Function k(x) is defined as the ratio of two continuous functions (with denominator x2 + 5 never equal to 0), is defined for all real values of x and therefore has no point of discontinuity.

g) l(x) = (x + 4)/(x + 4) = 1 . Hence lim l(x) as x approaches -4 = 1 = l(-4) . Function l(x) is continuous for all real values of x and therefore has no point of discontinuity.

Karnataka Class 12 Commerce Maths Continuity And Differentiability :

Example 2: Find b such that f(x) given below is continuous?

functions of question 2

Solution to Example 2
For x > -1, f(x) = 2 x

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2 + b is a polynomial function and therefore continuous.

For x < -1, f(x)= -x 3 is a polynomial function and therefore continuous.

For x = -1

f(-1) = 2(-1) 2 + b = 2 + b

let us consider the left and right hand limits

L1 = lim x→ -1 f(x) = -(-1) 3 = 1 (limit from left of -1)

L2 = lim x→ -1 + f(x) = 2(-1) 2 + b = 2 + b (limit from right of -1)

For function f to be continuous, we need to have

L1 = L2 = 2 + b

or 2 + b = 1 or b = -1.
Example 3: Find a and b such that both g(x) given below and its first derivative are continuous?

functions of question 3

Solution to Example 3

Continuity of function g

For x > 2, g(x) = a x 2 + b is a polynomial function and therefore continuous.

For x < 2, g(x) = -2 x + 2 is a polynomial function and therefore continuous.

let

L1 = lim x→ 2 g(x) = a (2) 2 + b = 4 a + b

L2 = lim x→ 2 + g(x) = -2(2) + 2 = -2

For continuity of g at x = 2, we need to have

L1 = L2 = g(2)

Which gives

4 a + b = -2

Continuity of the derivative g’

For x > 2, g ‘(x) = 2 a x is a polynomial function and therefore continuous.

For x < 2, g ‘(x) = -2 is a constant function and therefore continuous.

Let

l1 = lim x→ 2 g'(x) = 2 a (2) = 4 a

l2 = lim x→ 2 + g'(x) = – 2

For continuity of g’ at x = 2, we need to have

l1 = l2 or 4 a = – 2

The last equation gives: a = – 1 / 2. And substitute a by – 1 / 2 in the equation 4 a + b = -2 obtained above, we obtain b = 0.

Karnataka Class 12 Commerce Maths Continuity And Differentiability Complete Notes

Karnataka Class 12 Commerce Maths Continuity And Differentiability : In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

More generally, if x0 is a point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0f(x0)). The function f may also be called locally linear at x0, as it can be well approximated by a linear function near this point.

The most important use of limit is to find the derivative of any function.

If y = f(x) is a function and limδx0limδx→0f(x+δx)f(x)δxf(x+δx)−f(x)δx is the derivative of the given function at any point (x,f(x)). Then we can say that function f is differentiable at the point x if limits exists. If it does not exist, then we say that the function is not differentiable.
Differentiability

Differentiability at a Point

To know about differentiability at a point, first we have to know about the left hand side limit and right hand side limit.

Left Hand Limit or Derivative:

The LHD of f at a is defined as

where, h > 0, provided the limit exists.

In the above definition showing differentiability, substitute a + h = x, then h = x – a as

Rf ‘(a) can be rewritten as

Similarly, substitute a – h = x. Lf ‘(a) can be written as

Right Left Hand Limit or Derivative:

Let f be a function of x (y = f(x)). Let a be a point in the domain of f. The RHD of f at a is defined as

where, h>0, provided the limit exists.

So, we can say that diffferentiability at a point ‘a’ for a function f(x) if

  1. both Rf ‘(a) and Lf ‘(a) exists and finite.
  2. Rf ‘(a) = Lf ‘(a)

Consider the function y = |x|. This function is differentiable on (−∞,0) and (0,\infy\infy), but not differentiable at x = 0.

For x > 0, we have

Since the limit exists, f(x) is differentiable at x > 0. Similarly, we can show that f(x) is differentiable at x < 0.

We shall find RHD and LHD of f(x) at x = 0.

= -1

Therefore y = |x| is not differentiable at x = 0.

Continuity and Differentiability

If a function is differentiable at any point x, then that function is continuous at that point x. The following theorem says ‘Differentiability implies continuity”.

If a function f is differentiable at x = a, then it is continuous at x = a.

f is differentiable at x = a.

Now, we have

f(x) is continuous at x = a.

Every differential function is continuous, but every continuous function is not differential.

We have already shown that y = |x| is not differentiable at x = 0. It can be easily seen that
y = |x| is continuous at x = 0.

Download here Karnataka Class 12 Commerce Maths Continuity And Differentiability Notes In PDF Format 

Karnataka Class 12 Commerce Maths Continuity And Differentiability Complete Notes

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