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CBSE Class 12 Commerce Mathematics Continuity and Differentiability

CBSE Class 12 Commerce Mathematics Continuity and Differentiability

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:- we provide complete details of CBSE Class 12 Commerce Mathematics Continuity and Differentiability in this article.

CBSE Class 12 Commerce Mathematics Continuity and Differentiability

CBSE Class 12 Commerce Mathematics Continuity and Differentiability

This chapter is essentially a continuation of our study of differentiation of functions in Class XI. We had learnt to differentiate certain functions like polynomial functions and trigonometric functions. In this chapter, we introduce the very important concepts of continuity, differentiability and
relations between them. We will also learn differentiation of inverse trigonometric functions. Further, we introduce a new class of functions called exponential and logarithmic functions. These functions lead to powerful techniques of differentiation. We illustrate certain geometrically obvious
conditions through differential calculus. In the process, we will learn some fundamental theorems in this area.

Definition 1 Suppose f is a real function on a subset of the real numbers and let c be a point in the domain of f. Then f is continuous at c if. a function is continuous at x = c if the function is defined at x = c and if the value of the function at x = c equals the limit of the function at x = c.

Definition 2 A real function f is said to be continuous if it is continuous at every point in the domain of f.

This definition requires a bit of elaboration. Suppose f is a function defined on a closed interval [a, b], then for f to be continuous, it needs to be continuous at every point in [a, b] including the end points a and b.

As a consequence of this definition, if f is defined only at one point, it is continuous there, i.e., if the domain of f is a singleton, f is a continuous function.

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Algebra of continuous functions

In the previous class, after having understood the concept of limits, we learnt some algebra of limits. Analogously, now we will study some algebra of continuous functions. Since continuity of a function at a point is entirely dictated by the limit of the function at that point, it is reasonable to expect results analogous to the case of limits.

Theorem 1 Suppose f and g be two real functions continuous at a real number c.
Then
(1) f + g is continuous at x = c.
(2) f – g is continuous at x = c.
(3) f . g is continuous at x = c.
(4) f/g is continuous at x = c, (provided g (c) ≠ 0).

Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c. If g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.

Theorem 3 If a function f is differentiable at a point c, then it is also continuous at that point.

Derivatives of composite functions
To study derivative of composite functions, we start with an illustrative example. Say, we want to find the derivative of f, where f (x) = (2x + 1)3

Theorem 4 (Chain Rule) Let f be a real valued function which is a composite of two functions u and v; i.e., f = v o u.

Derivatives of implicit functions
Until now we have been differentiating various functions given in the form y = f (x). But it is not necessary that functions are always expressed in this form. For example, consider one of the following relationships between x and y:
x – y – π = 0
x + sin xy – y = 0

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Derivatives of inverse trigonometric functions

We remark that inverse trigonometric functions are continuous functions, but we will not prove this. Now we use chain rule to find derivatives of these functions.

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Exponential and Logarithmic Functions

Till now we have learnt some aspects of different classes of functions like polynomial functions, rational functions and trigonometric functions. In this section, we shall learn about a new class of (related) functions called exponential functions and logarithmic functions. It needs to be emphasized that many statements made in this section are motivational and precise proofs of these are well beyond the scope of this text.

Definition 3 The exponential function with positive base b > 1 is the function y = f (x) = bx The graph of y = 10x .it is advised that the reader plots this graph for particular values of b like 2, 3 and 4.

Following are some of the salient features of the exponential functions:
(1) Domain of the exponential function is R, the set of all real numbers.
(2) Range of the exponential function is the set of all positive real numbers.
(3) The point (0, 1) is always on the graph of the exponential function (this is a restatement of the fact that b0 = 1 for any real b > 1).
(4) Exponential function is ever increasing; i.e., as we move from left to right, the graph rises above.
(5) For very large negative values of x, the exponential function is very close to 0. In other words, in the second quadrant, the graph approaches x-axis (but never meets it).

Exponential function with base 10 is called the common exponential function. In the Appendix A.1.4 of Class XI, it was observed that the sum of the series

Definition 4 Let b > 1 be a real number. Then we say logarithm of a to base b is x if bx = a.

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Logarithmic Differentiation

In this section, we will learn to differentiate certain special class of functions given in the form y = f (x) = [u(x)]v (x) By taking logarithm (to base e) the above may be rewritten as log y = v(x) log [u(x)]

The main point to be noted in this method is that f (x) and u(x) must always be positive as otherwise their logarithms are not defined. This process of differentiation is known as logarithms differentiation

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Derivatives of Functions in Parametric Forms

Sometimes the relation between two variables is neither explicit nor implicit, but some link of a third variable with each of the two variables, separately, establishes a relation between the first two variables. In such a situation, we say that the relation between them is expressed via a third variable. The third variable is called the parameter. More precisely, a relation expressed between two variables x and y in the form x = f (t), y = g (t) is said to be parametric form with t as a parameter.

CBSE Class 12 Commerce Mathematics Continuity and Differentiability:-Mean Value Theorem

In this section, we will state two fundamental results in Calculus without proof. We shall also learn the geometric interpretation of these theorems

Theorem 6 (Rolle’s Theorem) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers. Then there exists some c in (a, b) such that f ′(c) = 0.

Theorem 7 (Mean Value Theorem) Let f : [a, b] → R be a continuous function on [a, b] and differentiable on (a, b).

CBSE Class 12 Commerce Mathematics Continuity and Differentiability

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