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Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE PDF Download

Definition : A function f is differentiable at a if f'(a) exists. It is differentiable on an open interval (a,b)   [or Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE ] if it is differentiable at every number in the interval.

Derivability Over An Interval : f(x) is said to be derivable over an interval if it is derivable at each & every point of the interval.  f(x) is said to be derivable over the closed interval [a, b] if :

(i) for the points a and b, f '(a+) & f '(b -) exist &
(ii) for any point c such that a < c < b, f '(c+) & f'(c -) exist & are equal .

How Can a Function Fail to Be Differentiable ? 

We see that the function y = |x| is not differentiable at 0 and Figure shows that its graph changes direction abruptly when x = 0. In general, if the graph of a function f has a "corner" or "kink" in it, then the graph of f has no tangent at this point and f is not differentiable there. [In trying to compute f '(a), we find that the left and right limits are different.]

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

There is another way for a function not to have a derivative. If f is discontinuous at a, then f is not differentiable at a. So at any discontinuity (for instance, a jump discontinuity), f fails to be differentiable.

A third possibility is that the curve has a vertical tangent line when at x = a, Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

This means that the tangent lines become steeper and steeper as x → a. Figure (a, b, c) illustrates the three posibilities that we have discussed.

 

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Right hand & Left hand Derivatives By definition : f '(a) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

(i) The right hand derivative of f ' at x = a denoted by f '+(a) is defined by :

f '+(a) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE , provided the limit exists & is finite.

(ii) The left hand derivative of f at x = a denoted by f '-(a) is defined by :

f ' -(a) =  Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE , Provided the limit exists & is finite. We also write f '+(a) = f '(a+) & f '-(a) = f '(a-) .

f'(a) exists if and only if these one-sided derivatives exist and are equal.

 

Ex.20 If a function f is defined by f(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE show that f is continuous but not derivable at x = 0

Sol. We have f(0 + 0) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE = 0

f(0 - 0) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE = 0

Also f(0) = 0 Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE f(0 + 0) = f(0 - 0) = f(0) ⇒ f is continuous at x = 0

Again f'(0 + 0) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE = 1

f'(0 - 0) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE = 0

Since f'(0 + 0) Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE f'(0 - 0), the derivative of f(x) at x = 0 does not exist.

Ex.21 A function f(x) is such that Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE, if it exists.

Sol. Given that =  Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

 Ex.22 Let f be differentiable at x = a and let f (a) ¹ 0. Evaluate Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE.

 Sol. lDerivability Over An Interval | Mathematics (Maths) Class 12 - JEE (1 form)

l = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE  (put n = 1/h)

 

Ex.23 Let f : R → R satisfying Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE then show f(x) is differentiable at x = 0.

Sol. Since, Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE f(0) = 0    ...(i)

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE   ....(ii) {f(0) = 0 from (i)}

Now, Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE → 0 ...(iii) {using Cauchy-Squeeze theorem}

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE from (ii) and (iii) , we get f'(0) = 0. i.e. f(x) is differentiable at x = 0.

F. Operation on Differentiable Functions

1. If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) - g(x), f(x). g(x) will also be derivable at x = a & if g (a) Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE 0 then the function f(x)/g(x) will also be derivable at x = a.

If f and g are differentiable functions, then prove that their product fg is differentiable.

Let a be a number in the domain of fg. By the definition of the product of two functions we have

(fg) (a) = f(a) g(a) (fg) (a + t) = f(a + t) g(a + t).

Hence (fg)' (a) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

The following algebraic manipulation will enable us to put the above fraction into a form in which we can see what the limit is:

f(a + t) g(a + t) - f(a) g(a) = f(a + t) g(a + t) - f(a) g (a + t) + f(a)g(a + t) - f(a) g(a)

= [f(a + t) - f(a)] g(a + t) + [g(a + t) - g(a)] f(a).

Thus (fg)' (a) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE .

The limit of a sum of products is the sum of the products of the limits. Moreover, f'(a) and g'(a) exist by hypothesis. Finally, since g is differentiable at a, it is continuous there ; and so Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE = f(a). We conclude that

(fg)'(a) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

= f'(a)g(a) + g'(a)f(a) = (f'g + g'f) (a).

2. If f(x) is differentiable at x = a & g(x) is not differentiable at x = a , then the product function F(x) = f(x) . g(x) can still be differentiable at x = a e.g. f(x) = x  and g(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE .

3. If f(x) & g(x) both are not differentiable at x = a then the product function ;

F(x) = f(x) . g(x) can still be differentiable at x = a e.g. f(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE & g(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

4. If f(x) & g(x) both are non-deri. at x = a then the sum function F(x) = f(x) + g(x) may be a differentiable function . e.g. f(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE & g(x) = - Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

5. If f(x) is derivable at x = a ⇒ f '(x) is continuous at x = a.

e.g. f(x) = Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

 

G. Functional Equations 

Ex.24 Let f(xy) = xf(y) + yf(x) for all x, Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE and f(x) be differentiable in (0, ∞) then determine f(x).

Given f(xy)= xf(y) + yf(x) 

Sol. Replacing x by 1 and y by x then we get x f(1) = 0 

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

On integrating w.r.t.x and taking limit 1 to x then f(x)/x - f(1)/1 = f'(1) (ln x – ln 1)

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

∴  f(1) = 0) ∴  f(x) = f'(1) (x ln x)

Alternative Method : 

Given f(xy) = xf(y) + yf(x)

Differentiating both sides w.r.t.x treating y as constant, f'(xy) . y = f(y) + yf'(x)

Putting y = x and x = 1, then

f'(xy). x = f(x) + xf'(x)

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Integrating both sides w.r.t.x taking limit 1 to x,

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE  Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Hence, f(x) =- f'(1)(x ln x).

Ex.25 If Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE and f'(1) = e, determine f(x).

Sol.

Given e–xy f(xy) = e–xf(x) + e–yf(y) ....(1)

Putting x = y = 1 in (1), we get f(1) = 0 ...(2)

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

 

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

On integrating we have e–xf(x) = ln x + c at x = 1, c = 0

∴  f(x) = ex ln x

Ex.26 Let f be a function such that f(x + f(y)) = f(f(x)) + f(y) x, y Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE  where ε > 0, then determine f"(x) and f(x).

Sol. Given f(x + f(y)) = f(f(x) + f(y))  .....(1)

Put x = y = 0 in (1), then f(0 + f(0)) = f(f(0)) + f(0)  ⇒    f(f(0)) = f(f(0)) + f(0)

∴ f(0) = 0 ...(2)

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE

Integrating both sides with limites 0 to x then f(x) = x   ∴ f'(x) = 1.

The document Derivability Over An Interval | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Derivability Over An Interval - Mathematics (Maths) Class 12 - JEE

1. What is derivability over an interval?
Ans. Derivability over an interval refers to the property of a function to have a derivative defined and continuous for all points within that interval. It means that the function is differentiable at every point within the given interval.
2. How can I determine if a function is derivable over a specific interval?
Ans. To determine if a function is derivable over a specific interval, you need to check if the function is continuous within that interval and if its derivative exists at every point within the interval. If both conditions are satisfied, then the function is derivable over the given interval.
3. What happens if a function is not derivable over an interval?
Ans. If a function is not derivable over an interval, it means that either the function is not continuous within that interval or its derivative does not exist at one or more points within the interval. In such cases, the function may have sharp corners, cusps, or vertical tangents, indicating a lack of smoothness or differentiability.
4. Can a function be derivable over a closed interval but not on the endpoints?
Ans. Yes, it is possible for a function to be derivable over a closed interval but not on the endpoints. In such cases, the function can have a derivative defined and continuous for all points within the interval, except at the endpoints. This situation often occurs when the function has a one-sided derivative at the endpoints.
5. What is the significance of derivability over an interval?
Ans. Derivability over an interval is significant as it indicates that a function is differentiable and smooth within that interval. It allows us to analyze the rate of change, slope, and behavior of the function at every point within the interval. Derivability also enables us to apply various calculus techniques, such as finding critical points, determining local extrema, and solving optimization problems.
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