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Important Questions: Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE PDF Download

Q1: If y= tan x + sec x , then show that d2.y / dx2 = cos x / (1-sin x)2
Ans:
Given that, y= tan x + sec x
Now, the differentiate wih respect to x, we get
dy/dx = sec2 x + sec x tan x
= (1/ cos2 x) + (sin x/ cos2 x)
= (1+ sinx)/ (1 + sinx)(1-sin x)
Thus, we get.
dy/dx = 1/(1-sin x)
Now, again differentiate with respect to x, we will get
d2y / dx2 = -(-cosx )/(1- sin x)2
d2y / dx= cos x / (1-sinx)2.

Q2: Verify the mean value theorem for the following function f (x) = (x – 3) (x – 6) (x – 9) in [3, 5]
Ans:
f(x) = (x−3)(x−6)(x−9)
= (x−3)(x− 15x + 54)
= x3−18x+ 99x − 162
fc∈(3,5)
f′(c) = f(5) − f(3)/5 − 3
f(5)=(5−3)(5−6)(5−9)
= 2(−1)(−4)=−8
f(3) = (3−3)(3−6)(3−9) = 0
f′(c) = 8−0/2 = 4
∴f′(c) = 3c2−36c + 99
3c2− 36c + 99 = 4
3c2−36c+95 = 0
ax+ bx + c = 0
a = 3
b = −36
c = 95
c = 36 ± √(36)2− 4(3)(95)/2(3)
= 36 ± √1296 − 1140/6
= 36 ± 12.496
c = 8.8&c = 4.8
c∈(3,5)
f(x)=(x−3)(x−6)(x−9) on [3,5]

Q3: Determine the points of discontinuity of the composite function y = f[f(x)], given that, f(x) = 1/x-1.
Ans
: Given that, f(x) = 1/x-1
We know that the function f(x) = 1/x-1 is discontinuous at x = 1
Now, for x ≠1,
f[f(x)] = f(1/x-1)
= 1/[(1/x-1)-1]
= x-1/ 2-x, which is discontinuous at the point x = 2.
Therefore, the points of discontinuity are x = 1 and x = 2.

Q4: Explain the continuity of the function f = |x| at x = 0.
Ans:
From the given function, we define that,
f(x) = {-x, if x < 0 and x, if x ≥ 0
It is clearly mentioned that the function is defined at 0 and f(0) = 0. Then the left-hand limit of f at 0 is
Limx→0- f(x) = limx→0- (-x) = 0
Similarly for the right-hand side,
Limx→0+ f(x) = limx→0+ (x) = 0
Therefore, for the both left hand and the right-hand limit, the value of the function coincide at the point x = 0.
Therefore, the function f is continuous at the point x = 0.

Q5: If f (x) = |cos x|, find f’(3π/4)
An
s: Given that, f(x) = |cos x|
When π/2 <x< π, cos x < 0,
Thus, |cos x| = -cos x
It means that, f(x) = -cos x
Hence, f’(x) = sin x
Therefore, f’(3π/4) = sin (3π/4) = 1/√2
f’(3π/4) = 1/√2

Q6: Explain the continuity of the function f(x) = sin x . cos x
Ans:
We know that sin x and cos x are continuous functions. It is known that the product of two continuous functions is also a continuous function.
Hence, the function f(x) = sin x . cos x is a continuous function.

The document Important Questions: Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Important Questions: Continuity & Differentiability - Mathematics (Maths) Class 12 - JEE

1. What is the difference between continuity and differentiability?
Ans. Continuity refers to the uninterrupted nature of a function, where there are no abrupt jumps or breaks in its graph. Differentiability, on the other hand, implies that the function has a well-defined derivative at every point within its domain.
2. Can a function be continuous but not differentiable?
Ans. Yes, it is possible for a function to be continuous but not differentiable. A classic example is the absolute value function, |x|. It is continuous for all values of x, but the derivative is not defined at x = 0.
3. How do you check if a function is differentiable at a point?
Ans. To check if a function is differentiable at a point, you need to verify two conditions: 1) The function must be continuous at that point, and 2) The derivative of the function must exist at that point.
4. Can a function be differentiable but not continuous?
Ans. No, a function cannot be differentiable but not continuous. Differentiability implies continuity, as the existence of a derivative at a point requires the function to be continuous at that point.
5. What are the common types of discontinuities in functions?
Ans. There are three common types of discontinuities: 1) Removable discontinuities occur when the function approaches a finite limit but has a different value at that point. 2) Jump discontinuities occur when the limit from the left and the limit from the right exist but are not equal. 3) Infinite discontinuities occur when the function approaches positive or negative infinity at a certain point.
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