2 Marks

Question 12 Marks

Define differentiability of a function at a point.

Answer

Let f(x) be a real valued function defined on an open interval (a, b) and let $\text{c}\in(\text{a, b}).$
Then f(x) is said to be differentiable or derivable at x = c iff $\lim_\limits{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}$ exists finitely.
or, $\text{f}'(\text{c})=\lim_\limits{\text{x}\rightarrow{\text{c}}}\frac{\text{f(x)}-\text{f(c)}}{\text{x}-\text{c}}.$

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Question 22 Marks

Is every continuous function differentiable?

Answer

No, function may be continuous at a point but may not be differentiable at that point .
For example: function f(x) = |x| is continuous at x = 0 but it is not differentiable at x = 0.

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Question 32 Marks

Write the points of non-differentiability of f(x) = |log |x||.

Answer

Here,
f(x) = |log |x||
f(x) will always positive and let two points x = 1 and x = -1
f(x) = 0
The function f(x) = |log |x|| is not differentiable at x = -1 and 1.

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Question 42 Marks

Discuss the continuity and differntiability of f(x) = |log |x||.

Answer

f(x) = |log |x||
Since, it is an absolute function. So, it is continuous function. The graph of the function is as below:-

From the graph, it is clear that f(x) is not differentiable at x = -1, 1 but continuous for all x.

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Question 52 Marks

If f(x) is differentiable at x = c, then write the value of $\text{f}\lim_\limits{\text{x}\rightarrow{\text{c}}}\text{f(x)}.$

Answer

Given,
f(x) is differentiable at,
$\lim_\limits{\text{x}\rightarrow{\text{c}}}\text{f(x)}=\text{f(c)}$

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Question 62 Marks

Is every differentiable function continuous?

Answer

Yes, every differentiable function is continuous.

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Question 72 Marks

Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.

Answer

We know that, modulus function

f(x) = |x| is continuous but not differntiable at x = 0.

So,

f(x) = |x| + |x - 1| + |x - 2| + |x - 3| + |x - 4| is continuous but not differentiable x = 0, 1, 2, 3, 4.

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