Differentiability - Maharashtra Board STD 12 Maths Questions

Q 1MCQ1 Mark

If $\text{f(x)}=\begin{cases}\frac{|\text{x}+2|}{\tan^{-1}(\text{x}+2)}, & \text{x}\neq-2\\2, & \text{x}=-2\end{cases},$ then $f(x)$ is:

A
Continuous at $x = -2$
✓
Not continuous at $x = -2$
C
Diffrentiable at $x = -2$
D
Continuous but nit derivable at $x = -2$

Answer: B.

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Q 2MCQ1 Mark

If $f(x) = |3 − x| + (3 + x),$ where $(x)$ denotes the least integer greater than or equal to $x,$ then $f(x)$ is:

A
Continuous and differentiable at $x = 3$
B
Continuous but not differentiable at $x = 3$
C
Differentiable nut not continuous at $x = 3$
✓
Neither differentiable nor continuous at $x = 3$

Answer: D.

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Q 3MCQ1 Mark

If $\text{f(x)}=\text{a}|\sin\text{x}|+\text{be}^{|\text{x}|}+\text{c|x|}^3$and if $f(x)$ is differentiable at $x = 0,$ then:

A
$\text{a}=\text{b}=\text{c}=0$
✓
$\text{a}=0,\text{b}=0;\text{c}\in\text{R}$
C
$\text{b}=\text{c}=0,\text{a}\in\text{R}$
D
$\text{c}=0,\text{a}=0,\text{b}\in\text{R}$

Answer: B.

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Q 4MCQ1 Mark

The set points where the function $f(x)$ given by $\text{f(x)=}|\text{x}-3|\cos\text{x}$ is diffrentiable, is:

✓
$R$
B
$R - \{3\}$
C
$(0,\infty)$
D
None of these.

Answer: A.

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Q 5MCQ1 Mark

If $\text{f(x)}=\begin{cases}\frac{1-\cos\text{x}}{\text{x}\sin\text{x}}, & \text{x}\neq 0\\\frac{1}{2} & \text{x}= 0\end{cases}$ then at $x = 0, f(x)$ is:

✓
Continuous and differentiable.
B
Differentiable but not continuous.
C
Continuous but not differentiable.
D
Neither continuous not differentiale.

Answer: A.

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Q 6Solve the Following Question.(2 Marks)2 Marks

Is every differentiable function continuous?

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Q 7Solve the Following Question.(2 Marks)2 Marks

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

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Q 8Solve the Following Question.(2 Marks)2 Marks

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

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Q 9Solve the Following Question.(2 Marks)2 Marks

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

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Q 10Solve the Following Question.(2 Marks)2 Marks

Define differentiability of a function at a point.

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Q 11Solve the Following Question.(3 Marks)3 Marks

Write the derivative of $f(x) = |x|^3 at x = 0$.

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Q 12Solve the Following Question.(3 Marks)3 Marks

Examine the differentialiblilty of the function f defined by $\text{f(x)}=\begin{cases}2\text{x}+3 & \text{if}-3\leq\text{x}\leq-2\\\text{x}+1 & \text{if} -2\leq\text{x}\leq0\\\text{x}+2&\text{if}\ 0\leq\text{x}\leq1\end{cases}$

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Q 13Solve the Following Question.(3 Marks)3 Marks

If f(x) = |x - 2| write whether f(2) exists or not.

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Q 14Solve the Following Question.(3 Marks)3 Marks

Write the value of the derivative of f(x) = |x − 1| + |x − 3| at x = 2.

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Q 15Solve the Following Question.(3 Marks)3 Marks

Give an example of a function which is continuos but not differentiable at at a point.

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Q 16Solve the Following Question.(5 Marks)5 Marks

If $\text{f(x)}=\begin{cases}\text{ax}^2-\text{b}, & \text{if |x|}<1\\\frac{1}{|\text{x}|}, & \text{if |x|}\geq1\end{cases}$ is differentiable at x = 1, find a, b.

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Q 17Solve the Following Question.(5 Marks)5 Marks

Show that the function $\text{f(x)}\begin{cases}\text{x}^\text{m}\sin(\frac{1}{\text{x}}), &\text{x}\neq0 \\0 ,& \text{x}=0\end{cases}$
Continuous but not diffierentiable at x = 0, if 0 < m < 1

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Q 18Solve the Following Question.(5 Marks)5 Marks

Finde the value of a and b, if the function f(x) defined by $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, &\text{x}\leq1\\\text{bx}+2, & \text{x}>1\end{cases}$is differentiable at x = 1.

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Q 19Solve the Following Question.(5 Marks)5 Marks

Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.

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Q 20Solve the Following Question.(5 Marks)5 Marks

Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$

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Differentiability question types

Differentiability questions

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Differentiability Maths - Differentiability

Differentiability question types

Differentiability questions

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