Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 14

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Earn 100

Let $f (x) = x + \frac{1}{2 x + \frac{1}{2 x + \frac{1}{2 x + . . . . . . \infty}}}$ . Compute the value of $f (100) . f^{'} (100)$ .

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Important Questions on Differential Coefficient

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 15

If

y = \tan^{- 1} \frac{u}{\sqrt{1 - u^{2}}}

and

x = \sec^{- 1} \frac{1}{2 u^{2} - 1}, u \in (0, \frac{1}{\sqrt{2}}) \cup (\frac{1}{\sqrt{2}}, 1)

; find the value of

|\frac{d y}{d x}|

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 16

If

y = \tan^{- 1} \frac{x}{1 + \sqrt{1 - x^{2}}} + \sin (2 \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}})

for

x \in (- 1, 1)

and if the value of

\frac{d y}{d x}

at

x = 0

is

k

, then find the value of

2 k

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 17

If

x = 2 \cos t - \cos 2 t

and

y = 2 \sin t - \sin 2 t

, find the value of

|\frac{d^{2} y}{d x^{2}}|

when

t = (\frac{π}{2})

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 18

Let

y = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{b x}{(x - b) (x - c)} + \frac{c}{(x - c)} + 1

, if

\frac{y^{'}}{y} = \frac{λ}{x} (\frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x})

, then find the value of

λ

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 19

Let

f (x) = x^{2} - 4 x - 3, x > 2

and let

g

be the inverse of

f

. Find the value of

g^{'}

at

f (x) = 9

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 20

Let

y = x \ln [{(a x)}^{- 1} + a^{- 1}]

, if

x (x + 1) \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} = y - μ

, then find the value of

μ

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 21

Suppose

f

and

g

are two functions such that

f, g : R \to R, f (x) = \ln (1 + \sqrt{1 + x^{2}})

and

g (x) = \ln (x + \sqrt{1 + x^{2}})

, then find the value of

x . e^{g (x)} {(f (\frac{1}{x}))}^{'} + g^{'} (x)

at

x = 1

.

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Beta Question Bank for Engineering: Mathematics>Chapter 28 - Differential Coefficient>EXERCISE-3>Q 22

If

f (x) = |\begin{matrix} {(x - a)}^{4} & {(x - a)}^{3} & 1 \\ {(x - b)}^{4} & {(x - b)}^{3} & 1 \\ {(x - c)}^{4} & {(x - c)}^{3} & 1 \end{matrix}|

then

f^{'} (x) = λ . |\begin{matrix} {(x - a)}^{4} & {(x - a)}^{2} & 1 \\ {(x - b)}^{4} & {(x - b)}^{2} & 1 \\ {(x - c)}^{4} & {(x - c)}^{2} & 1 \end{matrix}|

. Find the value of

λ

.