Q. A ladder, $5$ meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of $10 \,cm/sec$, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is $2$ metres from the wall is

Application of Derivatives Report Error

$ \frac{1}{10}$ radian/sec

$ \frac{1}{20}$ radian/sec

100%

$20$ radian/sec

$10$ radian/sec

Solution:

Given $ \frac{dy}{dt} =10\, cm/s$,
$x = 2m = 200\, cm$ and Length of ladder $= 5\, m = 500 \,cm$
$\Rightarrow $ Let $∠ACB = \theta$
Now $sin \,\theta = \frac{y}{500}$
Diff w.r.t. $t$, we get

$cos\,\theta \frac{d\theta}{dt} = \frac{1}{500} \frac{dy}{dt}$
$\Rightarrow \frac{200}{500} \times \frac{d\theta }{dt}$
$= \frac{1}{500} \times 10$
$\Rightarrow \frac{d\theta }{dt} = \frac{10}{200}$
$\Rightarrow \frac{d\theta }{dt} = \frac{1}{20}$ rad/s.

Questions from Application of Derivatives

1. Let $ 𝑓 \left(x\right) = 𝑥 + log_{e} x − 𝑥 log_{e} 𝑥, 𝑥 ∈ \left(0,∞\right).$
• Column 1 contains information about zeros of $𝑓 (𝑥) , 𝑓'(𝑥)$ and $f ''(𝑥)$.
• Column 2 contains information about the limiting behavior of $𝑓(𝑥) , 𝑓 ' (𝑥)$ and $f ''(𝑥)$ at infinity.
• Column 3 contains information about increasing/decreasing nature of $𝑓(𝑥)$ and $f' (𝑥)$ .

Column 1 Column 2 Column 3

(I) $𝑓 (𝑥) = 0$ for some $x\in\left(1,e^{2}\right)$ (i) $\lim_{x\to\infty}f \left(x\right)=0$ (P) $𝑓$ is increasing in ($0, 1$)

(II) $𝑓(𝑥) = 0$ for some $x\in\left(1, e\right)$ (ii) $\lim_{x\to\infty}f \left(x\right)=-\infty$ (Q) $f$ is decreasing in ($𝑒, 𝑒^2$)

(III) $𝑓(𝑥) = 0$ for some $x\in\left(0, 1\right)$ (ii) $\lim_{x\to\infty}f' \left(x\right)=-\infty$ (R) $f'$ is increasing in ($0, 1$)

(IV) $f ''(𝑥) = 0$ for some $x\in\left(1, e\right)$ (ii) $\lim_{x\to\infty}f'' \left(x\right)=0$ (S) $f'$ is decreasing in ($𝑒, 𝑒^2$)

Which of the following options is the only INCORRECT combination?

Column 1	Column 2	Column 3
(I) $𝑓 (𝑥) = 0$ for some $x\in\left(1,e^{2}\right)$	(i) $\lim_{x\to\infty}f \left(x\right)=0$	(P) $𝑓$ is increasing in ($0, 1$)
(II) $𝑓(𝑥) = 0$ for some $x\in\left(1, e\right)$	(ii) $\lim_{x\to\infty}f \left(x\right)=-\infty$	(Q) $f$ is decreasing in ($𝑒, 𝑒^2$)
(III) $𝑓(𝑥) = 0$ for some $x\in\left(0, 1\right)$	(ii) $\lim_{x\to\infty}f' \left(x\right)=-\infty$	(R) $f'$ is increasing in ($0, 1$)
(IV) $f ''(𝑥) = 0$ for some $x\in\left(1, e\right)$	(ii) $\lim_{x\to\infty}f'' \left(x\right)=0$	(S) $f'$ is decreasing in ($𝑒, 𝑒^2$)

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Then $\displaystyle \lim_{x \to\infty} \frac{f\left(2x\right)}{f\left(x\right)} = $

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