Application of Derivatives Test

Result

Application of Derivatives Test - 66

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Weekly Quiz Competition

Question 1
1 / -0

The function $$\frac{ln(1+x)}{x}$$ in $$(0,\infty )$$ is

A
increasing

B
decreasing

C
not decreasing

D
not increasing

Solution
Question 2
1 / -0

If the subnormal to the curve $${ x }^{ 2 }.{ y }^{ n }={ a }^{ 2 }$$ is a constant then n=

A
$$-4$$

B
$$-3$$

C
$$-2$$

D
$$-1$$

Solution

Given,

$$x^2y^n=a^2$$

$$\Rightarrow x=\sqrt{\dfrac{a^2}{y^n}}$$

$$\Rightarrow 2xy^n+x^2ny^{n-1}\dfrac{dy}{dx}=0$$

$$\Rightarrow \dfrac{dy}{dx}=-\dfrac{2y}{nx}$$

$$\Rightarrow \dfrac{dy}{dx}=-\dfrac{2y}{n\sqrt{\frac{a^2}{y^n}}}$$

$$\therefore \dfrac{dy}{dx}=-\dfrac{2}{na}y^{\frac{n+2}{2}}$$

Subnormal is given by,

$$SN=y \dfrac{dy}{dx}$$

$$=y\left ( -\dfrac{2}{na}y^{\frac{n+2}{2}} \right )$$

$$\therefore SN=-\dfrac{2}{na}y^{\frac{n+4}{2}}$$

$$\therefore n=-4$$
Question 3
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Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and g be the function satisfying $$f(x)+g(x)=x^2$$ the value of the integral $$\displaystyle\int^1_0f(x)g(x)dx$$ is?

A
$$\dfrac{1}{4}(e-7)$$

B
$$\dfrac{1}{4}(e-2)$$

C
$$\dfrac{1}{2}(e-3)$$

D
None of these

Solution

Given: $$f'(x)=f(x)$$, with $$f(0)=1$$ and $$f(x)+g(x)=x^2$$

To find: Value of $$\displaystyle \int'_0 f(x)g(x)dx$$.

Solution : $$f'(x)=f^0(x)=\dfrac{f'(x)}{f(x)}=1$$

$$\displaystyle \therefore \int \dfrac{f'(x)}{f(x)}dx=\int dx\Rightarrow \log |f(x)|=x+c$$

$$\Rightarrow f(x)=e^{x+c}$$

$$\therefore f(0)=e^e=1$$

$$\Rightarrow e=0$$

$$\therefore f(x)=e^x$$

Now, $$f(x)+g(x)=x^2$$

$$\Rightarrow g(x)=x^2-f(x)=x^2-e^x$$

$$\displaystyle \therefore \int^1_0 f(x) g(x)dx=\int^1_0 e^x(x^2-e^x)dx$$

$$\displaystyle =\int^1_0 e^x x^2 dx-\int^1_0 e^{2x}dx=I$$ (say)

Let $$\displaystyle I_1=\int e^x x^2 dx=x^2 \int e^x dx-\int [\dfrac{d}{dx}x^2\int e^x dx]dx$$

$$\displaystyle =x^2e^x-2\int xe^x dx+c_1$$

$$\displaystyle =x^2e^x-2x\int e^x dx+2\int [\dfrac{d}{dx} x.\int e^x dx]dx+c_1$$

$$\displaystyle =x^2e^x-2xe^x+2\int e^x dx+c_2$$

$$=x^2e^x-2xe^x+2e^x+c_3$$

$$=e^x[x^2-2x+2]+c_3$$

$$\therefore I=\left[e^x(x^2-2x+2)-\dfrac{e^{2x}}{2}\right]^1_0$$

$$=(e^1(1-2+2)-\dfrac{e^2}{2})-(e^0(0-0+2-\dfrac{1}{2}))$$

$$=e-\dfrac{e^2}{2}-\dfrac{3}{2}=\dfrac{1}{4}(4e-e^2-3)$$

Hence the answer is (D) None of these.
Question 4
1 / -0

Define $$f(x) = \dfrac{1}{2} [ |\sin x| + \sin x], 0 < x \le 2\pi$$. The $$f$$ is

A
increasing in $$\left(\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right)$$

B
decreasing in $$\left(0, \dfrac{\pi}{2}\right)$$ and increasing in $$\left(\dfrac{\pi}{2}, \pi\right)$$

C
increasing in $$\left(0, \dfrac{\pi}{2}\right)$$ and decreasing in $$\left(\dfrac{\pi}{2}, \pi\right)$$

D
increasing in $$\left(0, \dfrac{\pi}{4}\right)$$ and decreasing in $$\left(\dfrac{\pi}{4}, \pi\right)$$

Solution
Question 5
1 / -0

Let N be the set of positive integers. For all $$n \in N$$, let
$$f_n = (n + 1)^{1/3} - n^{1/3}$$ and $$A = \left\{n \in N : f_{n +1} < \dfrac{1}{3(n + 1)^{2/3}} < f_n \right\}$$
Then

A
A = N

B
A is a finite set

C
the complement of A in N is nonempty, but finite

D
A and its complement in N are both infinite

Solution
Question 6
1 / -0

A Stationary point of $$f(x)=\sqrt{16-x^{2}}$$ is

A
(4,0)

B
(-4,0)

C
(0,4)

D
(-4,4)

Solution
Question 7
1 / -0

If $$\int { \dfrac { { 2x }^{ 2 } }{ \left( { x }^{ 2 }+1 \right) ^{ 2 } } } dx=f\left( x \right) +c$$ where f (0) = 0, then

A
f(x) is an increasing function

B
x = 0 is point of extremum

C
f(x) is concave up for $$x\in \left( -\infty ,-1 \right) \cup \left( 0,1 \right) $$

D
the curve f(x) has 3 points of inflection

Solution
Question 8
1 / -0

$$A\;sationary\;po\operatorname{int} \;of\;f\left( x \right) = \sqrt {16 - {x^2}} \;is$$

A
$$\left( {4,0} \right)$$

B
$$\left( { - 4,0} \right)$$

C
$$\left( {0,4} \right)$$

D
$$\left( { - 4,4} \right)$$

Solution
Question 9
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The slope of the tangent to the curve at a point (x, y) on it is proportional to (x-2). If the slope of the tangent to the curve at (10,-9) on it -3. The equation of the curve is

A
$$y=k{ \left( x-2 \right) }^{ 2 }$$

B
$$y=\frac { -3 }{ 16 } { \left( x-2 \right) }^{ 2 }\ +1$$

C
$$y=\frac { -3 }{ 16 } { \left( x-2 \right) }^{ 2 }\ +3$$

D
$$y=k{ \left( x+2 \right) }^{ 2 }$$

Solution

$${\textbf{Step -1: Find the value of slope}}{\textbf{.}}$$
$${\text{The slope of the tangent to the curve at a point }}\left( {x,y} \right){\text{ on it is proportional to }}\left( {x - 2} \right).$$
$$ = \dfrac{{dy}}{{dx}} = m\left( {x - 2} \right)$$
$$ = \dfrac{{dy}}{{dx}} = k\left( {x - 2} \right)\left[ {{\text{where }}k{\text{ is a proportionality constant}}} \right]$$
$${\text{Slope of the tangent to the curve at }}\left( {10, - 9} \right){\text{ is }} - 3.$$
$$ \Rightarrow - 3 = k\left( {10 - 2} \right)$$
$$ \Rightarrow - 3 = 8k$$
$$ \Rightarrow k = - \dfrac{3}{8}$$
$$ \Rightarrow \dfrac{{dy}}{{dx}} = - \dfrac{3}{8}\left( {x - 2} \right)$$
$${\textbf{Step -2: Integrate the formed equation and solve it further}}{\textbf{.}}$$
$${\text{Integrating both sides,}}$$
$$ \Rightarrow y = - \dfrac{3}{8}\dfrac{{{{\left( {x - 2} \right)}^2}}}{2} + c{\text{ }}.......\left( i \right){\text{ }}\left[ {{\text{Eqaution of curve}}} \right]$$
$${\text{Also, }}\left( {10, - 9} \right){\text{ satisfy the equation of curve as it lies on the curve}}{\text{.}}$$
$$\therefore - 9 = - \dfrac{3}{8}\dfrac{{{{\left( {10 - 2} \right)}^2}}}{2} + c$$
$$ \Rightarrow - 9 = - \dfrac{3}{{16}} \times {8^2} + c$$
$$ \Rightarrow c = 3$$
$${\textbf{Hence, the equation of the curve is }}\mathbf{y= - \dfrac{3}{{16}}{\left( {x - 2} \right)^2} + 3.}$$
Question 10
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$$N$$ characters of information are held on magnetic tape, in batches of $$x$$ characters each, the batch processing time is $$\alpha +\beta x^2$$ seconds. $$\alpha $$ and $$\beta $$ are constants. The optical value of $$x$$ for last processing is,

A
$$\dfrac{\alpha}{\beta}$$

B
$$\dfrac{\beta }{\alpha}$$

C
$$\sqrt{\dfrac{\alpha}{\beta}}$$

D
$$\sqrt{\dfrac{\beta }{\alpha}}$$

Solution