Limits and Derivatives Test

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Limits and Derivatives Test - 11

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Weekly Quiz Competition

Question 1
1 / -0

Use limit properties to evaluate $$\displaystyle\lim_{x\to4}\dfrac{3x^2\tan \dfrac {\pi}{x}}x $$

A
$$12$$

B
$$14$$

C
$$16$$

D
$$18$$

Solution

$$\displaystyle\lim_{x\to4}\dfrac{3x^2\tan \dfrac {\pi}{x}}x =\lim_{x\to 4}\left(3x\tan\dfrac{\pi}{x}\right)$$, simplify

$$=3(4)\tan\dfrac{\pi}{4}=12(1)=12$$, [substitute the limit ]

The from of the limit is not indeterminate, so we can substitute the limit value
Question 2
1 / -0

Evaluate $$\underset{x \rightarrow 3}\lim \sqrt[4] {x^3}$$ using the properties of limits.

A
$$28^{1/4}$$

B
$$25^{1/4}$$

C
$$27^{1/4}$$

D
$$26^{1/4}$$

Solution

$$\displaystyle \lim _{ x\rightarrow 3 }{ \sqrt [ 4 ]{ { x }^{ 3 } } } \\={ (3^3) }^{ { 1 }/{ 4 } }={ (27) }^{ { 1 }/{ 4 } }$$
Question 3
1 / -0

What is $$\displaystyle\lim _{ x\rightarrow 0 }{ \frac { \cos { x } }{ \pi -x } } $$ equal to?

A
$$0$$

B
$$\pi $$

C
$$\dfrac { 1 }{ \pi } $$

D
$$1$$

Solution

Here, putting $$x=0$$ directly we have,
$$\displaystyle\lim _{ x\rightarrow 0 }{ \frac { \cos { x } }{ \pi -x } } $$

$$=\dfrac {\cos 0 }{ \pi -0 } =\dfrac { 1 }{ \pi } $$

Hence, C is correct.
Question 4
1 / -0

Find $$\dfrac{dy}{dx}$$ of function $$y= e^{x^3} +\dfrac{1}{2} \log x $$

A
$$2.e^{x^3}x^2+\dfrac {1}{2x}$$

B
$$e^{x^3}x^2+\dfrac {1}{2x}$$

C
$$3.e^{x^3}x^2+\dfrac {1}{2x}$$

D
$$3.e^{x^3}x^2+\dfrac {1}{x}$$

Solution

Given y= $$e^{x^3} +\frac{logx}{x}$$
$$(\frac{dy}{dx})=(\frac{de^{x^3}}{dx})+(\frac{d(\frac{logx}{2})}{dx})\\=(\frac{de^{x^3}}{dx^3})\times(\frac{dx^3}{dx})+(\frac{1}{2})(\frac{dlogx}{dx})\\=3x^2e^{x^3}+(\frac{1}{2x})$$
Question 5
1 / -0

Differentiate: $$x^{100} + \sin x - 1$$

A
$$100x^{99}-\cos x$$

B
$$100x^{99}+\cos x$$

C
$$x^{99}+\cos x$$

D
$$100x^{99}+\sin x$$

Solution

$$\cfrac{\mathrm{d} }{\mathrm{d} x}\left ( x^{100} +\sin x -1 \right )$$

$$=\cfrac{\mathrm{d} }{\mathrm{d} x}\left ( x^{100} \right )+\cfrac{\mathrm{d} }{\mathrm{d} x}\left ( \sin x \right )-\cfrac{\mathrm{d} }{\mathrm{d} x}\left ( 1 \right )$$

$$=100x^{100-1}+\cos x -0$$

$$=100x^{99}+\cos x$$
Question 6
1 / -0

Consider the differential equation $$\frac { d y } { d x } = \cos x$$ Then we observe that

A
$$y = \sin x$$

B
$$y = \sin x + 2$$

C
$$y = \sin x - \frac { 1 } { 2 }$$

D
$$y = \sin x + c$$

Solution

$$dy = \cos x\>dx\\\int dy=\int \cos x\>dx\\y=\sin x+C$$

We must add constant "C" as its family of curves.
Question 7
1 / -0

$$x^{\frac{1}{2}} + 1= t$$
differentiate w.r.t. x

A
$$\dfrac{dt}{dx} = \dfrac{1}{2\sqrt{x}}$$

B
$$\dfrac{dt}{dx} = \dfrac{1}{4\sqrt{x}}$$

C
$$\dfrac{dt}{dx} = \dfrac{1}{8\sqrt{x}}$$

D
$$\dfrac{dt}{dx} = \dfrac{1}{16\sqrt{x}}$$

Solution

$$x^{\frac{1}{2}} + 1= t$$

$$\dfrac{dt}{dx} = \dfrac{1}{2}(x) ^{\frac{1}{2}-1} + 0$$

$$\frac{1}{2}(x) ^{\frac{-1}{2}}$$

$$\dfrac{1}{2x^{\frac{1}{2}}}$$

$$\dfrac{dt}{dx} = \dfrac{1}{2\sqrt{x}}$$
Question 8
1 / -0

$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$ equals

A
$$\dfrac{7}{3}$$

B
$$\dfrac{10}{3}$$

C
$$\dfrac{14}{3}$$

D
$$\dfrac{1}{3}$$

Solution

We have,
$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\sin7x}{\sin3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\dfrac{\sin7x}{7x}\times 7x}{\dfrac{\sin3x}{3x}\times 3x}$$
$$\lim_{x\rightarrow\ 0}\dfrac{\dfrac{\sin7x}{7x}\times 7}{\dfrac{\sin3x}{3x}\times 3}$$
$$=\dfrac{1\times 7}{1\times 3}$$
$$=\dfrac{7}{3}$$

Hence, this is the answer.
Question 9
1 / -0

If a sequence $$< a_{n} >$$ is such that $$a_{1},a_{n+1}=\dfrac {2+3a_{n}}{1+2a_{n}}$$ and $$\displaystyle \lim_{n \rightarrow \infty}a_{n}$$ exists, then $$a_{n}$$ is equal to

A
$$\cos 36^{o}$$

B
$$2\cos 36^{o}$$

C
$$\sin 18^{o}$$

D
$$2\sin 36^{o}$$
Question 10
1 / -0

Evaluate the following limit :
$$lim_{x\rightarrow 0} \dfrac{1-\cos 2x}{x^2}$$

A
$$0$$

B
$$1$$

C
$$2$$

D
none of these

Solution

$$lim_{x\rightarrow 0} \dfrac{1-\cos 2x}{x^2}$$

$$=lim_{x\rightarrow 0} \dfrac{2\sin^2 x}{x^2}$$

$$=2lim_{x \rightarrow 0} (\dfrac{\sin x}{x} \times \dfrac{\sin x}{x})$$

$$=2lim_{x\rightarrow 0} \dfrac{\sin x}{x} \times lim_{x\rightarrow 0} \dfrac{\sin x}{x}=2(1)(1)=2$$

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Re-Attempt Weekly Quiz Competition

Limits and Derivatives Test - 11

Result

Limits and Derivatives Test - 11

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SHARING IS CARING

Question 1

$$12$$

$$14$$

$$16$$

$$18$$

Solution

Question 2

$$28^{1/4}$$

$$25^{1/4}$$

$$27^{1/4}$$

$$26^{1/4}$$

Solution

Question 3

$$0$$

$$\pi $$

$$\dfrac { 1 }{ \pi } $$

$$1$$

Solution

Question 4

$$2.e^{x^3}x^2+\dfrac {1}{2x}$$

$$e^{x^3}x^2+\dfrac {1}{2x}$$

$$3.e^{x^3}x^2+\dfrac {1}{2x}$$

$$3.e^{x^3}x^2+\dfrac {1}{x}$$

Solution

Question 5

$$100x^{99}-\cos x$$

$$100x^{99}+\cos x$$

$$x^{99}+\cos x$$

$$100x^{99}+\sin x$$

Solution

Question 6

$$y = \sin x$$

$$y = \sin x + 2$$

$$y = \sin x - \frac { 1 } { 2 }$$

$$y = \sin x + c$$

Solution

Question 7

$$\dfrac{dt}{dx} = \dfrac{1}{2\sqrt{x}}$$

$$\dfrac{dt}{dx} = \dfrac{1}{4\sqrt{x}}$$

$$\dfrac{dt}{dx} = \dfrac{1}{8\sqrt{x}}$$

$$\dfrac{dt}{dx} = \dfrac{1}{16\sqrt{x}}$$

Solution

Question 8

$$\dfrac{7}{3}$$

$$\dfrac{10}{3}$$

$$\dfrac{14}{3}$$

$$\dfrac{1}{3}$$

Solution

Question 9

$$\cos 36^{o}$$

$$2\cos 36^{o}$$

$$\sin 18^{o}$$

$$2\sin 36^{o}$$

Question 10

$$0$$

$$1$$

$$2$$

none of these

Solution

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