Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 33

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0

Let $$f(x + y) = f(x) f(y)$$ for all $$x$$ and $$y$$. If $$f(0) = 1, f(3) = 3$$ and $$f'(0) = 11$$, then $$f'(3)$$ is equal to

A
$$11$$

B
$$22$$

C
$$33$$

D
$$44$$

Solution

We have,
$$f'(x) = \underset{h\to 0}{\lim} \dfrac{f(x+h)-f(x)}{h}$$

$$\Rightarrow f'(3) = \underset{h\to 0}{\lim} \dfrac{f(3+h)-f(3)}{h}$$

$$=\underset{h\to 0}{\lim} \dfrac{f(3)f(h) - f(3+0)}{h}$$

$$=\underset{h\to 0}{\lim} \dfrac{f(3)f(h) - f(3)f(0)}{h}$$

$$=f(3) \underset{h\to 0}{\lim} \dfrac{f(h)-f(0)}{h}$$

$$=f(3) \underset{h\to 0}{\lim} \dfrac{f(0+h)-f(0)}{h}$$

$$= f(3)f'(0)$$

$$=3\times 11$$

$$=33$$
Question 2
1 / -0

Let $$f(x+y) = f(x) f(y) $$ and $$f(x) = 1+\sin(3x)g(x)$$, where $$g$$ is differentiable. The $$f'(x)$$ is equal to

A
$$3f(x)$$

B
$$g(0)$$

C
$$3f(x) g(0)$$

D
$$3g(x)$$

Solution

$$f'(x) = \underset{h\to 0}{\lim} \dfrac{f(x+h) - f(x)}{h}$$

$$=\underset{h\to 0}{\lim} \dfrac{f(x)f(h) - f(x)}{h}$$

$$= f(x) \underset{h \to 0}{\lim} \left(\dfrac{1+\sin 3h(g(h))-1}{h}\right)$$

$$3f(x) \underset{h\to 0}{\lim} \dfrac{\sin 3h}{3h} \underset{h\to 0}{\lim} g(h)$$

$$=3 f(x) \times 1 \times g(0) = 3f(x) g(0)$$
Question 3
1 / -0

If $$y=\cos^{-1}x^{3}$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{-1}{\sqrt{1-x^{6}}}$$

B
$$\dfrac{-3x^{2}}{\sqrt{1-x^{6}}}$$

C
$$\dfrac{-3}{x^{2}\sqrt{1-x^{6}}}$$

D
$$none\ of\ these$$

Solution
Question 4
1 / -0

If $$y=\cos^{-1}(4x^{3}-3x)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{3}{\sqrt{1-x^{2}}}$$

B
$$\dfrac{-3}{\sqrt{1-x^{2}}}$$

C
$$\dfrac{4}{\sqrt{1-x^{2}}}$$

D
$$\dfrac{-4}{(3x^{2}-1)}$$

Solution
Question 5
1 / -0

If $$y=\tan^{-1}\left(\dfrac{a\cos x-b\sin x}{b\cos x+a\sin x}\right)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{a}{b}$$

B
$$\dfrac{-b}{a}$$

C
$$1$$

D
$$-1$$

Solution

Let, $$\dfrac{a}{b}=\tan\theta$$

$$y=\tan^{-1}\left(\dfrac{a\cos x-b\sin x}{b\cos x+a\sin x}\right)\\\,\,=\tan^{-1}\left(\dfrac{\dfrac{a}{b}-\tan x}{1+\dfrac{a}{b}\tan x}\right)\\\,\,=\tan^{-1}\left(\dfrac{\tan \theta-\tan x}{1+\tan \theta \tan x}\right)$$
$$=\tan^{-1}\tan (\theta-x)\\=\theta-x\\=\left(\tan^{-1}\dfrac{a}{b}-x\right)$$.
$$\therefore \dfrac{dy}{dx}=-1$$.
Question 6
1 / -0

If $$y=\tan^{-1}\left(\dfrac{\sqrt{a+\sqrt{x}}}{1-\sqrt{ax}}\right)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{1}{(1+x)}$$

B
$$\dfrac{1}{\sqrt{x}(1+x)}$$

C
$$\dfrac{2}{\sqrt{x}(1+x)}$$

D
$$\dfrac{1}{2\sqrt{x}(1+x)}$$

Solution
Question 7
1 / -0

If $$y=\tan^{-1}(\sec x+6\tan x)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{1}{2}$$

B
$$\dfrac{-1}{2}$$

C
$$1$$

D
$$none\ of\ these$$

Solution
Question 8
1 / -0

If $$y=\sin^{-1}(3x-4x^3)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{3}{\sqrt {1-x^2}}$$

B
$$\dfrac{-4}{\sqrt {1-x^2}}$$

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

D
$$none\ of\ these$$

Solution
Question 9
1 / -0

If $$y=\tan^{-1}\left\{\dfrac{\cos x+\sin x}{\cos x-\sin x}\right\}$$ then $$\dfrac{dy}{dx}=?$$

A
$$1$$

B
$$-1$$

C
$$\dfrac{1}{2}$$

D
$$\dfrac{-1}{2}$$

Solution
Question 10
1 / -0

Which of the following statement is always true ([ .] represents the greatest integer function)

A
If f(x) is discontinuous, then $$ \left | f(x) \right | $$ is discontinuous

B
If f(x) is discontinuous , then f $$ \left ( \left | x \right | \right ) $$ is discontinuous

C
f(x) = $$ \left [g \left (x \right ) \right ] $$ is discontinuous when g(x) is an integer

D
None of these