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Limits And Derivatives

Multiple Choice Questions

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1.
$\lim_{x \to 0} \frac{1 - \cos^{3} (4 x)}{x^{2}} = ?$

0

12

24

36

C.

24

$\lim_{x \to 0} \frac{1 - \cos^{3} (4 x)}{x^{2}} \frac{0}{0} Form Apply' L' Hospital Rule, = \lim_{x \to 0} \frac{- 3 \cos^{2} (4 x) (- \sin (4 x)) 4}{2 x} = \lim_{x \to 0} \frac{12 \cos^{2} (4 x) \sin (4 x)}{2 x} \frac{0}{0} Form, Again Apply L' Hospital Rule, = \lim_{x \to 0} \frac{12 [2 \cos (4 x) . (- \sin (4 x) 4) \sin (4 x) + \cos^{2} (4 x) \cos (4 x) . 4]}{2} = \frac{12 (0 + 1 \times 1 \times 4)}{2} = \frac{48}{2} = 24$

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2.

$If f (x) = \sin (\cos (x)), then f' (x) = ?$

$\cos (\cos (x))$
$\sin (- \sin (x))$
$(\sin (x)) \cos (\cos (x))$
$(- \sin (x)) \cos (\cos (x))$

3.

$What is the derivative of the function f (x) = e^{\tan (x)} + \ln (\sec (x)) - e^{\ln (x)} at x = \frac{π}{4} ?$

$\frac{e}{2}$
e
2e
4e

4.

$What is \lim_{x \to 0} \frac{\tan (x)}{\sin (2 x)} = ?$

$\frac{1}{2}$
1
2
Limit does not exist

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5.

$What is \lim_{h \to 0} \frac{\sqrt{2 x + 3 h} - \sqrt{2 x}}{2 h} = ?$

$\frac{1}{2 \sqrt{2} x}$
$\frac{3}{\sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$
$\frac{3}{2 \sqrt{2} x}$

6.

What is the value of $\log_{7} (\sqrt{7 \sqrt{7 \sqrt{7}}}) = ?$

$3 \log_{2} (7)$
$1 - 3 \log_{2} (7)$
$1 - 3 \log_{7} (2)$
$\frac{7}{8}$

7.

$If f (x) = \sqrt{25 - x^{2}}, then what is \lim_{x \to 1} \frac{f (x) - f (1)}{x - 1} = ?$

$- \frac{1}{\sqrt{24}}$
$\frac{1}{\sqrt{24}}$
$- \frac{1}{4 \sqrt{3}}$
$\frac{1}{4 \sqrt{3}}$

8.

$What is \lim_{x \to 0} \frac{\sqrt{1 - \cos (θ)}}{θ} = ?$

$\sqrt{2}$
$2 \sqrt{2}$
$\frac{1}{\sqrt{2}}$
$- \frac{1}{2 \sqrt{2}}$

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9.

$If y = {|\sin (x)|}^{|x|}, then what is the value of \frac{dy}{dx} at x = - \frac{π}{6} ?$

$\frac{2^{- \frac{π}{6}} (6 \ln (2) - \sqrt{3} π)}{6}$
$\frac{2^{\frac{π}{6}} (6 \ln (2) + \sqrt{3} π)}{6}$
$\frac{2^{- \frac{π}{6}} (6 \ln (2) + \sqrt{3} π)}{6}$
$\frac{2^{\frac{π}{6}} (6 \ln (2) - \sqrt{3} π)}{6}$

10.

$What is \frac{d}{dx} \sqrt{1 - \sin (2 x)} = ? where \frac{π}{4} < x < \frac{π}{2} ?$

$\cos (x) + \sin (x)$
$- (\cos (x) + \sin (x))$
$\pm (\cos (x) + \sin (x))$
None of the above

1
2

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