CBSE - Class 11 Mathematics Limits and Derivatives Worksheet
1.
Differentiate each of the functions w. r. to x in Exercises 29 to 42. \n $ (x + \frac{1}{x})^3 $
2.
If $y = \frac{1-x^2/2}{1+x^2/2}$, then $\frac{dy}{dx}$ is
a.
$\frac{-4x}{(1-x^2)^2}$
b.
$\frac{-4x}{1-x^4}$
c.
$\frac{1-x^2}{4x}$
d.
$\frac{-4x}{1-x^2}$
3.
Differentiate each of the functions w. r. to x in Exercises 29 to 42. \n $ \frac{3x+4}{5x^2 - 7x + 9} $
4.
Evaluate each of the following limits in Exercises 47 to 53. \n $lim_{x->\pi} \frac{1-sin(\frac{x}{2})}{cos(\frac{x}{2})(cos(\frac{x}{4}) - sin(\frac{x}{4}))}$
5.
If $f(x) = \frac{x^n - a^n}{x-a}$ for some constant ‘a’, then $f'(a)$ is
a.
1
b.
0
c.
does not exist
d.
$\frac{1}{2}$
6.
Differentiate each of the functions w. r. to x in Exercises 29 to 42. \n $ \frac{x^4 + x^3 + x^2 + 1}{x} $
7.
Let $ f(x) = \begin{cases} \frac{k \cos(x)}{\pi - 2x} & \text{when } x \neq \frac{\pi}{2} \\ 3 & \text{when } x = \frac{\pi}{2} \end{cases} $ and if $lim_{x->\frac{\pi}{2}} f(x) = f(\frac{\pi}{2})$, find the value of k.
8.
$lim_{x->0} \frac{1-cos(4\theta)}{1-cos(6\theta)}$ is
a.
$\frac{4}{9}$
b.
$\frac{1}{2}$
c.
–$\frac{1}{2}$
d.
–1
9.
Evaluate : $lim_{x->a} \frac{sin(3x)}{sin(7x)}$
10.
If $lim_{x->1} \frac{x^4-1}{x-1} = lim_{x->k} \frac{x^3 - k^3}{x^2 - k^2}$, then find the value of k.
11.
If $y = \frac{sin(x+9)}{cos(x)}$, then $\frac{dy}{dx}$ at x = 0 is
a.
cos 9
b.
sin 9
c.
0
d.
1
12.
Differentiate each of the functions w. r. to x in Exercises 29 to 42. \n $ \frac{x^5 - cos(x)}{sin(x)} $
13.
$lim_{x->1} \frac{x^m-1}{x^n-1}$ is
a.
1
b.
$\frac{m}{n}$
c.
–$\frac{m}{n}$
d.
$\frac{m^2}{n^2}$
14.
Find 'n', if $lim_{x->2} \frac{x^n - 2^n}{x-2} = 80, n \in N$
15.
Let $ f(x) = \begin{cases} x+2 & x \leq -1 \\ cx^2 & x > -1 \end{cases} $, find ‘c’ if $lim_{x->-1} f(x)$ exists.
16.
Differentiate each of the functions w. r. to x in Exercises 29 to 42. \n $ \frac{x^2 cos(\frac{\pi}{4})}{sin(x)} $
17.
$lim_{x->\pi} \frac{sin(x)}{x-\pi}$ is
a.
1
b.
2
c.
–1
d.
–2
18.
Fill in the blanks in Exercises 77 to 80. \n $lim_{x->0} \frac{sin(mx)}{tan^3(2x)} = 2$, then m = ______________
