CBSE - Class 12 Mathematics Application of Derivatives Worksheet
1.
Show that the function given by $f(x) = \sin x$ is (c) neither increasing nor decreasing in $(0, \pi)$
2.
On which of the following intervals is the function $f$ given by $f(x) = x^{100} + \sin x –1$ decreasing ?
a.
$(0,1)$
b.
$(\frac{\pi}{2}, \pi)$
c.
$(0, \frac{\pi}{2})$
d.
None of these
3.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iii) $f(x) = 4x - \frac{1}{2}x^2, x \in [-2, \frac{9}{2}]$
4.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (ii) $g(x) = x^3 – 3x$
5.
The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is
a.
$(2\sqrt{2},4)$
b.
$(2\sqrt{2},0)$
c.
$(0, 0)$
d.
$(2, 2)$
6.
A particle moves along the curve $6y = x^3 +2$. Find the points on the curve at which the $y$-coordinate is changing $8$ times as fast as the $x$-coordinate.
7.
The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. The marginal revenue, when $x = 15$ is
a.
116
b.
96
c.
90
d.
126
8.
What is the maximum value of the function $\sin x + \cos x$?
9.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iv) $f(x) = \sin x – \cos x, 0 < x < 2\pi$
10.
Find the maximum value of $2x^3 – 24x + 107$ in the interval $[1, 3]$. Find the maximum value of the same function in $[–3, –1]$.
11.
Find the intervals in which the following functions are strictly increasing or decreasing: (b) $10 – 6x – 2x^2$
12.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (iv) $f(x) = (x-1)^2 + 3, x \in [-3, 1]$
13.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (iii) $h(x) = \sin x + \cos x, 0 < x < \frac{\pi}{2}$
14.
The interval in which $y = x^2 e^{–x}$ is increasing is
a.
(– \infty, \infty)
b.
(– 2, 0)
c.
(2, \infty)
d.
(0, 2)
15.
The rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is
a.
10$\pi$
b.
12$\pi$
c.
8$\pi$
d.
11$\pi$
16.
A wire of length $28$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
17.
The radius of a circle is increasing uniformly at the rate of $3$ cm/s. Find the rate at which the area of the circle is increasing when the radius is $10$ cm.
18.
Prove that the logarithmic function is increasing on $(0, \infty)$.
19.
The length $x$ of a rectangle is decreasing at the rate of $5$ cm/minute and the width $y$ is increasing at the rate of $4$ cm/minute. When $x = 8$cm and $y = 6$cm, find the rates of change of (b) the area of the rectangle.
20.
The volume of a cube is increasing at the rate of $8$ cm$^3$/s. How fast is the surface area increasing when the length of an edge is $12$ cm?
