Applications of derivatives - rate of change — practice questions

Question 1 · difficulty L2 · understanding

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after :

• A. 20 s
• B. 1 s
• C. 21 s
• D. 24 s

Question 2 · difficulty L3 · understanding

The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.

Question 3 · difficulty L3 · understanding

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ______________.

Question 4 · difficulty L3 · understanding

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 \pi} \mathrm{~cm} / \mathrm{min}$. The surface area (in $\mathrm{cm}^2$ ) of the chocolate ball (without the ice-cream layer) is :

• A. $128 \pi$
• B. $196 \pi$
• C. $225 \pi$
• D. $256 \pi$

Question 5 · difficulty L3 · understanding

Let $$f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$$, and $$g(x)$$ be a function such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. Then $$\frac{g(7)}{g^{\prime}(7)}$$ is equal to :

• A. 42
• B. 7
• C. 1
• D. 14

Question 6 · difficulty L3 · understanding

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

• A. 24
• B. 18
• C. 42
• D. 48

Question 7 · difficulty L3 · understanding

Water is being filled at the rate of 1 cm 3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm 2 / sec) at which the wet conical surface area of the vessel increases is

• A. 5
• B. $${{\sqrt {21} } \over 5}$$
• C. $${{\sqrt {26} } \over 5}$$
• D. $${{\sqrt {26} } \over {10}}$$

Question 8 · difficulty L3 · understanding

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

• A. 9
• B. 10
• C. 11
• D. 12

Question 9 · difficulty L3 · understanding

Let $$f(x)$$ be differentiable on the interval (0, $$\infty$$) such that $$f(1)=1$$, and $$\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1$$ for each $$x > 0$$. Then $$f(x)$$ is

• A. $${1 \over {3x}} + {{2{x^2}} \over 3}$$
• B. $$- {1 \over {3x}} + {{4{x^2}} \over 3}$$
• C. $$- {1 \over x} + {2 \over {{x^2}}}$$
• D. $${1 \over x}$$

Question 10 · difficulty L4 · understanding

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4 th day is 30, then number of infected students on 8 th day will be __________.