Calculus

Application of Derivatives

Easy

core

In a sphere the rate of change of volume with respect to(w.r.t) time is

π times the rate of change of radius w.r.t. time.

Surface area times the rate of change of radius w.r.t. time

Surface area times the rate of change of diameter w.r.t. time

$\frac{1}{3}\pi$ times the rate of change of radius w.r.t. time

✅ Correct Option: 2

With $V=\frac{4}{3}\pi r^3$ , the chain rule gives $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$ .

Since $4\pi r^2$ is the surface area, the rate of change of volume equals the surface area times $\frac{dr}{dt}$ .

2022: 25 May Shift 1

Match List I with List II

LIST I	LIST II
A. Maximum value of $f(x) = -\lvert x+1 \rvert + 3$	I. 6
B. Minimum value of $f(x) = (2x-1)^2 + 5$	II. 5
C. Maximum value of $f(x) = 6 - x^2$	III. no maximum value
D. Maximum value of $f(x) = x^3 + 1$	IV. 3

Choose the correct answer from the options given below:

2025: 22 May Shift 2

The curve $x = y^2$ and $xy = k$ cut orthogonally, then $k^2$ is equal to:

2023: 22 May Shift 3

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to :

2023: 15 June Shift 2

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$ . Then C is :

2026: 14 May Shift 2

The maximum value of the function $f(x)=\sin x(1+\cos x)$ in the internal $[0,\pi]$ is

2025: 14 May Shift 2

Match List-I with List-II

The function $f(x) = 2x^3 - 15x^2 + 36x + 5$ for $x \in [2,5]$ has

Choose the correct answer from the options given below:

Solution