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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-06-RAT-01
Rate of Change — Assignment
Chapter: Application of Derivatives
Topic: Derivative as a Rate of Change
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
If $A=\pi r^2$ and $\tfrac{dr}{dt}=2$, then $\tfrac{dA}{dt}$ at $r=3$ is:
  • A.$6\pi$
  • B.$12\pi$
  • C.$9\pi$
  • D.$3\pi$
2.
Marginal cost is:
  • A.$C(x)/x$
  • B.$C'(x)$
  • C.$\int C\,dx$
  • D.$xC(x)$
3.
For $V=\tfrac43\pi r^3$, $\dfrac{dV}{dr}=$
  • A.$4\pi r^2$
  • B.$2\pi r$
  • C.$\pi r^2$
  • D.$\tfrac43\pi r^2$
4.
A negative rate of change means the quantity is:
  • A.increasing
  • B.constant
  • C.decreasing
  • D.maximum
5.
Marginal revenue is:
  • A.$R(x)/x$
  • B.$\dfrac{dR}{dx}$
  • C.$\int R\,dx$
  • D.$R-C$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
The radius of a circle increases at $3$ cm/s. Find $\dfrac{dA}{dt}$ when $r=5$ cm.
7.
The side of a square grows at $2$ cm/s. Find $\dfrac{dA}{dt}$ when the side is $10$ cm.
8.
If $C(x)=x^2+4x$, find the marginal cost at $x=3$.
9.
If $R(x)=10x-x^2$, find the marginal revenue at $x=2$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
The volume of a sphere increases. Find $\dfrac{dV}{dt}$ when $r=2$ and $\dfrac{dr}{dt}=0.5$.
11.
The edge of a cube increases at $3$ cm/s. Find the rate of change of volume when the edge is $5$ cm.
12.
If $C(x)=0.005x^3-0.02x^2+30x$, find the marginal cost at $x=10$.
13.
The radius of a circle increases at $0.7$ cm/s. Find the rate of increase of its circumference.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
A spherical balloon's radius increases at $2$ cm/s. Find the rate of increase of its volume and surface area when $r=10$ cm.
15.
A man $2$ m tall walks away from a lamp post $6$ m high at $5$ km/h. Find the rate at which the length of his shadow increases.

Answer Key

Section A — Multiple Choice Questions
  1. (B) $12\pi$
  2. (B) $C'(x)$
  3. (A) $4\pi r^2$
  4. (C) decreasing
  5. (B) $\dfrac{dR}{dx}$
Section B — Short Answer (2 marks)
  1. $30\pi$ cm$^2$/s.
  2. $40$ cm$^2$/s.
  3. $10$.
  4. $6$.
Section C — Short Answer (3 marks)
  1. $8\pi$ cubic units/s.
  2. $225$ cm$^3$/s.
  3. $31.1$.
  4. $1.4\pi$ cm/s.
Section D — Long Answer (5 marks)
  1. $\dfrac{dV}{dt}=800\pi$ cm$^3$/s, $\dfrac{dS}{dt}=160\pi$ cm$^2$/s.
  2. $2.5$ km/h.
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