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## NCERT Class 12 Maths Textbook Chapter 6 With Answer PDF Free Download

### Chapter 5: Applications of Derivatives

#### 6.1 Introduction

In Chapter 5, we have learned how to find derivatives of composite functions, inverse trigonometric functions, implicit functions, exponential functions, and logarithmic functions.

In this chapter, we will study applications of the derivative in various disciplines, e.g., in engineering, science, social science, and many other fields.

For instance, we will learn how the derivative can be used (i) to determine the rate of change in quantities, (ii) to find the equations of tangent and normal to a curve at a point,

(iii) to find turning points on the graph of a function which in turn will help us to locate points at which the largest or smallest value (locally) of a function occurs.

We will also use derivatives to find intervals on which a function is increasing or decreasing. Finally, we use the derivative to find the approximate value of certain quantities.

Author | NCERT |

Language | English |

No. of Pages | 53 |

PDF Size | 3.8 MB |

Category | Mathematics |

Source/Credits | ncert.nic.in |

### NCERT Solutions Class 12 Maths Chapter 4 Applications of Derivatives

Question 1:

Find the rate of change of the area of a circle with respect to its radius *r* when

(a) *r* = 3 cm (b) *r* = 4 cm

ANSWER:

The area of a circle (*A)*with radius (*r*) is given by,

Now, the rate of change of the area with respect to its radius is given by,

- When
*r*= 3 cm,

Hence, the area of the circle is changing at the rate of 6π cm when its radius is 3 cm.

- When
*r*= 4 cm,

Hence, the area of the circle is changing at the rate of 8π cm when its radius is 4 cm.

Question 2:

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?

ANSWER:

Let *x* be the length of a side, *V* be the volume, and *s* be the surface area of the cube.

Then, *V* = *x*^{3} and *S* = 6*x*^{2} where *x* is a function of time *t*.

It is given that.

Then, by using the chain rule, we have:

∴

⇒

Thus, when *x* = 12 cm,

Hence, if the length of the edge of the cube is 12 cm, then the surface area is increasing at the rate of cm^{2}/s.

Question 3:

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.

ANSWER:

The area of a circle (*A)* with radius* (r*) is given by,

Now, the rate of change of area (*A*) with respect to time (*t)* is given by,

It is given that,

∴

Thus, when *r* = 10 cm,

Hence, the rate at which the area of the circle is increasing when the radius is 10 cm is 60π cm^{2}/s.

NCERT Class 12 Maths Textbook Chapter 6 With Answer PDF Free Download