Maharashtra Board Class 12 Maths Part 2 Chapter 2 Applications Of Derivatives PDF Download

Applications Of Derivatives

Let Us Study

Applications of Derivatives to Tangents and Normals

Derivative as a rate measure

Approximations

Rolle's Theorem and Lagrange's Mean Value Theorem

Increasing and Decreasing Functions

Maxima and Minima

Let Us Recall

Continuous functions.

Derivatives of Composite, Inverse Trigonometric, Logarithmic, Parametric functions.

Relation between derivative and slope.

Higher Order Derivatives.

2.1.1 Introduction

In the previous chapter we have studied the derivatives of various functions such as composite functions, Inverse Trigonometric functions, Logarithmic functions etc. and also the relation between Derivative and slope of the tangent. In this chapter we are going to study various applications of differentiation such as application to Geometry, Rate measure, Approximations, Rolle's Theorem and Lagrange's Mean Value Theorem, Increasing and Decreasing functions and Maxima and Minima.

Let Us Learn

2.1.2 Application of Derivative in Geometry

In the previous chapter we have studied the relation between derivative and slope of a line or slope of a tangent to the curve at a given point on it.

Let \(y = f(x)\) be a continuous function of \(x\) representing a curve in XY-plane and \(P(x_1, y_1)\) be any point on the curve.

Then \(\left[\frac{dy}{dx}\right]_{(x_1, y_1)} = [f'(x)]_{x_1, y_1}\) represents slope, also called gradient, of the tangent to the curve at \(P(x_1, y_1)\). The normal is perpendicular to the tangent. Hence, the slope of the normal at P will be the negative of reciprocal of the slope of tangent at P. Let m and m' be the slopes of tangent and normal respectively,

Teacher's Note

Tangent means a line that touches the curve at one point. In India, we see this in road design - a curved road connects to a straight road smoothly, like a tangent line.

Exam Trick

Remember: If slope of tangent is m, then slope of normal is -1/m. They are negative reciprocals, just like opposite directions.

Points to Remember

Derivative at a point gives the slope of the tangent at that point.

Slope of normal is negative reciprocal of slope of tangent.

Equation of tangent: \(y - y_1 = m(x - x_1)\) where m is slope.

Equation of normal: \(y - y_1 = m'(x - x_1)\) where m' is slope of normal.

This is a preview of the first 3 pages. To get the complete book, click below.