Tangents and Normals

Introduction

Derivatives play a significant role in mathematics. They refer to the rate of change of functions concerning variables. There is a particular branch of application of derivatives that is concerned with finding the tangents and normals to a given curve. 

This branch is of great significance. It helps to find the minima and maxima of a particular function. The same further analyses the directions of acceleration and velocity of a moving object. It also assists in finding the shortest distance between two curves and angles. Both tangents and normals have distinct properties. They help in the calculation of various curves and lines.

Tangents

A tangent is a straight line that touches the curve at a point without intersecting it. Its slope is equivalent to the derivative or gradient of the curve of that point. This definition also explains how to find the equation of the tangent to the curve at a particular point. So, if a function is y = f(x), the equation of the tangent to the curve is x = x0. Here is how you can find the equation:

1. Find out the derivative or gradient of the curve at the point x = x0. For this, you need to calculate dy/dx⌋x=x0. Here, the value is termed m, in analogy to the straight line slope.
2. Find out the straight line equation that passes through the point (x0, y(x0)) with slope m. For this, you need to calculate y – y1/x – x1 = m.

Properties of Tangents

1. Tangents touch the curve at the point of contact.
2. If the tangent to a curve is y = f(x), it makes the angle θ along with the x-axis. That means, dy/dx = slope of the tangent = tan θ.
3. If the slope of a tangent is zero, then tan θ = 0. Therefore, θ = 0. It implies that the line of the tangent is parallel to the x-axis.
4. If θ = π/2, then tan θ = ∞. It means that the line of the tangent is perpendicular to the x-axis.

Applications of Tangents

1. Suppose you are travelling by car around the corner. Suddenly, you decide to perform a drift. As a result, your car begins to skid. So, that will continue in a tangent direction to the curve.
2. Suppose you are holding a stone and swinging it in a circular motion. Once you let it go, the way it flies is tangent to the circular motion.

Normals

A normal refers to a line exactly perpendicular to the tangent. Suppose the given slope is n, and the tangent slope at that point is m. The same value also applies to the value of the derivative or gradient at that point. As a result, we get m × n = -1. So, the following are the steps to find the normal to a particular curve y = f(x) at a point x=x0.

1. Find out the derivative or gradient of the curve at the point x = x0. This step is equivalent to finding the equation to the tangent of a curve. It means m = dy/dx|x=x0.
2. Find the slope n of the normal. We know that the normal is once and for all perpendicular to the tangent. That means n = −1/m.
3. Find the equation of the straight line that passes through the (x0, y(x0)) with slope n. So, the equation is y – y1/x – x1= n.

Properties of Normals

1. A normal line may be present at any circle point. Moreover, it will always pass through the centre of that particular circle.
2. The normal to a curve is perpendicular to the tangent at any point on the given curve.

Applications of Normals

1. A centripetal force is there to act on the human body. It always moves in a circle. The same is normal to the point of contact at a time.
2. The spokes in a particular wheel are normal to the wheel’s rim at every point. It happens at a point when the spoke connects with the centre.

Difference between Tangents and Normals

Although there is a similarity in equations, tangents and normals also have significant differences. Firstly, a tangent is a straight line whose extension always takes place from a particular point on the curve. Its gradient equals the curve’s gradient that exists at a particular point. In contrast, a normal is also a straight line whose extension always takes place from a curve’s point. It happens so that it is perpendicular to that very point’s tangent. Whenever we analyse the forces that act on a moving body, we need to find the tangents and normals to curves.

Conclusion

It would help if you used differentiation to calculate the equations of both tangents and normals to a particular curve. To do the same, you must ensure that the equation of the straight line in use passes through the point with coordinates and has a gradient. The above study material notes on tangents and normals explain the properties and applications of the straight lines in detail. 

It would help to find tangents and normals to curves while analysing the forces acting on a moving body. It would help if you remembered that the normal is always perpendicular to the tangent line, no matter their differences. You must also understand the applications of derivatives to calculate these equations.