Tangents and Normals of Curves

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Tangents and Normals of Curves

If you differentiate the equation of a curve, you will get a formula

for the gradient of the curve. Before you learnt calculus, you would

have found the gradient of a curve by drawing a tangent to the curve

and measuring the gradient of this. This is because the gradient of a

curve at a point is equal to the gradient of the tangent at that

point.

Example:

Find the equation of the tangent to the curve y = x³ at the point (2,

8).

dy = 3x²

dx

Gradient of tangent when x = 2 is 3×2² = 12.

From the coordinate geometry section, the equation of the tangent is

therefore:

y - 8 = 12(x - 2)

so y = 12x - 16

You may also be asked to find the gradient of the normal to the curve.

The normal to the curve is the line perpendicular (at right angles) to

the tangent to the curve at that point.

Remember, if two lines are perpendicular, the product of their

gradients is -1.

So if the gradient of the tangent at the point (2, 8) of the curve y =

x³ is 8, the gradient of the normal is -1/8, since -1/8 × 8 = -1.

Integration

Introduction

Integration is the reverse of differentiation.

If y = 2x + 3, dy/dx = 2

If y = 2x + 5, dy/dx = 2

If y = 2x, dy/dx = 2

So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc.

For this reason, when we integrate, we have to add a constant. So the

integral of 2 is 2x + c, where c is a constant.

A 'S' shaped symbol is used to mean the integral of, and dx is written

at the end of the terms to be integrated, meaning 'with respect to x'.

This is the same 'dx' that appears in dy/dx .

To integrate a term, increase its power by 1 and divide by this

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