Starter

The line L passes through the points (0, 7) and (3, 19). Work out the equation of the line L.

Starter

The line L passes through the points (0, 7) and (3, 19). Work out the equation of the line L.Gradient =

Equation: y=4x+cPasses through (0,7) so y-intercept is 7Therefore the equation is y=4x+7

How would we have worked out c if we were not given the y-intercept?

Calculus - Differentiation

• Differentiation is a way of finding a gradient at a point on a curve.

• It is needed as curves have (by definition) a constantly changing gradient.

Differentiation

• Why and how differentiation works is not required knowledge for the Further Maths exam

• It will not be covered in this session (look up Differentiation from First Principles if you want some light summer reading)

Differentiation – How To…

OR:Multiply the whole thing by the power and reduce it by one

𝑦=𝑎𝑥𝑛❑⇒

𝑑𝑦𝑑𝑥=a 𝑥𝑛−1

Differentiation - Example

Differentiation – Try these

Differentiation – Try these

𝟓 𝒙𝟒

𝟏𝟐𝒙𝟑

+6x

Differentiation - Tangents

A tangent is a line that touches a curve at a single point.

The gradient of the tangent is equal to the gradient of the curve at that point.

Differentiation - Tangents

As a straight line the equation of the tangent is:

y=mx + c

This is equal to evaluated at the point P.𝒅𝒚𝒅𝒙

You will need to know a specific point on the line to find c.

Equations of Tangents

Find the equation of the tangent to , when x=2

Equations of Tangents

Find the equation of the tangent to , when x=2 when x=2 => y=7x+c

Then when x=2, So the line passes through (2 , 3)

Equations of Tangents

y=7x+c passing through (2,3)

So 3=7(2)+cc=-11

Equation is y=7x-11

Equations of Tangents

Complete the table (only the tangents side)

Tangents and Normals

• A normal is a line that is perpendicular to the tangent at a specific point.

• The gradient of a normal is the negative reciprocal of the tangent (-1/tangent gradient)

Tangents and NormalsTangent Gradient Normal Gradient

4 -1/4

-3 1/3

1/2 -2

-1/3 3

3/4 -4/3

-7/2 2/7

Examples of negative reciprocals

Once you have worked out the gradient, finding the equation is exactly the same as for the tangent

Tangents and Normals

Now complete the Normals side of the sheet