Block 1

Equations of Tangents

What is to be learned?

• How to use differentiation to find all the stuff we need to get equation of a tangent.

Differentiation gives us• Gradients of tangents to curves• Rates of change

Gradient of TangentCurve y = 4x2 – 6x Gradient of tangent at x = 5?

Find derivativedy/dx = 8x – 6

Substitute x value = 8(5) – 6

= 34

m

Gradient of TangentCurve y = x3 – 8x Gradient of tangent at x = 4?

Find derivativedy/dx = 3x2 – 8

Substitute x value = 3(4)2 – 8

= 40

m

tangent at x = 3?

For Equation needgradient mpoint (a , b)

3

y = x2

Equation of Tangent

y = x2

tangent at x = 3?

For gradient findThe Derivative

dy/dx = 2x

at x = 3

so m = 63

Equation of Tangentm = 6

so m = 2(3)

y = x2

tangent at x = 3?

Point?

(3 , ?)

Need y coordinate

3

Equation of Tangentm = 6

y = x2

tangent at x = 3?

Point?

at x = 3 (a , b) = (3 , 9)

Need y coordinate

3

Equation of Tangent

(3 , ?)

m = 6

y = 32

y = 9 9

y = x2

tangent at x = 3?

m = 6, (a , b) = (3 ,9) Need y

coordinateuse y – b = m(x – a)

3

Equation of Tangent

(3 , ?)

m = 6

9

(a , b) = (3 , 9)

Usually not given diagram!

Ex For curve y = x3 – 2x2 + 5Find equation of tangent at x = 3

Get mdy/dx = 3x2 – 4x

so at x = 3 m = 3(3)2 – 4(3)= 15

→ Need Derivative

Usually not given diagram!

Ex For curve y = x3 – 2x2 + 5Find equation of tangent at x = 3

Get pointso at x = 3 y = 33 – 2(3)2 + 5

= 14(a , b) = (3 , 14)Then y – b = m(x – a)

→ Use equation, y =

Equations of TangentsFor Equation need• gradient m• point (a , b)

Find DerivativeUse original equation

Then use y – b = m(x – a)

tangent at x = 4?

4

y = x2 – 6x

y = x2 – 6xtangent at x = 4?

dy/dx = 2x – 6at x = 4 m = 2(4) – 6

m = 2

4

For Gradient Find Derivative

y = x2 – 6xtangent at x = 4?

(4 , ?)

Need y coordinate

4

For Point Use Original Equation

at x = 4

(a , b) = (4 , -8)

y = 42 – 6(4)= -8

use y – b = m(x – a)

Key QuestionEx For curve y = x3 – 7x – 3

Find equation of tangent at x = 2

Get m dy/dx = 3x2 – 7

so at x = 2 m = 3(2)2 – 7= 5

Get pointso at x = 2 y = 23 – 7(2) – 3

= -9m = 5 (a , b) = (2 , -9)→ y + 9 = 5(x – 2) → y = 5x – 19