Differentiation Recap f(x) x The first derivative gives the ratio for which f(x) changes w.r.t. change in the x value.

F(x) – Non linear The closer the points are together the more accurate the approximation of the gradient For a Non linear function, we can not take just ANY two points If we wish to find the gradient at x=1 we must move the other point at (x=3) closer and closer to the point at x=1

The blue line is the approximation to the tangent x=3 moved to x=2 The approximation is more accurate then before The blue line is the approximation to the tangent

x=2 moved to x=1.5 Again the approximation is move accurate then before

x=1.5 moved to x=1.01 The approximation is now very accurate as the points are virtually coincident This red line through the point is called the TANGENT of f(x) at the point x=1 The tangent has the same gradient as the curve f(x) at the point in question and touches f(x) at x=1

The tangent has the same gradient as the curve f(x) at the point in question and touches f(x) at x=1 This red dashed line through the point is called the Normal of f(x) at the point x=1 It’s Perpendicular to the tangent

Definition of derivative We call this limit

The general rule is Some centuries ago Leibnitz and Isaac Newton Both independently applied this limit to many different functions And noticed a general pattern. This provided the basic rule of differentiation. And made the process very easy for polynomial functions The general rule is Otherwise written as

Increasing functions This function is increasing because Always +ve

Decreasing functions This function is decreasing because always -ve

Where the gradient is Zero In the above graphs the gradient passes through zero at x=0 we can write These points of zero gradient are very important in mathematics applications as at these points the rate of change of the function is zero

Consider this function The gradient is zero at these two points The tangents are horizontal The function does not change value when the gradient is zero And therefore with we can find local Maximum and Minimum values of functions

Importance Will tell us when a function is at a local maximum or minimum Can be used to find the maximum or minimum values in various questions involving rates of change But since BOTH Max and Min have gradient = 0 we need a way of distinguishing between the two

Function Max/Min We could plot the function and look at the graph But an easier way is to consider the 2nd derivative Consider the following function f(x) = 2x3-4x-4 f’(x) = 6x2-4 f’’(x)=12x

f’(x)=6x2-4 f’’(x)=12x f(x)=2x3-4x-4 f’(x)=0 f(x) decreasing as f’(x)<0 f ’’(x)<0 Concave up f(x) increasing as f’(x)>0 f ’’(x)>0 Concave down

First derivative: Second derivative: Curve is rising. is positive is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive Curve is concave up. (MIN) is negative Curve is concave down. (MAX)

Multiple choice Test

Question Show that the function is a decreasing function ANSWER

The diagram above shows part of the curve C with equation y = 9 - 2x - . The diagram above shows part of the curve C with equation y = 9 - 2x - The point A(1, 5) lies on C and the curve crosses the x-axis at B(b, 0), where b is a constant and b > 0. , (a) Verify that b = 4. (1) (1) The tangent to C at the point A cuts the x-axis at the point D, as shown in the diagram above. (b) Show that an equation of the tangent to C at A is y + x = 6. (4) (c) Find the coordinates of the point D. Find the Area of the ABD, assume it is a triangle

Answer (a) y = 9 – 2b - = 0 => b = 4 (c) Let y = 0 and x = 6 so D is (6, 0) (d) Area of shaded triangle is 5.0

Rates of Change The following function f(s) = 3t is shown below What does the gradient of the line represent ?

Rates of Change The following function f(s) = 3t is shown below What does the gradient of the line represent ?

The steeper the line the faster the velocity The Black line is fastest as it arrives at B the quickest The velocity is the Gradient

What about this graph? What does the gradient represent?

What does the gradient represent What about this graph? What does the gradient represent accelerating decelerating Constant Velocity

Summary We will need these formula for some rates of change Questions