Measuring Change: Differnetiation

First Derivative test

• Turning points (set equal to zero) - put y back into original equation to find the y coordinate

• Turning points can also be stationary points

• Looking for a minimum and a maximum

• if f’ is larger than zero then there is a maximum

• If f’ is smaller that zero then there is minimum

How do you find a local maximum

Find the first derivate, if larger that zero then there is a local maximum

How do you find a local minimum

Find the first derivative, if smaller than zero then there is a local minimum

Global Minimum

The value of y is the smallest attained in the functions domain

Global maximum

The value of y is the greatest attained on the entire domain

Local maximum

A TURNING POINT where the gradient (slope) is positive

Local minimum

Where the TURNING POINT (going to be on a gradient) has a negative sign

(Test) The second derivative

• To find the points of inflection

• Find the maxima and minima

• Using concavity - determined on positive or negative to the direction of concavity

How do you know if your function is concave up (local minimum)

If f’’ is smaller that 0 then there is a local minimum (a minimum on a concavity)

How do you know if your function is concave down

f’’ will be larger than zero (you will have a local maximum)

What do you do if f’’ is = 0

Inconclusive - look the first derivative f’

Inflection points

Corresponds to a change in concavity - placed where there is a change in concavity

Concave down (from inflection point)

When f’’ is smaller than zero

Concave up (from inflection point)

When f’’ is larger than 0

Horizontal point of inflection

When f’ =0 the slope (gradient) is parallel to the x axis

Curve sketching

Label your graph:

• axes

• Intercepts (x) and y

• Turning points and points of inflection

• asymptotes

Optimization

Finding the Maximum or Minimum of something ( a shape) (they can ask for volume, area, perimeter, height and more)

Kinematics

The movement of objects you need to describe the potion

Kinematics - path

A set of points along which an object travels

Kinematics - Distance traveled

An object is the total length of the path followed by the object after it leaves the starting point (it can “re track” its steps, so some lengths may be doubled depending on a turn)

Kinematics - Displacment

An objects is the difference between the objects posits and a fixed point. The fixed point is called the origin, but an origin may not be the same thing as the starting point.