Graphs and Derivatives

   On the following page, we will look at graphs and derivatives. We will get some practice in sketching gradient functions and we will carefully consider stationary points (maximum, minimum and points of inflexion) as well as non-stationary points of inflexion.


Key Concepts

On this page, you should learn about

  • graphs of \(f,f'\ \mathrm{and} \ f''\)
  • local maximum and minimum points
  • stationary points of inflexion
  • non-stationary points of inflexion

Essentials

The following videos will help you understand all the concepts from this page

Sketching the Gradient Function

In the following video we shall see how to sketch graphs of a gradient function

You can get some practice of sketching gradient functions by trying the activity below. This works best if you have a device with a tactile screen so that you can sketch the graphs on screen. If this is not the case, you can print a copy from here and draw with a pencil.

  • Choose the pen tool and sketch the gradient function
  • Check your answer by clicking in the check box
  • Turn off the answer and delete your sketch before moving on to the next question

Stationary Points

In the following section we consider stationary points, that is, points on a graph when the gradient = 0.

There are three different types of stationary points

  1. Local Maxima
  2. Local Minima
  3. Point of Inflexion

Local Maxima and Local Minima

Drag the point and explore the gradient of this function

 

Stationary Point of Inflexion

Drag the point and explore the gradient of this function

In the following video, we shall summarize the main features of graphs with stationary points and how we use the gradient function (and the gradient of the gradient function) to determine the nature of the stationary points.

Stationary Points Example

In the following video, we will look at an example in which we are required to find the coordinates of stationary points

Find the co-ordinates of the stationary points on the curve \(y = x^4 – 4x^3\) and determine their nature. Sketch the curve.

Notes from the video

Non-stationary Points of Inflexion

Non-stationary points of inflexion exist where \(\frac{d^2y}{dx^2}=0\) and \(\frac { dy }{ dx } \neq 0\)

In the following video, we see why this is the case and we look at the following example in which we are required to find the coordinates of a point of inflexion.


Find the coordinates of the point of inflexion on the curve \(y=x^3-6x^2+13x-9\)

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

The graph of y = f(x) is shown below, where B is a local maximum and C is a local minimum

Sketch a graph of y = f'(x), clearly showing the images of the points B and C labellling them B' and C' respectively

Hint

Full Solution

 

Question 2

A function is given by \(f(x)=-x^3+6x^2+4\)

a) Find the coordinates of any stationary points and describe their nature

b) Determine the values of x such that f(x) is a increasing function

c) Find the coordinates of the point of inflexion

Hint

Full Solution

 

Question 3

The following diagram shows the graph of \(f'\), the derivative of f

On the graph below, sketch the graph of y = f(x) given that f(0) = 0. Mark the images of A , B and C labelling them A' , B' and C'.

Hint

Full Solution

 

Question 4

Consider the function \(f(x)=-x^3-3x^2+9x\)

a) Find the coordinates of any stationary points and determine their nature

b) Find the equation of the straight line that passes through both the local maximum and the local minimum points.

c) Show that the point of inflexion lies on this line.

Hint

Full Solution

 

MY PROGRESS

How much of Graphs and Derivatives have you understood?