Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
a) Solve \(\left(e^{x}-1\right)\left(x-5\right)=0 \)
b) Solve \(3x^{2}-12=0\)
Solutions
Get Ready
Questions
Find the coordinates of the point on the graph \(y=2x^{2}-6x+1\) where the gradient of the tangent is equal to \(0\).
State the equation of the tangent and normal to the graph at this point.
Solutions
Notes
At the vertex of a parabola, the tangent is horizontal.
The gradient of the tangent is \(0\).
Points on a graph of a function with gradient 0 are known as Stationary Points.
We have that$$f’\left(a\right)=0 \iff y=f\left(x\right)\text{has a stationary point at }x=a$$
Examples and Your Turns
Example
Locate the vertex of the parabola \(y=x^{2}-6x+7\).
Your Turn
Find the coordinates of the stationary points of the curve \(y=x^{3}-6x^{2}+9x-1\).
Notes
There are three types of stationary point:
local maximum
local minimum
stationary points of inflection
Local maxima and local minima are also known as turning points. Can you explain why?
Global maxima and global minima are not always stationary points.
We can use sign diagrams to determine the nature of a stationary point.
| \(x\) | SP | ||
|---|---|---|---|
| \(\frac{dy}{dx}\) | \(-\) | \(0\) | \(+\) |
| \(\backslash\) | --- | \(/\) | |
| Min | |||
| \(x\) | SP | ||
|---|---|---|---|
| \(\frac{dy}{dx}\) | \(+\) | \(0\) | \(-\) |
| \(/\) | --- | \(\backslash\) | |
| Max | |||
| \(x\) | SP | ||
|---|---|---|---|
| \(\frac{dy}{dx}\) | \(-\) | \(0\) | \(-\) |
| \(\backslash\) | --- | \(\backslash\) | |
| PoI | |||
| \(x\) | SP | ||
|---|---|---|---|
| \(\frac{dy}{dx}\) | \(+\) | \(0\) | \(+\) |
| \(/\) | --- | \(/\) | |
| PoI | |||
Examples and Your Turns
Example
Find and classify the stationary points of the function \(y=x^{4}-4x^{3}+4x^{2}\).
Your Turn
Find and classify the stationary points of the function \(y=x^{4}+2x^{3}\).
Your Turn
Find and classify the stationary points of the function \(y=x^{2}e^{-x}\).
Your Turn
Find and classify the stationary points of the function \(y=\sin x +\cos x\) in the interval \(o\le x\le \pi\)..
Your Turn (HL)
Find the stationary points for \(x^{2}+xy+y^{2}=12\).
Your Turn
\(f\left(x\right)=2x^{3}+ax^{2}-24x+1\) has a local minimum at \(x=-4\). Find \(a\).
Your Turn
\(f\left(x\right)=x^{3}+ax+b\) has a stationary point at \(\left(-2,3\right)\).
a) Find the values of \(a\) and \(b\).
b) Find the coordinates and nature of all stationary points.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why do we call points where the gradient of the tangent is \(0\) Stationary Points?
Which stationary points are classified as turning points? Why?
If a function is always increasing (like \(y=e^{x}\)), can it have any stationary points?
Common Mistakes / Misconceptions
The most common mistake is to not check the nature of a stationary point. Remember, it could be a stationary point of inflection and not a maximum nor a minimum.
If you are asked to find the coordinate, make sure you give both the \(x\) and \(y\) values for the point. If it asks for the maximum value, it is the \(y\) coordinate they are looking for.
Connecting This to Other Skills
You could have to find the stationary points for any Functions (Unit 2) we have learned to differentiate, such as Polynomials (4.5), Exponentials and Logarithms (4.11) and Trigonometric Functions (4.12) as well as those requiring the Chain Rule (4.7), Product Rule (4.8), Quotient Rule (4.9) or even Implicit Differentiation (4.10).
In the next skill we will learn about the related idea of Concavity and Points of Inflection (4.15).
The main application of stationary points is Optimisation (4.18)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?