Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
For each function \(f\left(x\right)\) below, find \(f’\left(x\right)\):
a) \(x^{3}+\frac{2}{x}-4 \)
b) \(2\sqrt{x}+x\)
c) \(-\frac{2}{\sqrt{x}}\)
d) \(3x^{2}-x\sqrt{x}\)
e) \(\frac{3}{x^{3}\sqrt{x}}\)
f) \(2x-\frac{4}{3x^{2}}\)
Solve the following:
a) \(3x^{2}+5x-2=0\)
b) \(3x^{2}-12x=0\)
c) \(4x^{2}-25=0\)
d) \(2x-10\lt 0\)
e) \(x^{2}-10x+16\ge 0\)
f) \(9-x^{2}\lt 0\)
Solutions
Get Ready
Questions
Imagine a rollercoaster. When it gets to the very peak of the climb, at that split second, is it moving up or down?
Notes
\(f’\left(x\right)\) is a function so we can input values of \(x\) to get an output.
At \(x=c\), \(f’\left(c\right)\) tells us:
the gradient of \(y=f\left(x\right)\) at \(x=c\)
the rate of change of \(y\) with respect to \(x\) at \(x=c\)
When using \(\frac{dy}{dx}\) notation, we write \(f’\left(c\right)\) as $$f’\left(c\right)=\left.\frac{dy}{dx}\right\rvert_{x=c}$$
If
\(f’\left(c\right)\gt 0\) then \(f\left(x\right)\) is increasing at \(x=c\);
\(f’\left(c\right)\lt 0\) then \(f\left(x\right)\) is decreasing at \(x=c\)
\(f’\left(c\right)= 0\) then \(f\left(x\right)\) is stationary at \(x=c\)
Examples and Your Turns
Example
Find the gradient of the graph \(y=4x^{3}\) at the point where \(x=2\). State whether the function is increasing, decreasing or stationary at this point.
Your Turn
Find the gradient of the graph \(y=x^{2}-\frac{4}{x}\) at the point where \(x=-2\). State whether the function is increasing, decreasing or stationary at this point.
Your Turn
Find the gradient of the graph \(y=3-\frac{2}{x^{2}}\) at the point where \(x=1\). State whether the function is increasing, decreasing or stationary at this point.
Your Turn
Find the gradient of the graph \(y=\sqrt{x}\) at the point where \(x=9\). State whether the function is increasing, decreasing or stationary at this point.
Practice
| Function | Derivative | \(x\)-coordinate | Gradient at Point | Nature |
|---|---|---|---|---|
| \(f(x) = 4x^3\) | \(f'(x) = 12x^2\) | \(x = 2\) | \(f'(2) = 12(2)^2 = 48\) | Increasing |
| \(f(x) = 4x^3\) | \(f'(x) = 12x^2\) ? |
\(x = -2\) | \(f'(-2) = 12(-2)^2 = 48\) ? |
Increasing ? |
| \(f(x) = 4x^3\) | \(f'(x) = 12x^2\) ? |
\(x = 0\) | \(f'(0) = 0\) ? |
Stationary ? |
| \(f(x) = 3x^4\) | \(f'(x) = 12x^3\) ? |
\(x = 2\) | \(f'(2) = 96\) ? |
Increasing ? |
| \(f(x) = 3x^4\) | \(f'(x) = 12x^3\) ? |
\(x = -2\) | \(f'(-2) = -96\) ? |
Decreasing ? |
| \(f(x) = 3x^4\) | \(f'(x) = 12x^3\) ? |
\(x = 0\) | \(f'(0) = 0\) ? |
Stationary ? |
| \(f(x) = 2x^2 - 3x + 7\) | \(f'(x) = 4x - 3\) ? |
\(x = -1\) | \(f'(-1) = -7\) ? |
Decreasing ? |
| \(f(x) = 5x - x^3\) | \(f'(x) = 5 - 3x^2\) ? |
\(x = 1\) | \(f'(1) = 2\) ? |
Increasing ? |
| \(f(x) = 5 - 3x - x^3\) | \(f'(x) = -3 - 3x^2\) ? |
\(x = 0\) | \(f'(0) = -3\) ? |
Decreasing ? |
| \(f(x) = 5 - 3x\) | \(f'(x) = -3\) ? |
\(x = 0\) | \(f'(0) = -3\) ? |
Decreasing ? |
| \(f(x) = 5 - 3x\) | \(f'(x) = -3\) ? |
\(x = 2\) | \(f'(2) = -3\) ? |
Decreasing ? |
| \(f(x) = \frac{8}{x^2}\) | \(f'(x) = -\frac{16}{x^3}\) ? |
\(x = 9\) | \(f'(9) = -\frac{16}{729}\) ? |
Decreasing ? |
| \(f(x) = 4x^3 + C\) ? |
\(f'(x) = 12x^2\) | \(x = -1\) | \(f'(-1) = 12\) ? |
Increasing ? |
| \(f(x) = 6x^2 + C\) ? |
\(f'(x) = 12x\) | \(x = -1\) | \(f'(-1) = -12\) ? |
Decreasing ? |
| \(f(x) = x^2 + 5x + 6\) | \(f'(x) = 2x + 5\) ? |
\(x = -2\) ? |
\(f'(a) = 1\) | Increasing ? |
| \(f(x) = -3x^2 - x + 2\) | \(f'(x) = -6x - 1\) ? |
\(x = 4\) ? |
\(f'(a) = -25\) | Decreasing ? |
| \(f(x) = x^2 + 6x - 1\) | \(f'(x) = 2x + 6\) ? |
\(x = -3\) ? |
\(f'(a) = 0\) | Stationary ? |
| \(f(x) = x^3 - 7x + 1\) | \(f'(x) = 3x^2 - 7\) ? |
\(x = \pm 2\) ? |
\(f'(a) = 5\) | Increasing ? |
| \(f(x) = \frac{1}{x^2}\) | \(f'(x) = -\frac{2}{x^3}\) ? |
\(x = -1\) ? |
\(f'(a) = 2\) | Increasing ? |
| \(f(x) = 2x^2 + \frac{1}{x}\) | \(f'(x) = 4x - \frac{1}{x^2}\) ? |
\(x = -1\) ? |
\(f'(a) = -5\) | Decreasing ? |
| \(f(x) = 2x^3 - 5x^2 + C\) ? |
\(f'(x) = 6x^2 - 10x\) | \(x = 2, -\frac{1}{3}\) ? |
\(f'(a) = 4\) | Increasing ? |
| \(f(x) = -2x^2 + 12x - 3\) | \(f'(x) = -4x + 12\) ? |
\(x = 3\) ? |
\(f'(3) = 0\) ? |
Stationary |
| \(f(x) = x^2 - 7x + 2\) | \(f'(x) = 2x - 7\) ? |
\(x = 3.5\) ? |
\(f'(3.5) = 0\) ? |
Stationary |
| \(f(x) = 4x^2 + \frac{1}{x}\) | \(f'(x) = 8x - \frac{1}{x^2}\) ? |
\(x = 0.5\) ? |
\(f'(0.5) = 0\) ? |
Stationary |
| \(f(x) = 2x^2 + 3x - 4\) | \(f'(x) = 4x + 3\) ? |
\(x = -0.75\) ? |
\(f'(-0.75) = 0\) ? |
Stationary |
| \(f(x) = 2x^2 + 3x - 4\) | \(f'(x) = 4x + 3\) ? |
\(x > -0.75\) ? |
\(f'(x) > 0\) ? |
Increasing |
| \(f(x) = 3x^2 - 12x + 1\) | \(f'(x) = 6x - 12\) ? |
\(x < 2\) ? |
\(f'(x) < 0\) ? |
Decreasing |
| \(f(x) = 2x^3 - 6x\) | \(f'(x) = 6x^2 - 6\) ? |
\(-1 < x < 1\) ? |
\(f'(x) < 0\) ? |
Decreasing |
| \(f(x) = x^4 - 4x^3\) | \(f'(x) = 4x^3 - 12x^2\) ? |
\(x > 3\) ? |
\(f'(x) > 0\) ? |
Increasing |
Notes
\(\frac{dy}{dx}\) is the derivative of \(y\) with respect to \(x\).
There is nothing special about the letters \(x, y\).
\(\frac{dV}{dt}\) is the derivative of \(V\) with respect to \(t\).
That is, it is the rate of change of \(V\) with respect to \(t\).
Your Turn
Given that \(a=\sqrt{S}\) find the rate of change of \(a\) when \(S=9\).
Your Turn
Given that \(T=p^{3}+2p\) find the rate of change of \(T\) when \(P=-2\).
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
If \(f’\left(x\right)=0\) does the function have to be at a peak at that moment?
What is the difference between “increasing” and “strictly increasing”?
Common Mistakes / Misconceptions
The biggest problem with this skill is not being able to solve inequalities correctly.
Connecting This to Other Skills
In order to access this skill you need to be able to Differentiate Polynomials (4.5), and also Solve Quadratic Equations (PK3) and Quadratic Inequalities (2.15). At times you will have to be able to solve other Equation (2.23) and Inequalities (2.24).
We will be building on these ideas in Stationary Points (4.14) and then applying the ideas in Optimisation (4.18) and Kinematics (5.15)
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?