There are many physical phenomena that can be modelled using curves. Not only does a curve give us a visual representation of the rise and fall of a variable's amount or size over time, but we can also investigate its slope to gain an understanding of how quickly that change is happening or, predict possible future change.
Some real life applications we may wish to investigate include:
- Temperature change during a season
- Velocity of a falling object at a particular time
- Current that passes through a circuit at a particular time
- Variation in stock market prices throughout the year
- Population growth in a town, city or whole country
Average and instantaneous rate of change
Let's consider the following graph, which shows the number of international students that have been studying in Australia over the last two decades.
We can see that the number of students rose from approximately $100000$100000 in 1994 to approximately $300000$300000 in 2003. Thus during the first decade, the number of students was increasing at an average rate of $20000$20000 per year (since the change in $y$y values divided by the change in $x$x values of the graph is $20000$20000).
Measuring the instantaneous rate of increase, however, will provide us with more detail in our investigation. We can see that in fact, the number of international students was quite stable in between 1996 and 2000. It then increased rapidly until 2003. An instantaneous rate will allow us to see when the biggest changes occurred; that is, when the rate of change was highest. To do this we have added a curve to the graph and simplified the bar chart.
To find the instantaneous rate requires tangents to be drawn at various points. The gradient of the tangent is the instantaneous rate of increase or decrease of a function at any time $x.$x. The gradient of the curve $y=f(x)$y=f(x) at any point is equivalent to the gradient of the tangent at that point and called the derivative, which is written mathematically as $f'(x)$f′(x).
Measuring the gradients at the marked points $A$A, $B$B and $C$C gives a table of values of the derivative:
| $x$x | $A$A | $B$B | $C$C |
|---|---|---|---|
| $f'(x)$f′(x) | $10000$10000 | $80000$80000 | $50000$50000 |
Later in this chapter, we will learn mathematical methods to determine the gradient of a function at any single point, giving us enough information to determine the equation of the tangent. For now, it is important that we become familiar with the language and notation of the gradient and the derivative.
The derivative $f'(x)$f′(x) is the gradient of the tangent to $y=f(x)$y=f(x) at each point on the curve.
Increasing, decreasing and stationary at a point
We will now learn about some language associated with the gradient of a curve.
At a point where a curve is sloping upwards, the tangent has a positive gradient, and $y$y is increasing as $x$x increases.
At a point where it is sloping downwards, the tangent has a negative gradient, and $y$y is decreasing as $x$x increases.
At a point where the tangent is horizontal, that is, its slope is $0$0, the curve is considered to be stationary.
Let $f(x)$f(x) be a function defined at $x=a$x=a. Then:
| If $f'(a)>0$f′(a)>0, then $f(x)$f(x) is called increasing at $x=a$x=a |
| If $f'(a)<0$f′(a)<0, then $f(x)$f(x)is called decreasing at $x=a$x=a |
| If $f'(a)=0$f′(a)=0, then $f(x)$f(x) is called stationary at $x=a$x=a |
For example, the curve in the diagram below is:
- increasing at $A$A, $G$G and $I$I
- decreasing at $C$C and $E$E
- stationary at $B$B, $D$D, $F$F and $H$H.
Practice questions
question 1
Consider the graph of the quartic function, $f\left(x\right)$f(x), graphed below.
What are the $x$x-values of the stationary points of $f\left(x\right)$f(x)?
Write all of the values on the same line separated by commas.
What are the regions of the domain where $f\left(x\right)$f(x) is increasing?
Write all of the regions in interval notation separated by commas.
What are the regions of the domain where $f\left(x\right)$f(x) is decreasing?
Write all of the regions in interval notation separated by commas.
question 2
What is the gradient of the tangent at the given point?
question 3
Consider the graph of $y=x$y=x below.
What is the gradient of the line at $x=4$x=4?
What is the gradient at any value of $x$x?
Which of the following is a true statement?
If $f\left(x\right)$f(x) is a linear function, the derivative $f'\left(x\right)$f′(x) depends on the value of $x$x.
AA linear function has a constant gradient.
BThe gradient of a linear function is always $1$1.
C