Differentiation

New Zealand

Level 7 - NCEA Level 2

Lesson

So what are the key features we may be interested in on a graph of a function?

- How steep is the function?
- Is it increasing or decreasing?
- Are there flat spots?
- Are there any local or global maximums or minimums?

The derivative function can help us with all of these. Let's explore how.

This applet shows you a number selection of curves. (linear, quadratic, cubic and simple inverse function of $\frac{1}{x}$1`x`. You also have the capacity to move a point along each of the curves, and as you do so you can see the tangent to the curve at that point.

Move the point to interesting parts of the graph and see if you can identify how the tangent (or maybe more specifically, the gradient of the tangent can help you identify these key features.

- The steepness of the curve
- Where it is increasing or decreasing?
- What happens at any flat spots?
- Are there any local or global maximums or minimums?

This next applet shows the same curves as we just explored above, as well as the plotted curve of the gradient function. This means that the value of the gradient function actually tells you the value of the gradient of the original function. Can you see any connections now? Look for the same features as above.

- The steepness of the curve
- Where it is increasing or decreasing?
- What happens at any flat spots?
- Are there any local or global maximums or minimums?

Hopefully were you able to find some of these connections. Let's see how you did.

We can tell how steep a function is by looking at the value of the derivative. The larger the value the steeper the original function is.

At this point on the curve, we can see it is fairly steep and the gradient value of 8.39 confirms this.

By looking at whether the gradient function is positive or negative we can tell if the original function is increasing or decreasing.

We can do that both algebraically using the gradient function, or by looking for values of the gradient function above or below zero on the graph of the derivative.

In this image you can see that I have shaded where the original function is increasing, this corresponds to where the gradient function is above the x axis - indicating that it has a positive value.

The function of the derivative is $f'(x)=\frac{3}{4}x^2-3x+\frac{5}{4}$`f`′(`x`)=34`x`2−3`x`+54. We can also identify algebraically where this is greater than zero by using inequalities, (or using your CAS).

**Are there flat spots?**

Flat spots on the curve occur where the gradient is $0$0, (this would create a horizontal tangent). We can find flat spots by solving the derivative. We do this the same way we solve any function, set it equal to $0$0 and solve it algebraically, or we look for the roots of the gradient function on the graph.

In the image on the left, we can see the horizontal tangent, meaning that the gradient of the function at that point is $0$0. In the image on the right, we can see that this same point, corresponds to the root of the derivative function (where it crosses the $x$`x`-axis).

Are there any local or global maximums or minimums?

A maximum or minimum on the graph (whether local or global) occur where the curve changes from increasing to decreasing. ( a maximum); or from decreasing to increasing (a minimum). Right at the point where it is neither positive or negative the gradient is zero. This means we solve the derivative (find the zeros of the gradient function) and we find a maximum or minimum. To determine which we would either need to check the graph, or check the signs either side - checking for whether it is a increasing-decreasing change or decreasing-increasing change. We do more work on Maximums and minimums a bit later here.

In this same image from before, we can see that the minimum value of the quadratic occurs at the point where the gradient has a value of zero.

Likewise in the maximum and minimum of this cubic are also at the same points as the zeros of the derivative function.

Consider the function $y=4x-3$`y`=4`x`−3.

Find the gradient function of $y=4x-3$

`y`=4`x`−3.Hence graph the gradient function.

Loading Graph...

Consider the function $y=\left(x-5\right)^2-3$`y`=(`x`−5)2−3 graphed below.

Loading Graph...

State the $x$

`x`-coordinate of the $x$`x`-intercept of the gradient function.For $x<5$

`x`<5, are the values of the gradient function above or below the $x$`x`-axis?Above

ABelow

BFor $x>5$

`x`>5, are the values of the gradient function above or below the $x$`x`-axis?Above

ABelow

B

Consider the function $y=-\left(x+7\right)^3-3$`y`=−(`x`+7)3−3 graphed below.

Loading Graph...

State the coordinates of the point of inflection.

Give your answer in the form $\left(a,b\right)$(

`a`,`b`).What is the gradient of the curve at $x=-7$

`x`=−7?Note: the gradient is an integer.

What feature of the gradient function does $x=-7$

`x`=−7 represent?the $x$

`x`-interceptAthe $y$

`y`-interceptBFor $x<-7$

`x`<−7, are the values of the gradient function above or below the $x$`x`-axis?Above

ABelow

BFor $x>-7$

`x`>−7, are the values of the gradient function above or below the $x$`x`-axis?Above

ABelow

BThe results from the previous parts indicate that $x=-7$

`x`=−7 also represents the $x$`x`-coordinate of:a maximum turning point of the gradient function

Aa minimum turning point of the gradient function

Ban inflection point of the gradient function

C

Sketch the graphs of functions and their gradient functions and describe the relationship between these graphs

Apply calculus methods in solving problems