Essential Knowledge 2.2A3

Essential Knowledge 2.2A3 Students will know that key features of the graphs off, f ', and f '' are related to one another.

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These questions are very popular on calculus exams. These questions really see if you know the meaning of a derivative. We can only learn part for now, and we'll learn the rest in the last Big Idea.

Example

If the function shown in the graph below is f(x), draw the graph of f'(x).

Since this curve is not a smooth curve, we will need to determine the derivative piece by piece

From A to B

The segment from A to B is a straight line. This makes finding the derivative simple, because straight lines have a constant slope. Since this line goes up 4 over 1, the slope is 4. This the derivative from A to B is 4.

Let's begin graphing the derivative.

From B to C

The segment from B to C is a horizontal line. Horizontal lines have a slope of 0, so f'(x) from B to C is 0. Let's add this to our graph.

From C to D

This segment has a slope of 1. Add it on.

(I realized I should have been using hollow points on the derivative all along, oops. This is important because derivatives do not exist at corners.)

From D to E

This one is a bit different. The curve is not linear, so the derivative will change. Fortunately they told us the function, so we can just differentiate using differentiation rules.

y = -(x-2)² + 5

y' = -2(x-2) = -2x + 4

So we can simply graph the line y = -2x + 4

And voila, we have our graph of f'(x). Here are a few key points to notice

• When f(x) rises, the derivative is positive. When f(x) falls, the derivative is negative.

• When f(x) is horizontal, the derivative is zero.

• When f(x) is linear, the derivative is a horizontal line segment

• When f(x) is curved, the derivative is not a horizontal line segment