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## Definition of Derivative. Example Problem 1

Order the slopes of the tangent lines at the given points for the graph below.

[Graph]

Let's consider a function f. We'll plot the function on the standard coordinate plane and

consider the slopes of its tangent lines at various points.

Basically, we are going to order the slopes of the lines tangent to the graph of fat given

points from least to greatest.

Of course, one can be negative and one can be positive.

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But it is best to start drawing

tangent lines. When choosing between two options, it is best to choose the one with

the greatest potential.

Let's look at the graph off and see where its slope is the steepest. The steepest slope

is at point D, so we'll put a dot there. Next, let's draw lines that are tangent to the graph

at A, B and C. Let's say we have a negative slope at A and B, but not at C; therefore C

will come third in our list.

Finally, since all four lines in our diagram have positive

slopes, D will come last in our list.

Now we are looking at the graph between B and C. You will notice that at C, the slope

of the line is zero (or horizontal). This means that C is the only point on the graph with

a horizontal tangent, which means that it must be point 3 on our number line. Next, we

see that B has a positive slope and no horizontal tangent point.

This means that B

must be point 2 on our number line. Finally, A has no slope, so it must be point 1 on

our number line. Therefore, the answer to this question is D, A. C. B.

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