## Graphing the Derivative miscellaneous on-line topics for Calculus Applied to the Real World

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Utility: Function Evaluator & Grapher

To begin, we recall two basic facts about the derivative f'(x) of a function f(x):

1. The value f'(a) of f'(x) at x = a is the slope of the tangent to the graph of the function f at the point where x = a.
2. f'(x) is a function of x: the slope at a point on the graph depends on the x-coordinate of that point.

The graph of the derivative function f'(x) gives us interesting information about the original function f(x). The following example shows us how to sketch the graph of f'(x) from a knowledge of the graph of f(x).

Example 1 Sketching the Graph of the Derivative

Let f(x) have the graph shown below.

Give a rough sketch of the graph of f'(x).

Solution

Remember that f'(x) is the slope of the tangent at the point (x, f(x)) on the graph of f. To sketch the graph of f', we make a table with several values of x (the corresponding points are shown on the graph) and rough estimates of the slope of the tangent f'(x).

 x 0 0.5 1 1.5 2 2.5 3 f'(x) 3 0 4 3 0 1 0

(Note that rough estimates are the best we can do; it is difficult to measure the slope of the tangent accurately without using a grid and a ruler, so we couldn't reasonably expect two people's estimates to agree. However, all that is asked for is a rough sketch of the derivative.) Plotting these points suggests the curve shown below.

Notice that the graph f'(x) intersects the x-axis at points that correspond to the high and low points on the graph of f(x). Why is this so?

Here is a more interactive example.

Example 2 Graph of Derivative

Let f(x) have the graph shown below.

Complete the following table, giving rough estimates of the slope of the tangent f'(x) at the given values of x.

 x 3 2 1 0 1 2 3 f'(x) Peek at Answers

Now plot these points, and hence make a rough sketch of the graph of f'(x). Which of the following best approximates your sketch of the graph of f'(x)? (click on one)

Last Updated:November, 1997