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

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Utility: Function Evaluator & Grapher
Español 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