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Read this section for an introduction to functions and their graphs. Work through practice problems 1-5.

Calculators and computers can help students, reporters, business people and scientific professionals create graphs quickly and easily, and because of this, graphs are being used more often than ever to present information and justify arguments. And this
text takes a distinctly graphical approach to the ideas and meaning of calculus. Calculators and computers can help us create graphs, but we need to be able to read them carefully. The next examples illustrate some types of information which can be
obtained by carefully reading and understanding graphs.

**Example 1: **A boat starts from St. Thomas and sails due west with the velocity shown in Fig. 7

(a) When is the boat traveling the fastest?

(b) What does a negative velocity away from St. Thomas mean?

(c) When is the boat the farthest from St. Thomas?

Solution: (a) The greatest speed is 10 mph at .

(b) It
means that the boat is heading back toward St. Thomas.

(c) The boat is farthest from St. Thomas at . For the boat's velocity is positive, and the distance from the boat to St. Thomas is increasing. For the boat's velocity is negative, and the distance from the boat to St. Thomas is decreasing.

**Practice 3: **You and a friend start out together and hike along the

same trail but walk at different speeds (Fig. 8).

(a) Who is walking faster at ?

(b) Who is ahead at ?

(c) When are you and your friend farthest apart?

(d) Who is ahead when ?

**Example 2: **In Fig. 9, which has the largest slope: the
line through the points and , the line through and , or the line through and ?

Solution: The line through C and P has the largest slope: .

**Practice 4:** In Fig. 10, the point on the curve is fixed, and the point is moving to the right along the curve toward
the point . As moves toward :

(a) the x–coordinate of is Increasing, Decreasing, Remaining constant, or None of these.

(b) the x–increment from to is Increasing, Decreasing, Remaining constant, or None of these

(c) the slope from to is Increasing, Decreasing, Remaining constant, or None of these.

**Example 3:** The graph of is shown in Fig. 11. Let be the slope of the line tangent to the graph of
at the point .

(a) Estimate the values , and .

(c) At what value(s) of is largest?

(d) Sketch the graph of .

Solution: (a) Fig. 11 shows the graph with several tangent lines to the graph of . From Fig. 11 we can estimate that (the slope of the line tangent to the graph of f at (1,0) )
is approximately equal to 1. Similarly, and .

(b) The slope of the tangent line appears to be horizontal () at and at .

(c) The tangent line to the graph appears to have greatest slope
(be steepest) near .

(d) We can build a table of values of and then sketch the graph of these values.

x | f(x) | g(x) = tangent slope at (x, f(x) ) |
---|---|---|

0 | -1 | .5 |

1 | 0 | 1 |

2 | 2 | 0 |

3 | 1 | -1 |

4 | 0 | -1 |

5 | -1 | 0 |

6 | -.5 | .5 |