MA005: Calculus I

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 $t = 3 \; hours$.
(b) It means that the boat is heading back toward St. Thomas.
(c) The boat is farthest from St. Thomas at $t = 6 \; hours$. For $t < 6$ the boat's velocity is positive, and the distance from the boat to St. Thomas is increasing. For $t > 6$ 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 $t = 20$?
(b) Who is ahead at $t = 20$?
(c) When are you and your friend farthest apart?
(d) Who is ahead when $t = 50$?

Example 2: In Fig. 9, which has the largest slope: the line through the points $A$ and $P$, the line through $B$ and $P$, or the line through $C$ and $P$?

Solution: The line through C and P has the largest slope:  $m_{PC} > m_{PB} > m_{PA}$.

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

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

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

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


Example 3: The graph of $y = f(x)$ is shown in Fig. 11. Let $g(x)$ be the slope of the line tangent to the graph of $f(x)$ at the point $(x,f(x))$.

(a) Estimate the values $g(1)$, $g(2)$ and $g(3)$.

(b) When does $g(x) = 0)$?

(c) At what value(s) of $x$ is $g(x)$ largest?

(d) Sketch the graph of $y = g(x)$.

Solution: (a) Fig. 11 shows the graph $y = f(x)$ with several tangent lines to the graph of $f$. From Fig. 11 we can estimate that $g(1)$ (the slope of the line tangent to the graph of f at (1,0) ) is approximately equal to 1. Similarly, $g(2) ≈ 0$ and $g(3) ≈ –1$.
(b) The slope of the tangent line appears to be horizontal ($slope = 0$) at $x = 2$ and at $x = 5$.
(c) The tangent line to the graph appears to have greatest slope (be steepest) near $x = 1.5$.
(d) We can build a table of values of $g(x)$ and then sketch the graph of these values.