§5.1 Constructing accurate graphs of antiderivatives (continued)

Summary

A moving particle has its velocity given by the quadratic function $v$ pictured in Figure 5.1. In addition, it is given that $A_1 = \frac{7}{6}$ and $A_2 = \frac{8}{3}$, as well as that for the corresponding position function $s$, $s(0) = 0.5$.

• Use the given information to determine $s(1)$, $s(3)$, $s(5)$, and $s(6)$. What do these values mean in the context of the moving particle? - On what interval(s) is $s$ increasing? That is, when is the particle moving forward? On what interval(s) is $s$ decreasing? That is, when is the particle moving backward? - On what interval(s) is $s$ concave up? That is, when is the particle accelerating? On what interval(s) is $s$ concave down? That is, when is the particle decelerating? - Sketch an accurate, labeled graph of $s$ on the axes at right in Figure 5.1. - Note that $v(t) = -2 + \frac{1}{2}(t-3)^2$. Find a formula for $s$.

A person exercising on a treadmill experiences different levels of resistance and thus burns calories at different rates, depending on the treadmill's setting. In a particular workout, the rate at which a person is burning calories is given by the piecewise constant function $c$ pictured in Figure 5.1. Note that the units on $c$ are “calories per minute.”

• Let $C$ be an antiderivative of $c$. What does the function $C$ measure? What are its units? - Assume that $C(0) = 0$. Determine the exact value of $C(t)$ at the values $t = 5, 10, 15, 20, 25, 30$. - Sketch an accurate graph of $C$ on the axes provided at right in Figure 5.1. Be certain to label the scale on the vertical axis. - Determine a formula for $C$ that does not involve an integral and is valid for $5 \le t \le 10$.

Consider the piecewise linear function $f$ given in Figure 5.1. Let the functions $A$, $B$, and $C$ be defined by the rules $A(x) = \int_{-1}^{x} f(t) \, dt$, $B(x) = \int_{0}^{x} f(t) \, dt$, and $C(x) = \int_{1}^{x} f(t) \, dt$.

• For the values $x = -1, 0, 1, \ldots, 6$, make a table that lists corresponding values of $A(x)$, $B(x)$, and $C(x)$. - On the axes provided in Figure 5.1, sketch the graphs of $A$, $B$, and $C$. - How are the graphs of $A$, $B$, and $C$ related? - How would you best describe the relationship between the function $A$ and the function $f$?