§4.1 Determining distance traveled from velocity

Introduction

Area under the graph of the velocity function

Two approaches: area and antidifferentiation

When velocity is negative

Suppose that a person is taking a walk along a long straight path and walks at a constant rate of 3 miles per hour.

On the left-hand axes provided in the following figure, sketch a labeled graph of the velocity function $v(t) = 3$. Note that while the scale on the two sets of axes is the same, the units on the right-hand axes differ from those on the left. The right-hand axes will be used in question (d).

Using the grid, graph, and given data appropriately, estimate the distance traveled by the walker during the two hour interval from $t = 0$ to $t = 2$. You should use time intervals of width $\Delta t = 0.5$, choosing a way to use the function consistently to determine the height of each rectangle in order to approximate distance traveled.

How could you get a better approximation of the distance traveled on $[0,2]$? Explain, and then find this new estimate.

Now suppose that you know that $v$ is given by $v(t) = 0.5t^3-1.5t^2+1.5t+1.5$. Remember that $v$ is the derivative of the walker's position function, $s$. Find a formula for $s$ so that $s' = v$.

Based on your work in (c), what is the value of $s(2) - s(0)$? What is the meaning of this quantity?

For what values of $t$ is the velocity of the ball positive? What does this tell you about the motion of the ball on this interval of time values?

Find an antiderivative, $s$, of $v$ that satisfies $s(0) = 0$.

Compute the value of $s(1) - s(\frac{1}{2})$. What is the meaning of the value you find?

Using the graph of $y = v(t)$ provided in the following figure, find the exact area of the region between the velocity curve and the $t$-axis between $t = \frac{1}{2}$ and $t = 1$. What is the meaning of the value you find?

Answer the same questions as in (c) and (d) but instead using the interval $[0,1]$.

What is the value of $s(2) - s(0)$? What does this result tell you about the flight of the ball? How is this value connected to the provided graph of $y = v(t)$? Explain.

Determine the total distance traveled and the total change in position on the time interval $0 \le t \le 2$. What is the object's position at $t = 2$?

On what time intervals is the moving object's position function increasing? Why? On what intervals is the object's position decreasing? Why?

What is the object's position at $t = 8$? How many total meters has it traveled to get to this point (including distance in both directions)? Is this different from the object's total change in position on $t = 0$ to $t = 8$?

Find the exact position of the object at $t = 1, 2, 3, \ldots, 8$ and use this data to sketch an accurate graph of $y = s(t)$ on the axes provided at right in the figure. How can you use the provided information about $y = v(t)$ to determine the concavity of $s$ on each relevant interval?