Chapter 3.4, Problem 26E

a.

To determine

To graph: the truck’s velocity v=ds/dt for $0\le t\le 15.$ Then repeat the process, with the velocity curve, to graph the truck’s acceleration dv/dt.

Answer to Problem 26E

Explanation of Solution

Given information: The graph shows the position s of truck traveling on a highway. The Truck starts at t =0 and return 15 hours later at t =15.

  

Calculation:

The velocity function v (in dark blue) is the derivative of the position function is s. The derivative at point t gives the slope of the tangent line to the original function at point t.

  

Starting at t = 0. The tangent to s is horizontal so v=0. Following along the curve of s up to about t = 5, the slope of the tangent lines are increasing, so v increases in the same interval.

After t =5, the slope of the tangent lines starts to decrease until it gets horizontal again at t =10. So from t =5 to t = 10, v decreases and goes back to 0.

After t = 10, the slope of the tangent lines becomes increasingly negative all the way through t = 15. So after t = 10, v is increasingly negative.

  

Acceleration a (in red) is the derivative of velocity because the derivative is just the slope of the tangent lines. Before t = 5, the slope of the tangent to v are positive but decreasing to 0. So a is positive and decreasing to 0 at t = 5.

After t = 5. The slope of the tangent to v is negative and continues to decrease at the same rate.

So a is negative and continues in the same direction.

b.

To determine

To graph: ds/dt and $d^{2}s/d^{2}t$ , and compare graphs with in part (a) where $s=15t^2-t^3.$

Answer to Problem 26E

  $\begin{array}{l}\frac{ds}{dt}=30t-3t^2.\\ \frac{d^{2}s}{d^{2}t}=30-6t.\end{array}$

Explanation of Solution

Given information: The graph shows the position s of truck traveling on a highway. The Truck starts at t =0 and return 15 hours later at t =15.

  

Calculation:

Given s, so find the first and second derivative using power rule.

  $\begin{array}{l}s=15t^2-t^3.\\ \frac{ds}{dt}=30t-3t^2.\\ \frac{d^{2}s}{d^{2}t}=30-6t.\end{array}$

The graph of s, v=ds/dt and $a=d^{2}s/d^{2}t$ are shown below.

  

For all of them the horizontal axis is t in hours, but for v the vertical axis is km per hours and for a if is km/ $hou{r}^2$.