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2-81.

A helium balloon is released from the ground and floats upward. The height of the balloon is shown at the following times:

 Time (s) $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ Height (feet) $0$ $50$ $98$ $144$ $188$ $230$ $270$ $308$ $344$ $378$ $410$
1. What is the average velocity over the first $10$ seconds of the balloon’s flight? Over the first $5$ seconds?

In the case of the helium balloon.

$\text{average velocity }=\frac{\Delta\text{height}}{\Delta\text{time}}$

average velocity between $t = 0$ and $t = 10$:

$\frac{h(10)-h(0)}{10-0}=\frac{410-0}{10-0}=41 \text{ ft/sec}$

Now find the average velocity between $t = 0$ and $t = 5$.

2. Calculate the finite differences for the heights. How is the velocity changing?

This is an example of a finite difference table.

Use the eTool below to explore this.
Click on the link below to view the full version of the eTool. Calc 2-81 HW eTool

 $\boldsymbol x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $\boldsymbol{f(x)}$ $9$ $4$ $1$ $0$ $1$ $4$ $9$ $\boldsymbol{\Delta{y}}$ $\underset{\LARGE{\smile}}{-5}$ The difference between 9 and 4 is minus 5          $\underset{\LARGE{\smile}}{-3}$  The difference between 4 and 1 is minus 3         $\underset{\LARGE{\smile}}{-1}$ The difference between 1 and 0 is minus 1.         $\underset{\LARGE{\smile}}{+1}$The difference between 0 and 1 is positive 1.           $\underset{\LARGE{\smile}}{+3}$ The difference between 1 and 4 is positive 3.           $\underset{\LARGE{\smile}}{+5}$ The difference between 4 and 9 is positive 5.

In this story, finite differences $(\Delta y)$ represent the average velocity over consecutive one second intervals.

3. What do the finite differences tell you about the height function for the balloon?

Is the height of the balloon increasing, decreasing or both? How do finite differences help you answer this?