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

1. What is the average velocity over the first $10$ seconds of the balloon’s flight? Over the first $5$ seconds?

   Hint (a):

   In the case of the helium balloon.

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

   Step 1(a):

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

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

   Step 2(a):

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

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

   Hint (b):

   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.

   Hint 2(b):

   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?

   Hint (c):

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