## Average and Instantaneous Rate of Change

HELP

A man stands on a cliff and watches a hot air balloon (the balloon is far away, it's not just very small). You can click the play/pause button in the lower left-hand corner of the applet to watch the balloon rise and fall. Two important questions we'll probe:
1. What is the average rate of change of the balloon's height, between two different moments in time?
2. What is the instantaneous rate of change of the balloon's height, at one particular moment in time?

#### Average Rate of Ascent

1. Watch the animation and see how the movement of the balloon is related to the graph. Time moves at a steady rate, but the balloon rises and falls at different rates throughout its trip.
2. How would you describe those parts of the graph where the balloon is rising? falling?
3. What quality does the graph have at times when the balloon is moving quickly? slowly?
4. Stop the animation, uncheck the "Show balloon" checkbox, and check the "Visualize average rate of change" checkbox. The formula for average rate of ascent is given at the bottom of the applet, and you can move points on the `x`-axis to set two different times.
5. Is it clear when the average rate of ascent should be positive or negative?
6. Check the "Show secant line" checkbox. What is the relationship between the average rate of ascent between two times and the slope of the associated secant line?
7. Suppose we don't want just the average rate of ascent for the balloon between two different times, but instead we want to compute the instantaneous rate of ascent at exactly time = 10 minutes. Why can't we compute this by dragging the points for `t_1` and `t_2` to 10?

#### Instantaneous Rate of Ascent

Consider again the height function of the balloon.

1. Click the checkbox to show the secant line, and confirm that the average rate of change of the balloon is the slope of the secant line.
2. Click the checkbox to show the line tangent to the curve at time `t = t_1`. It's reasonable to believe (really?) that the slope of the tangent line is the instantaneous rate of change of the balloon height, because as you slowly drag `t_2` to meet `t_1` you see that the secant line is a better and better approximation to the tangent line.
3. Sadly, we can't compute the slope of the tangent line simply by using the same formula we did for the secant line (why not?). So here's the REALLY BIG IMPORTANT QUESTION FOR CALCULUS: How do we compute the slope of the tangent line?