Snow is starting to fall with a rate at any time after the start being inches per hour for in (i.e., the snow falls for hours- from noon until PM). There were already inches on the ground when the storm started (What does this statement say notation-wise?).

A natural question would be to ask how much snow fell during the storm (what is the calculus notation for this?). But because the rate is always changing, this is a difficult question to answer (Yet, we will eventually answer it!). Let’s take what we know about constant rates and amounts and use that to help us answer our question (i.e., once again, taking what we know and using it to find out something about what we don’t know).

(Side note: We will be saying “how much snow fell”, but what we’re really asking is how the depth of snow on the ground changed - with being the rate at time of how the depth is changing at time . In this problem, these likely will mean the same thing- Why?)

Assume the rate stays the same as it was at the start of the storm. How much snow fell? Is this a realistic estimate?
Now assume the rate is the same as it is at the start for the first two hours, then changes to what it is at PM for the final two hours. How much snow fell? Is this a realistic estimate? Is it likely to be better or worse than that of the first question?
Now assume the rate stays constant by the hour (i.e., it only changes on the hour to its rate at those times of noon, PM, PM, and PM). How much snow fell?
Now do the same, but it changes on the half-hour.
What would we need to do to find the exact amount that fell?
Graph and interpret what you did in terms of the curve (i.e., geometrically) for all the parts.
What would you need to do geometrically to find the exact amount that fell?
If you knew the exact amount that fell, what would you need to do to determine how much snow is on the ground? Also, write the notation for the amount of snow on the ground (think about the “side note” given at the end of the intro and fanfare).
2024-10-10 13:53:07