The Good Ol’ Days (Or “History Repeats Itself”)

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Our goal in this activity is not to find the net change in an “amount” or “position” function $f(x)$ , but rather to find $f(x)$ itself so that we can find specific amounts (or positions) at any instant of time.

But first, we go down memory lane….

(a): Way back when, in that galaxy (or middle/high school) far far away and long long ago, you may have been given this problem in algebra class: Find the equation of the line with slope $16$ . Then use the equation to find the point on that line when $x = 7$ . What were the answers to those questions?
(b): Perhaps at another time, you were asked to find the equation of the line that contains the point $(3, 83)$ . Then use the equation to find the point on that line for when $x = 7$ . What were the answers to those questions?
(c): What is wrong with both of the above problems? Why do they not geometrically make sense? In each case, what is the least amount of additional information that you would need to find the requested answers?
Let us return to our days gone by….
(d): You may recall Free Lance Freddy Kroger, who stalks…errr…stocks groceries. He brought some lunch money to work, which he started at noon today. Let’s assume he has $\$83$ at $3$ PM and is earning money at $\$16$ per hour. How much money will he have at $7$ PM? Solve this in two ways: One way without ever finding his lunch money and one way by finding his lunch money first. When you are done with that, predict how much money he’ll have at $x$ hours after noon. Call this function $f(x)$ , which is the amount of money Freddy has at precisely $x$ PM!
(e): OK youse guys, that’s enough of this fond reminiscing! It’s time to snap out of it and return to the cold hard reality of 2018! Here, in the future, our friend Freddy is still kicking around to annoy calculus students. He still brings lunch money to work, which he started at noon today. Assume again he has $\$83$ at $3$ PM. However, now he earns money at a variable rate that is dependent on the time after noon, which we’ll call $t$ . Freddy is earning money at $4t^3+e^t$ dollars per hour. We again want to know how much money he’ll have at $7$ PM. Solve this in two ways: One way without ever finding his lunch money and one way by finding his lunch money first. When you are done with that, predict how much money he’ll have at $x$ hours after noon. Call this function $f(x)$ , which is the amount of money Freddy has at precisely $x$ PM!

Let’s do the same thing (i.e., find $f(x)$ given $f'(x)$ ), only this time we’ll do it graphically. In each of the following, Given the graph of $f'(t)$ , sketch $3$ possible graphs for $f(t)$ .

(a)
(b)

2024-10-10 13:52:54