Pedro Project Part 1 Solution

Hey guys! The problem I came up with is a review of the relationships between parent functions, first derivative functions and second derivative functions. I highly recommend that you keep the story page open in another tab/window so that you can refer to the graphs all the time. In this post, I will:

(1) Identify the parent graph, first derivative graph and second derivative graph.
(2) Eliminate the three irrelevant graphs.
(3) Confirm/justify the answers.

Remember when we first did one of those group in-class assignments? It was about Willy the Wallaby, I think. Mr. K said that it's good to have the answers, but it's just a minimal part of the education process. It's how we got to our answers that matter. SO, here's the solution =)

Solution:

Well to start with, we could look at the details of the story and find anything that stands out. For me, it's when the donkey exploded. When this incident occured, Pedro immediately came to a halt, and his velocity dropped to zero. We've dealt with graphs before where, at the end, they gradually plateau (e.g. figure 1), but for this problem, it shouldn't. Pedro and the donkey didn't slow down then screeched to a halt; the donkey just went boom in the middle of its gallop! So, graphs A, B, D or F can't be the position-time graph. This leaves C and E. Well, if you take a look at E, it's decreasing throughout. We know that Pedro starts the journey off by going in the positive direction, goes back, and dashes away on a positive position again, so E can't be it. Therefore, C is the position-time graph.

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It's been said that Pedro just abruptly stops, right? That means that his velocity just dropped to zero, and A, B, C or E can't be the velocity-time graph. Well, if you take a look at graph D, at the end of it is a removable discontinuity. The output just suddenly became zero, and that's promising. If you take a look at graph F, it appears to just drop to zero, but if it is a vertical line, then it wouldn't even be a function. So, graph F isn't the velocity-time graph, but D.

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Now comes the actual calculus solutions. So far, we've been able to determine the parent and first derivative graphs simply through common sense. Well, we'll get to the core of derivatives shortly.

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We learned that a derivative is rate of change, or the slope of a tangent line on a given graph. If we don't, don't, don't forget that a derivative is a slope, then we should be fine.

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Let's confirm our answers above first (f is graph C, f' is graph D). Well, at the minimum or maximum of C, there should be a root on D, right? If you take a look at the graphs, this is indeed the case. Also, Where C is decreasing, D should be below the x-axis, and where C is increasing, D should be above the x-axis, and that is true for the graphs.

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When it comes to the second-derivative graph, we can first observe the slopes of D. Where there are minima or maxima in D, there should be a root in f", right? Well, which among A,B,E and F fit this? Seems like only F fits this. Assuming that F is the second derivative graph, wherever it has a root, there should be a point of inflection on C. Wherever C is concave up, F must be below the x-axis, and wherever C is concave down, F must be below the x-axis. If we check, we'll see that what I've described does occur for the two graphs. Therefore, F is the acceleration-time graph.

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That's it for me! Please make suggestions on how to improve either the solutions or my part of the story. While I was solving the question, it almost seemed too easy. I hope you guys found it more challenging.