Introduction to Related Rates Problems

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Misc

SI

The next session will be Sunday 6:00 - 7:30. It will review this week, and talk about problem set questions if time permits.

Mid-Semester Feedback

Some themes that showed up in responses (sometimes from multiple classes):

• Could we record classes or make them available via Zoom for people not attending in person? That would be technically and possibly legally challenging. But it might be possible to record small focused examples (e.g., how to do certain things with Mathematica) outside of class.
  • Might it be possible to do Zoom classes when we’re online after Thanksgiving? My experience with such things last spring was that they reduce students’ engagement and make it hard to tell how well people really understand things, but if enough students want them I would consider them for a short period (e.g., after Thanksgiving). Let me know how you feel.
• Can there be more lecture & examples from me, and less use of in-class problem-solving? There is actually research evidence that students feel like they are learning less from highly active-learning classes like mine than from lectures, while in fact exactly the opposite is true. So I will keep emphasizing active learning, although I can try to scaffold it a bit more so it’s easier to get into.
• Canvas discussions don’t contribute very much. This is true, but I want to keep them around, because if we go online on short notice (for example, if I have to quarantine) they will suddenly become one of the only ways to have multi-person interactions over class material.
• Can we move at a slower pace? I will try to slow down a little.

Rejuvenation Day

Next Tuesday is the second rejuvenation day.

This doesn’t directly affect us, but...

...I’m discouraging individual meetings with me that day.

...I’ll also ease off on homework that week, so you can use the week to catch up on grading, questions, etc.

Related Rates

From section 4.1 in the textbook and the related rates discussion.

Squares

(From the discussion)

Suppose you’re drawing a square in such a way that the height and width are both changing at a rate of 2 cm/sec. How fast is the area changing when the sides are both 5 cm long?

What makes this a “related rates” problem? The essence of the problem is to figure out one rate of change (in this case, the area’s) from another (in this case, the side’s).

How would you go about answering it? Write down the equation for area in terms of the length of a side, and then differentiate both sides of that equation. This is, in fact, a good example of the general process for solving related rates problems:

1. Draw a picture of the situation!
2. Set up an equation between the quantities in the problem
3. Differentiate that equation
4. Rearrange to isolate the derivative you want
5. Plug in numbers from the problem

So what’s the answer for the square problem?

Following the process outline, start by drawing a picture that shows the relevant quantities and the relationships between them:

The write down the equation for area in terms of side length:

Differentiate both sides of that equation…

Those derivatives already have the derivative of area isolated on the left, so there’s no need for the fourth step in this problem. Go straight to plugging in the numbers for L and dL/dt from the problem. Notice that when we do so the units on the answer come out right, which is nice reassurance that we’ve done everything approximately right:

Street Crossings

Suppose you and a car are both approaching a crosswalk. How fast are you and the car closing in on each other if you are 20 feet from the crosswalk walking at a speed of 3 ft/sec, and the car is 100 feet away moving at 22 ft/sec?

Once again, start by drawing the situation:

Then use the Pythagorean Theorem to relate the car and pedestrian distances to the crosswalk and the distance between car and pedestrian:

Next, differentiate both sides of the equation. Notice that there are a lot of variables in the equation that aren’t the one we’re differentiating with respect to, and we need to use the chain rule when differentiating expressions involving those variables. This is typical of related rates problems.

Next, people wanted to plug known numbers into the equation. This is a different order for doing things than in the outline above, but it actually doesn’t matter which order you do the “rearrange” and “plug in numbers” steps.

Finally, do the arithmetic and rearrange to isolate dL/dt:

Next

Continue related rates examples.

Read “Examples of the Process” in section 4.1 of the textbook by class time tomorrow.

Keep posting questions, ideas, etc. in the related rates discussion.

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