Homework #6 – due March 9, 10, & 11 (Updated)

WebWork sets Derivatives-ChainRule, Derivatives-InverseTrig, and Derivatives-Exponential-Logs are all due Sunday night. Don’t forget to set your clock FORWARD. Otherwise, these sets will be due an hour before you expect them to be.

There is no written component this week. Instead, you’ll complete the Test #1 Review assignment. Instructions are here. Don’t forget, before you publish your post, add the category “Test #1 Review.” As long as everyone remembers that part, you’ll be able to see all the submissions here.

Update: Since I droned on and on about old homework in today’s class (sorry) we didn’t get to derivatives of inverse trig functions, so the due date for Webwork set Derivatives-InverseTrig will be postponed, now due Tuesday night. The questions aren’t too hard, once you have the tools, though, and questions of this type may appear on Wednesday’s test, so you’re more than welcome to try this Webwork set now. All you’ll need is what’s contained in Theorem 3 on page 180.

I just realized that I may have written the wrong section numbers on the board today. We covered part of Section 3.8 and part of Section 3.9. From Section 3.8 all that’s left is the derivatives of the inverse trigonometric functions. Section 3.9 includes a discussion of hyperbolic trigonometric functions, which I plan on skipping (but you’re welcome to read yourself), so we’re officially done with 3.9.

My plan for Monday’s class is to finish off 3.8 and then go over the review questions you guys are putting up on the OpenLab, and tie up any other loose ends that need tying up before Wednesday’s test.

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Two interesting events on campus tomorrow – Thursday, February 27

These two events might compete for your attention during club hours tomorrow. I’ll be in a meeting, but if I weren’t, I’d be interested in attending either the Math Club talk or the Physics Club talk myself! Details for both are below.

MATH CLUB
Title: “Conway’s RATS”
Speaker: Dr. Johann Thiel (NYCCT)
Date/Room: Thursday Feb 27, 2014, 12:45-2pm, Namm N719
Abstract: John H. Conway is known for several mathematical games, including the well-known Game of Life. In this talk we will discuss Conway’s RATS game. To play, just pick a number. Reverse the digits of the number, Add the reversed number to the original, Then sort the digits in increasing order. The game ends if repeating this process enough times gives you the same number twice. As simple as this sounds, very little is known about this game. 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778,…

PHYSICS CLUB
Title: Falling into a Black hole
Speaker: Dr. Justin Vazquez-Poritz (NYCCT)
Date/Room: February 27, 1-2pm, room: 804
Abstract: This talk will answer the question: what would it be like to fall into
a black hole? Various effects of Relativity, such as how time freezes
at the event horizon of a black hole, will be illustrated with stories
involving animated sequences. There are three ways by which falling
into a black hole could lead to a tragic ending, and we will discuss
how two out of the three ways might be prevented. If one is able to
survive the trip going inside a black hole, bizarre properties of
space and time emerge, such as being able to travel backwards in time.
We will discuss how the quantum vacuum could be used in order to
escape from a black hole.
We will also discuss plans for attending Brookhaven National
Laboratory. A free guided tour of an exciting research institution.

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Homework #5 – due March 2, 3, & 4 (Updated)

WebWork sets Derivatives-ProductQuotient and Derivatives-Trigonometric are due Sunday night. These sets are a little long but should be straightforward after tomorrow’s class.

At the beginning of Monday’s class, please hand in the written component:
3.3 #64, 65, and 66
3.6 #49

An optional interesting problem is 3.6 #58. You might like to play around with this one, and we can talk about what you come up with, but don’t hand it in with the other exercises.

UPDATE: WebWork set Derivatives-Trigonometric is now due TUESDAY, MARCH 4 at 11:59pm. You can certainly go through most of the exercises now, but we’ll go through a bit more of the theory together on Monday in class. You can also omit 3.6 #49 from the written component. We can talk about this one in class together on Monday.

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Homework #4 – resubmit Wednesday if you like

A few people had asked permission to resubmit the homework they handed in today, based on the feedback they got on the homework that was returned today. If anyone else wants to resubmit, you can do so and hand in the updated version in class on Wednesday.

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Exercise for Wednesday

At the end of today’s class we saw that we could use \lim_{h\to 0}\frac{e^h-1}{h}=1 to prove that the derivative of e^x is e^x. Try using this fact to determine the derivative of b^x, where b is any positive number.

Hint 1: There’s a clever way to rewrite b^x that will help you see what to do.

Hint 2: Remember how, given \lim_{x \to 0}\frac{\sin(x)}{x} =1, you were able to evaluate, for example \lim_{x \to 0}\frac{\sin(kx)}{x}, where k is just some number? Try using that knowledge to evaluate \lim_{h\to 0}\frac{e^{kh}-1}{h}. Then try to see how this actually helps you differentiate b^x.

This exercise is technically optional, but I think you might like to think about it and play around before Wednesday’s class, even if you don’t get anywhere. If you’d like to share what you came up with, you’re welcome to write it on the board before class. You can also post a comment here, but try not just posting your answer until Tuesday night or so…it’s less fun for the others to work on it if you’ve already spoiled the answer for them!

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Homework #4 update

The update to Homework #4 is that there is no update. I just checked, and given what we talked about today, you should be able to complete all the Webwork for Sunday night, and all the written work for Monday morning. There’s more to talk about from Section 3.2, but you won’t need it for the homework exercises.

Most of the questions assigned to you from 3.2 involve the relationship between the graph of a function and the graph of its derivative. This can take some getting used to at first, but once you get the hang of it, it’ll be easy. It might be helpful for you to explore the example I showed in class today more yourself. Click below.

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Slopes of tangent lines – practice for tonight

At the end of today’s class each group used the limit formula to compute slopes of tangents for f(x)=\frac{1}{x} for points of tangency with x-coordinates latex 1, 2, 3, 4, and 5. The slopes were -1, -\frac{1}{4}, -\frac{1}{9}, -\frac{1}{16}, and -\frac{1}{25} respectively. From this, you all guessed that the slope of the line tangent to the graph y = f(x) at the point where x=a was -\frac{1}{a^2}. (In class I called it “i” instead of “a” but this isn’t a significant difference.) You were correct and it’ll be almost trivial to see this formally, which is roughly the content of the first part of Section 3.2 and tomorrow’s lesson.

If you’ve got time tonight, it’d be worth practicing a few more examples like this. Some suggested functions are: f(x)=x^2, f(x)=x^3, f(x)=\sqrt{x}, f(x)=|x|. For each of the functions,

  • select a few different points of tangency (a,f(a)) on the graph y=f(x),
  • use the limit definition to compute the slopes of the tangent lines at each of the points you’ve chosen,
  • from these results, guess what the slope of the tangent at (a, f(a)) would be, for general a.

If anyone does this tonight, it’d be great to see your results. You can share them as a comment on this post, or put them up on the board before tomorrow’s class begins. Just make sure you’re clear each time what your choice of f(x) is and what your x=a is.

Just a reminder, given a function f(x) and a point x=a, the slope of the line tangent to the graph y=f(x) at x=a is given by

\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}.

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Homework #4 (copied from Blackboard)

(Originally posted on Blackboard on February 14)

Webwork: DerivativeDefinition and DerivativeFunction due February 23

Handwritten component due February 24:

2.7 #23-29 odd (these are fairly computational; there is no WebWork set corresponding to section 2.7) and #43

3.1 #49

3.2 #43, 44, 45, 66, 67, 68, 69, and 98

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Homework #3 (copied from Blackboard)

(Originally posted on Blackboard on February 14)

Webwork: LimitsAlgebraic Trig due February 16)

Handwritten component due February 20:

2.5, Preliminary questions, #2

2.5, Further insights and challenges, #55

2.6, Exercises, #3

2.6, Exercises, #33

2.6, Exercises, #51

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Homework #2 (copied from Blackboard)

(Originally posted on Blackboard on February 3.)

Webwork: LimitsIntro and LimitsContinuity due February 9.

Handwritten due February 10.

2.2 #37

2.3 #31

2.4 #81, 84, 88

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