Tips for Max/Min Problems

Your assignment is p.229/43, 45, 46. 
43.)  Already discussed.  If the minimum of a function is positive, can the function ever be negative?  Find the exact minimum.

45.)  a.)  At the moment that the springs have the same position, they are passing each other.  No calculus required - just solve a trig equation.  You will have to use the identity in the hint.  Remember that, to solve trig equations, a common strategy is to get all terms to one side (set = 0) and factor out the GCF. 

b.)  Calculus required.  You have to maximize the distance between them.  SO, you must find an equation for that distance.  You should be tempted to use absolute values here but you don't have to.  Consider this "simple sample":

The points -3, -2, 0, 1 and 5 are on a number line and you want to know the greatest distance between any two numbers.  If you subtract in one direction, you get -3 - 5 = -8.  If you subtract 5 - (-3), you get 8.  The maximum distance is 8 and the minimum distance is its opposite.  Maximizing (or minimizing) will still get you that -3 and 5 are the farthest apart.

Same with this spring problem.  Subtract the positions to find distance, m1 - m2.  Differentiating and finding the 0's will give you a max, a min, or both.  Use the First Derivative Test.  When you find the actual distances by plugging the critical values back in, ignore any negative signs in the distance values.

If you have trig questions, post them and I'll check back later.

46.)  a.)  Same as the last one.  Set up and solve an equation.  Remember the Sum Identity:

sin (a + b) = sin a cos b + cos a sin b

You should get an equation that looks something like.....  b sin t = d cos t  (b, d are numbers).  Divide both sides by (b cos t) so that you can solve a tangent equation. 

b.)  Also, same as above.  Subtract positions and call that D (distance between them).  Differentiate D.  Find 0's.  First Derivative Test.  Plug values of t back in to find greatest distance.

c.)  When is the distance between the particles changing the fastest??  I don't want to spoil the satisfaction of figuring this out.  What are you trying to maximize?  What equation must be differentiated?

To warm yourselves up for the graphing assignment,click on the link below.  It is an applet where you can practice graphing derivatives from the graphs of functions.  Don't rush through it.  Take the time to jot down your thoughts before you check your answer.  After using the applet for at least 10 functions, answer the following questions.

Don't forget to use the uphill/downhill analysis, bowl analysis, and shape predictions !!

     1.)  On the graph of a derivative, we see zeroes (on the x-axis) where the original graph has a max or min.  You will see some zeroes in this applet that do not represent max or min.  What do you think they represent?  How can you tell the difference between the "max-min zeroes" and these impostors??

     2.)  Remember "bowl up" or "bowl down"?  That is technically called concavity.  When a graph is "bowl up" or concave up, its derivative graph will always be going in which direction? Is the inverse of that true?

     3.)  In your own words, describe where points of inflection occur on the original graph.  What are the ways to show a point of inflection on a derivative graph? (Hint:  See problem 1.)

Derivative Graphs - Practice Applet

Power Point Links

I uploaded the PPTs to Google Docs and got the links to share with you.  However, I just tested it and no go - the slides are all messed up.  You guys have been taking notes since starting the PPT's - study your notes and you'll be fine.  Sorry about that.  I'll work on ways to do this for the future.

Introducing the Derivative

Before watching this video (which is a little dry, I warn you), you should keep in mind the ultimate "prize" --

                THE SLOPE OF THE TANGENT LINE

Why?  Because tangent lines tell us a lot about curves, even complicated ones.  On a piece of paper, draw a curve that goes up and comes down a few times.  Choose a bunch of points on the curve and draw the tangent lines at those points?  What do you notice?

1.  When the curve seems to be going uphill (looking from left to right), what do you notice about the tangent line?  Positive or negative slope?

2.  Likewise, as the curve goes downhill, the slopes will have the same sign.  Do all downhill slopes have the same value?

3.  At the points where the curve turns around, what are the slopes of the tangent lines?

You should notice that the tangent lines are useful in describing the curve.  That's what mathematicians are all about... finding numerical ways to describe behavior, in this case, function behavior.  Instead of saying that the curve seems to be "going uphill" sharply at a certain point, we can say that the slope of the tangent line is "positive 4" or "positive 20" or whatever.  This tells us that, yes, the curve is going uphill and it also tells us exactly how steep it is.  Efficient, huh?!

Watch the video link below, close to 10 minutes long.  At about 4 minutes into the video, the teacher will bring up an example of an object in free fall.  Answer these questions for this week's post:

a.  Why does the speed of the object differ on time intervals (2, 3) and (3, 4)?

b.  What rudimentary method does he use to find the slope of the tangent line?  Why is this inadequate?

c.  In BC times (before calculus), you needed two points to find a slope.  First, the man uses 3 and 3.1 as the two x-values; then, he uses 3 and 3.01.  What concept allows us to use smaller than 3.000000001 as an x-value?

So, here's the very important point:  the DERIVATIVE of a function = SLOPE OF THE TANGENT LINE.

 
Well, here's the link.  Get some popcorn:   Instantaneous Rate of Change

Limits of Rational Functions

After watching the 5 minute video linked below, answer the following questions:

1.  The first function had an error message in the table at x = 0.  Why?

2.  Does this problem matter when "eye-balling" the limit?

3.  If you were using your calculator to assist you in drawing a graph, what would you put on your graph that is not on the calculator screen?

4.  The second function had error messages at x=1 and at x= -1.  What is the difference between them?

5.  If you're approaching a hole in the graph of a rational function, the limit still exists.  How about if you're approaching a vertical asymptote?


Rational Functions - Use a calculator??

More on Limits

Last week, I posted about the idea of a limit.  Your posts were heading in the right direction but "not quite there."  I found out that you haven't been introduced to this concept yet and it's very important that you understand limits.

We'll talk about limits in class but I found an 8-minute video on Teacher Tube that I'd like you to watch.  It's well-made and easy to understand.

A Good Look at Limits

 Remember these three things:

• The word limit (or lim - the abbreviation) must be followed by a function. 
• Beneath the word limit, you will find the part of the graph you're supposed to focus on.  It will be described by an x-value. 
• The answer to limit problems, if the limit exists, is always a y-value.  It describes the intended destination of the function.  

For your post this week, after watching the video, I'd like you to think about the function below:

lim  (x^3 + x - 1)
x---> 4

Evaluate this limit.  What would happen to the limit if I restricted the domain to [-4, 4]?

Jake's function last week was y = sqrt(x)?  Where, in its domain, does the limit not exist and why?

Have a good weekend... see you Tuesday.

                

What's a Limit?

Once we delve into calculus, understanding the notion of a limit will be critical.  In case you've never discussed limits before, first consider the everyday meaning of the word.  The Speed Limit is a decent analogy to a mathematical limit, in that you can legally drive at a speed up to or including the actual limit. You don't have to drive the actual speed limit. The key is in not exceeding it.  In calculus, you'll be finding the "speed limit" and the function is like the car.  Whether or not the function actually reaches the limit is not important.  The limit will be the y-value that the function targets.   


Here's an example.  I'll have to give you a location on the function (an x-value) for you to mentally zoom in on.  Consider the function y = x.  Picture the graph and focus on the piece near the origin.

As x approaches 0, either from the left or from the right, what y-value is the graph moving toward?


You should be getting 0 and, in this case, the function actually reaches its limit.

For your post, I'd like you to consider any function that you know.  Everyone's function should be different.  State the function (use ^ for exponents), tell me where to zoom in on the x's, and give the y-value that the function approaches.