Another problem
f´´(x)=5x²+11/x²
f´(1)=3
f(1)=7

And another ...
f´´(x)=3cos(4x)+8sin(x)
f´(5Pi)=3
f(4Pi)=-7

And another ...
Just first order this time:
f´(x)=sqrt(1+x³)
f(0)=2
Can't antidifferentiate in terms of familiar functions, so ... a qualitative study.

Just one more
f´(x)=x/(1+x³)
f(2)=3
Vocabulary
Direction field; solution curve

An actual solution can be written ...

Numbers?
f´(x)=sqrt(1+x³)
f(0)=2
How can we get f(1), then?

Problem #2

General facts

Yet another problem

Second attempt (zooming in)
(Bumps)

More zooming
(Wiggles)

The whole thing

Moving on: a max/min problem

The Consumer Price Index
I wrote the changes in the consumer price index during three periods of U.S. history, and tried to explain how these numbers, reflecting data which was gathered by looking at prices, implied information about the shape of the CPI over time. In particular, these shapes showed that the CPI increased and decreased, and that the CPI graph also "curved". This curving is called concavity. Logically the concavity and the {in|de}creasing nature of the curve are not connected. I tried to show this in the diagram below.

Concavity and {In|de}creasing

Graphing e^-x2
I asked students to have and use a graphing calculator. The result was sort of like what is shown to the right. Realize that looking at the graph is not "cheating" and I use graphing software whenever possible. At the same time, the graphing software can be fooled -- deceptive results can occur, and sometimes you need to look really closely at the calculus side of things.

If f(x)=e^-x2 then f'(x)=(e^-x2)(-2x). The exponential function is very nice. It is never 0 and always positive. Therefore the only x for which f'(x)=0 is when -2x=0. So x=0 is the only critical number. Now reasoning using the Intermediate Value Theorem says that f (which is certainly continuous!) can have only one sign for x<0 and one sign for x>0 (or else f(x) would have to have to be 0 again). We can check signs at, say. x=1 and x=-1. f is increasing in (-infinity,0) and f is decreasing in (0,infinity). Naturally 0 represents a local (and indeed, absolute!) maximum.

What information can we get from the second derivative? If we use the product and the chain rule correctly, then f''(x)=(e^-x2)(4x²-2). Logic similar to the preceding asserts that this is 0 exactly when the non-exponential factor is 0. But 4x²-2=0 when x=+/-sqrt(2). Again, we can check signs in between the 0's of f'', and f will be concave up for x<-1/sqrt(2) and for x>1/sqrt(2). For x between -1/sqrt(2) and +1/sqrt(2), the graph will be concave down. The points where x=+/-1/sqrt(2) are where the concavity of f changes: these are called inflection points. These particular inflection points are related to the standard deviation, which represents dispersal from an average when this function is used in statistics.

Graphing a degree 3 polynomial from the book
I looked at y=2-15x+9x²-x³, which is problem #3 in section 4.5. This is a problem in the textbook. Again we began with a graphing calculator result. We computed the first and second derivatives and found out where the signs of these functions changed. From this we were able to get information about increasing and decreasing and inflection points.

Graphing x³(x-1)⁴
Here I invited students again to use a graphing calculator and try to see what y=x³(x-1)⁴ "looked like". I did remark that the "action" took place somewhere between -1 and 2. The result of this was something like the graph shown to the right. I believe that calculators and graphing devices are wonderful, but sometimes they almost conceal what's going on.

Here f(x)=x³(x-1)⁴. If we want to find out where f is increasing and decreasing, we really should look at f´(x). For this we need the product rule and the chain rule. So:
f´(x)=3x²(x-1)⁴+x³4(x-1)³.
Generally I am against "simplifying" because I view it as mostly a chance to make lots of mistakes. But here some simplifying will reveal structure in the derivative. So please notice the common factors, and what you get is as follows:
f´(x)=x²(x-1)³(3(x-1)+4x)=x²(x-1)³(7x-3).

What can we tell about where the derivative is 0 and where it is positive and where it is negative? Well, the different factors allow us to deduce that the derivative is 0 at x=0 and x=1 and x=3/7.

If x is very large positive, say, then f(x) is a product of three factors, all of which are positive. And if x is very large negative, then the x² is positive and the (x-1)³ is negative and the 7x-3 is negative. Therefore f(x) in that range is positive also. So we have learned (using logic from the Intermediate Value Theorem as before) that the derivative is positive on at least the intervals (-infinity,0) and (1,infinity). There is a chance for the derivative to change signs at x=0, but the factor which controls sign change there is x²: since 2 is even, there is no sign change at x=0. But there is a sign change at x=3/7 and at x=1. So now we have broken up the real line into pieces

    Deriv is +     Still +          Now it is -        Back to + here 
<---------------0-----------3/7--------------------1------------------->
 Func increases   increases     The func decreases   Here it decreases

Then we considered the concavity of this function, and made some guesses about the number and location of inflection points. We can be sure about this if we find the second derivative.
I will use the product rule, and make the second factor, a product itself. So:
f´(x)=x²(x-1)³(7x-3)= x²((x-1)³(7x-3))
f´´(x)=2x((x-1)³(7x-3))+x²(3(x-1)²(7x-3)+(x-1)³(7))
So let me try to "simplify" f´´(x). We will get:
x(x-1)²(2(x-1)(7x-3)+3x(7x-3)+x(x-1)7)= x(x-1)²(2(7x²-10x+3)+21x²-9x+7x²-7x)=x(x-1)²(42x²-36x+6)

So the second derivative is 0 at x=0 and at x=1 and at the roots of 42x²-36x+6=0: those are x=(3/7)+/-sqrt(2)/7 (approximately .227 and .631). The second derivative does not change sign at 1 because the factor is (x-1)², an even power. It does change sign at 0 and the two other numbers which are on either side of the local max. There are three inflection points.

I needed to work fairly hard to get everything correct in the preceding example.

Francine on the New Jersey Turnpike

Driving inside the speed limit
(always valuable!)Displacement from velocity information

The important qualitative consequences
#1:Deriv always 0. Antiderivatives. Why? How?

#2 & #3: increasing and decreasing

An easy example

Perhaps a more complicated example?

Velocity information geometrically
Acceleration and displacement

Repeat of L'H
Suppose we want to compute lim_x-->a[f(x)/g(x)].
Eligibility criterion Suppose you know that f(a)=0 and f(b)=0 (this is called an indeterminate form of the type 0/0).
If the eligibility criterion is valid, and if lim_x-->a[f´(x)/g´(x)] exists, then the value of this limit is the same as lim_x-->a[f(x)/g(x)].

Other indeterminate forms
I remarked that the same result is true if, for example, the "plug in" values resulted in a quotient of the type infinity/infinity. This quotient arises quite frequently in applications, for example, in comparing rates of different kinds of chemical reactions or comparing how various algorithms behave as the amount of input varies (sorting).

A simple example
What about lim_x-->infinityx/e^x? The plug in gives exactly infinity/infinity, so this quotient is eligible for L'H. If we differentiate the top and bottom, we then need to consider lim_x-->infinity1/e^x and that certainly has limit equal to 0.
To the right is a picture of the situation, showing both x (in red) and e^x (in green) on only the domain [0,4]. Although I love pictures, asymptotic information is sometimes not easy to see in them. Here the trouble is that the exponential function is so much bigger than x that an undistorted picture is already almost ludicrous. So, sure, as x gets large, the exponential function gets much bigger faster.

More complicated examples
I think I tried to analyze lim_x-->infinityx^5,000/e^.0007x. I began by "evaluating" this quotient at x=2, just to try to understand what's going on. I put quotes around the word because I will only try to approximate the quotient.

The top is 2^5,000, and how big is this in terms of powers of 10? Well, as a math grad student reminded me after the lecture, we know that 2¹⁰=1,024, and 10³=1,000 and so maybe we see that 10 powers of 2 give 3 powers of 10. And so 5,000 powers of 2 give about 3·[5,000/10]=1500. Actually, Maple told me that 2^5,000 is about 1.41246·10^1,505. So the other estimate is pretty good. How about e^.00007·2? Well, this is e^x when x is quite close to 0, and f(x)=f(0)+f´(0)x+(tiny error), Here f(0)=e⁰=1 and f´(0)=e⁰=1, so e^.00007·2=e^.00014 is about 1+.0014. Actually Maple reports a 10 digit answer of 1.000140010, so this estimate is also good.

This f(2)/g(2) is about 10^1,500/1.00014, which certainly looks good for the top -- it is much larger than the bottom. Maple really didn't want to draw this function on the interval [0,1.1]: it objected with the remark "Y axis values too large". So to the right is my diagram of what the graph might look like. x⁵⁰⁰⁰ is in green. When x is between -1 and +1, the high power drives the values down down down to 0 (at least as far as the scale in this picture is concerned). But still the value at +1 is +1, so the green graph must go through (1,1). And the darn thing must be continuous. For x>1, the values get really large, very very very fast. So the green graph seems to shoot straight up. What about e^.00007x, in red? On the scale shown, this graph seems to be almost a horizontal line at height 1. The thickness of the drawn line conceals any tilt the graph might have. So this is all quite strange, especially if we evaluate the limits.

1. We consider lim_x-->infinityx^5,000/e^.0007x. Plugging in gets us infinity/inifinity, so we can try L'H. The result is lim_x-->infinity5,000x^4,999/.0007e^.0007x.
2. Now plug in to try to get lim_x-->infinity5,000x^4,999/.0007e^.0007x. The result is again infinity/infinity, so again the quotient is eligible for L'H. We must consider lim_x-->infinity(5,000)(4,999)x^4,998/(.0007)²e^.0007x.

Quite a lot of time goes by ...

5000. After 5,000 uses of L'H, we must consider lim_x-->infinityConstant₁/Constant₂e^.0007x. The alert student may be able to identify the constants: Constant₁ is (5,000)!, five thousand factorial, the product of the integers from 1 to 5,000. And Constant₂ is (.00007)^5,000. But ... I don't need to know this. I can tell you now that the bottom "wins": it eventually gets bigger faster, no matter what the constants are. The limit will be 0.

Log vs polynomials
I did not do this is class, but, time and energy permitting, I should have.
What is the lim_x-->infinity[ln(x)]/[x^1/300]?
Before using L'H we must check that some appropriate set of hypotheses are satisfied. Here certainly ln(x) gets larger when x gets large "enough" and so does x^1/300. That is, the limit of the top as x-->infinity is infinity, and the bottom has the same limiting behavior. It isn't hard, by the way, to see that both the top and the bottom get large. Try a big number (?), like 10¹⁰⁰. Then ln(10¹⁰⁰)=100ln(10) is approximately 230, and (10¹⁰⁰)^1/300=10^1/3 is about 2.15. Well they do get big, but they get big "slowly" and this limit is essentially asking which of the two, the top or the bottom, gets bigger slower.
In any case, this is an indeteriminate form of the type, "infinity/infinity" and we can try to use L'H.
lim_x-->infinity[ln(x)]/[x^1/300]=(using L'H)=lim_x-->infinity[1/x]/[(1/300)x^(1/300)-1]. Let's get rid of the compound fraction in the last expression. We then need to evaluate lim_x->300x^-1+1-1/300. I think I did the exponents correctly, and the result is the limit as x-->infinity of a constant (300) multiplying a negative (-1/300) power of x. The result is certainly 0.
Therefore we conclude that the limit is 0 and x^1/300 gets bigger faster than ln(x)! But what about the numbers? We just didn't take numbers that are big enough to see what's happening. Try (10¹⁰⁰)¹⁰⁰. A little bit of juggling with exponents will show that ln((10¹⁰⁰)¹⁰⁰)=10,000ln(10) which is about 23,000. And ((10¹⁰⁰)¹⁰⁰)^1/300 is (10¹⁰⁰)^1/3 which is about 10³³. And it is true that 10³³ is much much bigger that 23,000.

Binary entropy function
Here is one reference defining the binary entropy function. Google has over one and half million references to this. The letter p is usually used for the input to H because the p stands for probability.

H(p)=-p·ln(p)-(1-p)·ln(1-p) is called the binary entropy function. It is used when investigating the amount of information which can be sent through certain communications channels. Usually the log's involved are logs base 2, but that really doesn't matter here.

Even on a fairly simple level, there are already difficulties: what is the domain of H? Since part of H's formula involves ln(p), certainly p>0. But part of the formula also involves ln(1-p), so 1-p>0 or p<1. So the actual domain of H is just the open interval from 0 to 1, not including its endpoints. A Maple graph of this function is shown to the right. (I think I had this upside down in class, which just shows how badly my brain is functioning).

The point in my discussing this function at this stage of the course is to look at the edges of this graph. The edges seem to suggest that certain limits of H(p) exist and even suggest the values of these limits. For example, it certainly looks like if p gets close to 0 (more officially, p-->0⁺) then H(p)-->0. Diary entry in progress!

Demons, human beings, and angels
I mentioned this in class but did not go into detail. Here is more detail, beginning with a metaphor.

The world is made up of a hierarchy (spelling?) of demons and humans and angels. All the demons are less than the humans and all the humans are less than the angels. The "internal" arrangements of {demon|human|angel} society are quite complex, but between the societies things are rather simple.

Now polynomials are human. What are angels? An angel is a sum of constants multiplying exponentials with constants. So angels are 2e^{3x} and 5e^{.0003x}+99e^{-99x}. These functions also have rates of growth as x-->+infinity, and we can compare two of them in a similar fashion, only here the comparison is first look for a positive coefficient multiplying an exponential with a positive growth number. So .99e^{.03x} is eventually bigger than 9999999999e^{.0003x}.

How are polynomials and exponentials related? Let me stick to things with positive coefficients. A very tiny exponential, say .00001e^{.00000000001x}, compared to a huge polynomial, say 10,000,000,000x^{100,000,000,000}, is bigger, as x-->infinity. Eventually, all angels outrank humans.

Now, continuing in our development of function growth via analogy and idiotic metaphor, let's consider polynomials of log functions: these are functions like 33(ln(x))^{30} and sums of them. Well, these are the demons. EVERY demon is eventually less than EVERY human.

Let me "compare" P(x)=33(ln(x))^{300} with, say, Q(x)=x^{.0001}. Poor Q(x) is a very weakly growing human, as x-->+infinity. And, wow, P(x) is rather a strong demon. Indeed, P(10) is about 1.5 times 10 to the 110th power, and Q(10) is about 1.00023. But let me investigate their "ultimate strength". The simplest way is to consider the limit of P(x)/Q(x) as x-->+infinity. Certainly this is a limit of the form infinity/infinity. so I should L'Hop the whole mess. If I do, the result seems to be:

Top=33(300)(ln(x))^{299}(1/x).
Bottom=.0001x^{-.9999}

I hope I did this correctly. Now let's do some algebra to this quotient. I will put all of the x powers downstairs, and push the constants out to the front. So the result is (I hope):

(ugly constant)
33(300)/.0001 multiplying  (ln(x))^{299} divided by x^{.0001}.

Essentially all I have done is lowered the degree of the demon by 1, and I still want the limit as x-->infinity. I hope you can convince yourself that eventually (after another 299 L'Hops?) that the limit will be 0. This "miserable" human (?) eventually defeats a very powerful demon. It may take a while but this really really happens. For example, I bet (I just experimented in another window of my computer!) that if x is greater than 10^8, P(x) is LESS THAN Q(x). If you object that 10^8 is large, my response will be that there is just as much "room" between 10^8 and infinity and there is between, say, 17 and infinity. And scale of action doesn't matter to demons and humans and angels, only what EVENTUALLY happens.

Sigh. I hope this does help you. It may be more than you want to know but it really is more or less the mathematical truth. The metaphor is just there to help. People who study theoretical computer science and algorithms really worry about the growth of functions. They have the families of functions we have just discussed (called Exp and Poly and Log) but also many others. Sigh. You can look at their zoo if you like, to check that I'm not kidding: the complexity zoo.

A string of definitions with some examples

A situation ...

Rolle's Theorem

Newton's Method
Newton's method is a way to (try to) improve a guess at a root of f(x)=0 when f is a differentiable function. The guess, G, is (hopefully!) improved with the following process (as you read this, please glance at the picture to the right). First, go up from G until you "hit" the graph of y=f(x). The point will be (G,f(G)). Then "slide down" the tangent line of the graph at the pointwhose slope is f´(G). The point that this line hits the x-axis is the new guess, N. The picture is rather simple and shows the new guess closer to a root of the function. This picture is rather simple, and is the way we would like the method to work. I will discuss more horrible possibilities later, but right now I would like to get a formula for N in terms of G. Well, the slope is f´(G), but this slope is also equal to OPPOSITE over ADJACENT. Here OPPOSITE is f(G) and ADJACENT is G-N. Therefore
f´(G)=f(G)/(G-N) which is the same as G-N=f(G)/f´(G), so that

For square roots
Here f(x)=x²-A and f´(x)=2x, so that G-{f(G)/f´(G)} becomes G-{[G²-A]/2G} which is {2G²-[G²-A]}/2G which is [G²+A]/2G which is (1/2)[G+{A/G}]. That is, "Improve the guess to a new guess, N, by taking the average of the old guess, G, and A/G: N=(1/2)(G+[A/G])."

Some numerical results
Here is an example, computed by Maple: I wanted to compute the square root of 2. As is well-known (but not by me!) this is 1.4142135623730950488 to 20-digit accuracy. My first guess, A, was 3. Here is a table of next guesses using the Babylonian method, and the difference between the guess and the true value of sqrt(2):

Approximating 7^1/5
Suppose we wanted to compute the fifth root of 7. Sigh. Well, just suppose. I would like a "simple" function whose root is 7^1/5. I should be able to evaluate the function and its derivative "easily", and this certainly suggests polynomials. I think of f(x)=x⁵-7 which has exactly one root at 7^1/5. Then f´(x)=5x⁴. The Newton's method "scheme" of N=G-{f(G)/f´(G)} becomes N=G-[(G⁵-7)/5G⁴]. My faithful silicon companion (?) "simplifies" this to (4G⁵+7)/(5G⁴).

Some numerical results
7^1/5 is 1.475773161594552069276917 (to 25 digits of accuracy). Suppose I use Newton's method with an initial guess of 1. Here are some numerical results:

More phenomena
Newton's method can give some surprising answers. It definitely should be used, since under good circumstances it converges very rapidly, but you should be warned that it can also misbehave.

An example
Here is one example of the use of Newton's method with quite a simple function, selected to show some problems. I hope the analysis will not be too hard to understand. If you try a random polynomial you will probably get very messy results, much more difficult to understand and analyze than this example. But the example does show behavior that does happen in general.
The function is f(x)=x/(1+x²). Let's look at the derivative: f'(x)=[1·(1+x²)-2x(x)]/[(1+x²)²]. Here it is worthwhile to simplify, since I would like to work with f'(x). The result is f'(x)=(1-x²)/[(1+x²)²]. The Newton's method iteration for going from an old guess, G, to a new guess, N, is to take G to G-f(G)/f'(G). In this particular case, N=G-f(G)/f'(G) is G-[x/(1+x²)]/[(1-x²)/[(1+x²)²]]. The second term is a compound fraction. The expression can be simplified quite a lot with some algebraic work. In fact, for going from an old guess, G, to a new guess, N, the formula is: N=(2G³)/(G²-1).

Also notice that y=x/(1+x^2/) has a root at exactly one value of x, x=0. What I would like to do now is color the points of the curve green if starting from that point the sequence of Newton's method iterations converges to the only root, 0. I would like to color the points red if the iterations do not converge to 0. I would also like to discuss any weird behavior that occurs.

Here is a picture (qualitatively correct) of x/(1+x2).

Probably geometry may be more of a help to us than the algebra alone. In fact, they should be used together, to reinforce each other. The curve drawn is symmetric with respect to the origin. It has pieces only in the first and third quadrants. So the colors I draw should reflect that symmetry. The analysis I present here uses only two colors and is therefore a bit more simple than what I said in class. I just distinguish between points whose "iterates" approach the only root of 0 and those which get large (both +/-infinity). So here we go:

A green point, then some more green points Certainly, starting with G=0 gets N=0, so that the root itself is green. Are there other green points? Well, drawing a few pictures near O should convince you that if you try starting near 0, the Newton's method iteration rapidly approaches 0. The picture here tries to show that with a few iterations. The interval around 0 is an example of what is called a basin of attraction (really!).

Some red, then some more red points Suppose we try x=10 as G, our initial guess. Then the formula gives N=(2G3)/(G2-1). How does N compare with G? G seems fairly "large", and the top has twice G cubed, while the bottom only has G squared. I bet that the new guess, N, is to the right of G. In fact, the top of the bump (the maximum, we will officially call it on Monday) is located at (1,1/2) (just see where f'(x)=0 -- remember that f'(x) is (1-x2)/[(1+x2)2], so it is 0 only when the top, 1-x2, is 0, and that is at x=+/-1. So 10 is far to the right. And if G=10, N is even larger, more to the right. For me, looking at the picture is much easier than trying to understand the algebra. And if you look to the right of the top bump, you can see that the tangent lines will all slant "down" (have negative slope) and that therefore for guesses which are bigger than x=1, the new guesses are being driven away from 0, the root. So all of the points on the curve which are bigger than 1 must be red. Of course, a symmetric conclusion is reach for points less than -1, where the iteration drives guesses to the left.

Weird things or not so weird at all The algebraic formula N=(2G3)/(G2-1) isn't so good when G=+/-1. We are not supposed to divide by 0. What happens in the picture? The tangent lines at those points are horizontal, and they never intersect the x-axis. So maybe I should color those two points some special color just to show that the red/green distinction doesn't apply to them. I guess I will color them brown, a silly color for a silly point.

More red in towards 0 Suppose now we look at the curve a tiny bit to the left of where x=1. Then the tangent line has very small positive slope, and will hit the x-axis on the far left. Above that point, the curve is red, and redness spreads contagiously backwards so the point we started with is red. So some part of the left-hand side of the top bump is red, and similarly, by (anti)symmetry, something on the right-hand side of the bottom bump must be red.

Brown: how can there be more brown? Keep traveling left on the top bump, in the red stuff. The next new guess will begin traveling right on a part of the curve where x<0. Eventually, though, the line will intersect the x-axis at x=-1. Do you believe me? well, let us solve N=(2G3)/(G2-1) when N=-1. Then 2G3=1-G2 (by cross-multiplying) and so we need to find numbers for which 2G3+G2-1=0. There is exactly one such number (verify this on your own graphing calculator, please) and its approximate value is .657298, so I think since this number is shoved into -1 it should be brown also. And so should its negative, since that gets pushed into +1.

And more red Push to the left from the previous point. Then the Newton's method iteration takes us into the red region directly to the left of the point on the curve where x=-1. And then, if you can imagine it, further to the left we come to a point which gets pushed into the second brown point on the left. So that point should be brown also, and then there should be a part of the curve which is red and then ... This is very complicated. There are actually on each side of the curve infinitely many intervals of red, and each red interval has brown points on its edge, so there are also infinitely many brown points. The red intervals are getting shorter and shorter, though, until: Something new happens If you look at the picture closely, you will see that there is a positive number G so that when we compute N and then compute N again, the two tangent lines are parallel! For this point Newton's method does not go to +/-infinity and does not go to 0: it oscillates repeatedly. We can even compute this number. By symmetry, at this point, N should be -G, so the equation N=(2G3)/(G2-1) becomes -G=(2G3)/(G2-1) and if we cross-multiply we will eventually get 3G3=G. One root is G=0 which gives us nothing new. The other roots are G=+/-sqrt(3). So I have shown these two points in pink: pink for periodic. In general, Newton's method does have points that sort of wander around, and don't converge to anything, and don't "escape" to +/-infinity. Newton's method can be very complicated. I have also indicated the behavior of infinitely alternating red strips and brown points with !? because I (and all other people!) can't draw this very well. This behavior is an example of a fractal set.

This picture tries to display a completely colored graph. The green points are attracted to the root, 0, by Newton's method. The red points go out to +/- infinity (alternate strips of red go to alternate infinities [here by infinity I just mean eventual travel steadily out to the left or the right]), and each red strip is bordered by brown points which eventually under iteration don't go anywhere (!): they get sent to the wastebasket, outside of the domain of Newton's method. The region labelled !? is too hard to draw, with infinitely many alternations of color. Finally, the two pink points represent points which are alternated with each other periodically by Newton's method. This is already complicated. A "random" function would likely have even more complicated behavior.

L'H
L'Hopital's (also L'Hôpital's or L'Hospital's) Rule is a method of evaluating certain very special limits. It is worth showing to you because it is another neat application of "local linearization", and some really important limits become easy to evaluate. However, it must be used with some care. One quote I found on the Internet declared, "Giving l'Hopital's Rule to a calculus student is like handing a chainsaw to a three year old."

A motivating example
Let's look at lim_x-->0[e^4x-1]/sin(3x). Of course I began my consideration of the limit, which is made up of familiar functions, by just plugging in x=0 in the hope that the functions, which are individually continuous, will be continuous near and at 0. Since this is an invented example, we get 0/0: no hope.

We can do the following. Remember that f(0+h)=f(0)+f'(0)h+Err_f(h)·h where the error term, Err_f(h), goes to 0 as h-->0. Also g(0+h)=g(0)+g'(0)h+Err_g(h)·h where that error term, Err_g(h), goes to 0 also as h-->0.

Now look at the quotient:

 f(x)     f(0)+f'(0)h+Errf(h)·h
------ = -----------------------------
 g(x)     g(0)+g'(0)h+Errg(h)·h

Then lim_x-->0[e^4x-1]/sin(3x) becomes lim_x-->0[0+4x+Err·x]/[0+3x+Err·x] which (factoring out an x everywhere) is lim_x-->0[4+Err]/[3+Err]. The error terms are different, but qualitatively they have this common property: as x-->0, both error terms-->0. Now we see "easily" that the limit is 4/3.

The preceding discussion is not a proof, but merely presents heuristic evidence that the result below is true.
L'Hopital's Rule (version 1)
Suppose that f and g have continuous derivatives, and that f(a)=g(a)=0 (the eligibility criterion). If lim_x-->af'(x)/g'(x) exists, then lim_x-->af(x)/g(x) exists and is the same value.

1. L'H resembles the way we might like to differentiate quotients in our dreams. Please try not to get confused:
   • L'H is a method to evaluate limits.
   • The quotient rule is a method to compute derivatives.
2. You must check the eligibility criterion of L'H. A simple limit like lim_x-->0[x^2-3x-4]/[x³+5x-6] is not equal to lim_x-->0[2x-3]/[3x²+5], the limit of the quotient of the derivatives. The first limit can be evaluated by direct substitution, due to the continuity and nonvanishing of the denominator: it is 4/6. The second limit can be evaluated similarly and is -3/5. These are not equal. You must check the hypotheses and in work submitted for this class you must show that you have checked them.

To analyze this kind of limit, we usually look at lim_x-->infinity. It turns out that L'H works for many different "indeterminate forms": 0/0 is one, and infinity/infinity is another. There are more (please see your text, and see there for more formally stated versions of L'H). I will do more next time.

Today we will investigate two consequences of linearization. Linearization, geometrically, is the idea that if you "zoom in" enough on the correct kind of curve, the result is that the graph of the function will more and more nearly resemble a non-vertical line. Algebraically this corresponds to the definition of the derivative. But (see the September 27 lecture, please) we can "unroll" the definition to a remarkable equation.

Now I considered again a remarkable equation.

A silly example
In this example, f(x)=x². Let us look at f(a+h). In this case we can compute f(a+h) exactly. It is a²+2ah+h². We can "match up" the terms here with parts of the REMARKABLE equation above. So f(a)=a², and since f'(x)=2x, the term f'(a)h is exactly 2ah. The "Err·h" term is h², and certainly when h is small, this gets smaller faster than b. A numerical example is something like (5.01)² which we compute exactly as 25.1001, and here a=5 and h=.01. This splits up as 25+.1+.0001=5²+2(5)(.1)+(.01)², and the result of omitting the last term is a very small error.

The official name
The use of f(a)+f'(a)h in place of f(a+h) is called the linear approximation to f at a. There are several reasons it is called the linear approximation. First, algebraic formulas (here thinking of h as the variable) which only involve constant and first degree terms are called linear. Second, consider the tangent line to y=f(x) at x=a. The slope of that line is f'(a). It goes through (a,f(a)). If we increase a by an amount h, the line's height is increased by f'(a)h (because OPP/ADJ is f'(a) and ADJ is h in the right triangle shown). So the linear approximation gives a formula which instead of using the correct value of f(a+h) substitutes the value obtained by "extrapolating" using the value on the tangent line.

Another example, perhaps less silly
What is the square root of a number close to 1? Since square root is a continuous function, such a value of square root should be approximately the square root of 1, which is 1. But how should we "correct" the answer if we want sqrt(1+x) for x small? Here we have f(x)=sqrt(x) and a=1. So we need f'(1) which is easy, since f'(x) is (1/2)x^-1/2 and f'(1)=1/2. Therefore the linear approximation to f(a+h), that is, to sqrt(1+h), is just 1+(1/2)h. Now I tried to do some arithmetic. The following table is somewhat more extensive than I computed in class with student help. Here my help was Maple and the computations were done to 30 digits accuracy. (!)

Values of h	"True" values of sqrt(1+h)	Linear approximation values: 1+(1/2)h	3rd column - 2nd column
1.5	1.2247448713915890491	1.25	-0.0252551286084109509
1.05	1.0246950765959598383	1.025	-0.0003049234040401617
1.005	1.0024968827881710675	1.025	-0.0000031172118289324
1.0005	1.0002499687578100594	1.00025	-0.0000000312421899406
1.00005	1.0000249996875078123	1.000025	-0.0000000003124921877
1.000005	1.0000024999968750078	1.0000025	-0.0000000000031249922
1.0000005	1.0000002499999687500	1.0000005	-0.0000000000000312499

Richard Hamming, a very famous applied mathematician of the 20^th century, declared that

• Consider the number of zeros in the first column and in the last column. In the first column, the transition from one number to the next is gotten by "adding" one 0. So the a+h becomes a+(h/10). The last column shows the discrepancy, the difference, between the linear approximation and the true value. Look at the number of zeros. In each step (and this is true data!) the number of zeros increases by two, so while h changes to h/10, the difference gets divided by 100. In other words, the term "Err·h" is actually about a constant multiplied by h²: it is a second-order term. This is generally true.
• What about the sign in the last column. It is always negative. Why is that? The curve lies under the tangent, which means that the linear approximation will be an overestimate of the true value.

One more example
Finally, I asked what happened to the three-halves powers of numbers near 9. Here f(x)=x^3/2, so f'(x)=(3/2)x^1/2. If a=9 and h is a small number, we can hope that (9+h)^3/2 is close to f(9)+f'(9)h. Since f(9)=27 (hey, I chose 9 because the 3/2's power of 9 is "simple") and f'(9)=(3/2)(3)=4.5, we now know that (9+h)^3/2 is approximately 27+(4.5)h. This may or may not be interesting to you, but if you need to really think about many values of x^3/2 near 9, this approximation provides some insight. I also can tell you that this approximation is an overestimate of the true value, since x^3/2 curves "up" and the tangent line at (9,27) is under the curve. A picture that Maple drew of this situation is to the right. We will soon learn how to tell if the tangent line is underneath or over the curve without looking at the graph. As one numerical example, the linear approximation to (8.96)^3/2 is 26.820 because this is 27+(8.96-9)(9/2), and the "true" value seems to be about 26.82020, just a bit more (as the picture shows). An implicit function example

How your calculator computes square roots: some possibilities
I asked how a calculator computes square roots. Most of the answers I got were, "By pushing the button," but I repeated my question. Here are my suggested answers.

1. Since World War II smaller and smaller genetically modified people have been raised. I think in '47 the two foot high person was the standard of excellence in this direction, and then in '53, the first 1/4 inch person ... and then in the 70's really teensy people were created and put inside calculators and they just used clever guessing to get square roots. They were also put on super-amphetamines to work really fast.
2. If your calculator can manipulate 10 decimal digit numbers, then (since each number position can hold one of 10 choices of digits) the calculator can only be asked to find the square root of 10¹⁰ numbers. These square roots were all computed in 1950's and 60's with U.S. Department of Defense function, a massive research project called "Square Roots for Freedom!" The cost was about 47 billion dollars to compute all these square roots and the job was finally completely in 1964 when but the task was finally completed in 1965 when ten digits of the square root of 37 were announced: 6.0827625302.
3. The calculator uses ancient Babylonian mysteries to compute each square root very efficiently.

Babylonian square roots
Here is a complete description of the process:
How to compute the square root of A to a specified amount of accuracy

• First guess the answer: G.
• Improve the guess to a new guess, N, by taking the average of the old guess, G, and A/G: N=(1/2)(G+[A/G]).
• See if the new guess is close enough (that is, compute N² and compare to A) otherwise repeat the previous step.

The method initially looks quite weird: replace G by (1/2)(G+[A/G]. Well, is it? First, if your "guess" is too large (so G²>A) then G>sqrt(A) and (G/A)>(1/sqrt(A)) so that (A/G)<sqrt(A). We are averaging numbers that are on either side of sqrt(A). So maybe we will get closer. The amazing result is that we get very much closer, very fast, and I will substantiate this with some numbers next time.

Mostly descriptive but now analytical.

puerile trivial, childish, immature.

A cube with edge length changing

A rain drop evaporating

Running on the square root curve

Leaving Hoboken

Now: a rectangle that does everything!

The exam comes back

• "Precalculus"
  • Algebra, including especially algebraic manipulations of various kinds.
  • Functions, with definitions such as doamin & range, and knowledge of specific functions (exp, ln, trig functions, etc.)
  • Graphing, and graphs of the standard functions
• Limits
  • Intuitive understanding of the concept
  • Precise definition of limit; working with this definition. Although certainly a precise defintion was stated, not much emphasis or class time was given. I would not regard this as an eligible topic for an exam.
  • Numerical evidence for limits. Although this was covered in several important examples in class, and in some homework problems, since students will not be allowed to use calculators on the exam, I would find such questions not very appropriate on an exam.
  • Limits done by "plugging in": when is this appropriate?
  • Limits using algebraic manipluation, changing them so the previous method can be used.
  • Limits with "special" significance (most promiently inthis course are the limits as h-->0 of [sin(h)/h] and [{e^h-1}/h]
  • Limts done with geometric information.
• Continuity
  • Precise definition and use of this definition
  • Diagnosis by algebra
  • Diagnosis by geometric information (graphs)
  • The most important consequence so far in the course: the Intermediate Value Theorem,
• Differentiation
  • Precise definition of derivative and verifying this with specific examples
  • Differentiability implies continuity (the connection between these two properties)
  • Simple uses of the derivative
    Algorithmic computation of derivatives Not including the Chain Rule and implicit differentiation since the exam covers only up to Lecture 9.
  • Recognition of differentiability on a graph. Sketching a derivative from the graph of a function.

Maintained by greenfie@math.rutgers.edu and last modified 10/16/2006.