?!

Where's the union notation?

Where's the mathematical notation for what just happened? [1,inf)U(-inf,5]=[1,5] Missed opportunity.

Identify

n being a positive integer for number 8, right?

Where's Theorem 3 ? In fact, where are Theorems 1-5?

Where's Theorems 1 and 3?

Where's Theorem 3?

Where's Theorem 1 and 2?

There's a bunch of proof math missing here that explains why these things are true.

because x>=0

because 1-x^2=0, x=+-1

I get a domain of (-inf, inf), x cannot equal 1, with interval (-inf,0)U(0,inf)

identify

brackets [a,b]

bracket notation amiss: [0,1), etc

so, [-2,1)U[-1,0)U[0,1)U[1,2)U[2,3)

Evaluate

Also, 3. fails and the explanation is that lim x--> dne for integer values. State that explicitly please.

All f(c) exist.

In other words, limits only exist when c is not an integer.

The notation is not intuitive. It should be stated that the notches are only at the bottom, as in a floor, and there should be a statement that there is also a ceiling function with the notches at the top. I've never heard about this until today. I'm 52 y/o.

You need to explicity say x cannot equal 1, not that f cannot be continuous at x=1. The sentence starts out letting you think one thing and then changes it's mind.

Shouldn't the bracket notation be [0,1)U(1,3]?

when

They haven't failed to exist, they do not exist. What's with your English? Standardization standards.

We're not listing the answers by number any more?

x-->1-=1, x-->1+=2 x-->1- does not equal x-->1+, so the limit does not exist.

Well-behaved

This is a strange structure for a sentence!

we call these "well-behaved"

Which one is Theorem 3?

Where's Theorem 3?

Where's this Theorem 3?

allow, require...

with ", x not equal to 1"

So why isn't the limit a negative number?

at the same rate

ratios are so important, and understated right now.

You just said that.

proves

OK, why do we want to create less constraint on the tolerance? Is it just to improve the math situation for exponentiating?

Graphs please. Put the pictures in the book!

Why do I want to replace -ln(1+epsilon) with ln(1-epsilon)? It's just a more negative value than the previous statement and gives less constraint to the x-value when we're trying to find less tolerance to narrow in on the x-value?

graphs for all of the ln functions would be extremely helpful!

A graph of ln(1+x) with |ln(1-x)| on the same graph showed me which one had smaller values, more constraint, and less tolerance.

estimate

I don't follow why you chose x^2+x-1<5? Is it because of the -1?

It was easiest when I started with two columns, one for x^2+-4, and one for x+-delta=2 and worked them mathematically to |x^2-4|<epsilon and |x-2|<delta. Then let delta=epsilon|x+2|, and delta<1 and |x+2|<1, finding boundary conditions 1<x<3, and then computing the |x+2| boundary conditions and then inverting, or taking the reciprocal of, to find the smaller tolerance with more constraint to be 1/5. Then I followed the math for the suggested delta<=epsilon/5 and concluded that |x^2-4|=|f(x)-L|<epsilon.

But I found that earlier when I calculated f(x)+-epsilon=L to be |x^2-4|, so I'm not sure why I did the rest, other than to find delta, but I found the delta limit expressed when I calculated x+-delta=c to be |x-2|<delta.

?

So far with the first two examples, I'm seeing circular arguments that the tolerances exist and what the minimum tolerances should be based on a generalized equation. The specific equation shows what the tolerances are for delta and epsilon. The parts that explain how the limits are approaching that which the limit says it is, like a table, aren't included and need to be.

The talk of tolerances is what cinches this for me. Tolerances of f(x)+-epsilon=L, and x+-delta=c. from there: +-epsilon=x^2-4, +- delta=x-2, -epsilon<x<epsilon, -delta<x<delta, and finally |x^2-4|<epsilon, |x-2|<delta.

The talk of tolerances is what cinches this for me. Tolerances of f(x)+-epsilon=L, and x+-delta=c. from there: +-epsilon=x^2-4, +- delta=x-2, -epsilon<x<epsilon, -delta<x<delta, and finally |x^2-4|<epsilon, |x-2|<delta.

Evaluate using the definition, or analyze.

Evaluate using the definition, or analyze.

If epsilon/5<1/|x+2|, and delta=1/|x+2|, why would I set delta<=epsilon/5 instead of >=?

something arbitrary is going on with these absolute value signs.

are we choosing 1/5 because it has a smaller tolerance and more constraint?

are we calling this a boundary condition? I have never seen this done before.

approximate

In other words, because epsilon and delta are both small values, we can let delta be less than 1.

a table of values for epsilon, delta, |x-4|<delta (or 4-delta<x<4+delta) , and |f(x)-2|<epsilon (or 2-epsilon<f(x)<2+epsilon) would be useful here to demonstrate that the limit=2. Maybe instead of reiterating the information, you could compute .001=epsilon and show the above mentioned values to see how it really is honing in on x-->4 and f(x)-->2. This is proving the definition of a limit.

Nowhere do I see that this equals 2. I see reasons why the tolerances are valid.

Just do the math that 4epsilon-epsilon^2=0, so epsilon =4 and epsilon^2+4epsilon=0, so epsilon=-4, and we choose epsilon>0, so it must be 4epsjilon-epsilon^2.

I would formalize this sentence with math: lower bound: 4-delta=2.25, delta<1.75 upper bound: 4+delta=6.25, delta<2.25 Delta's tolerance has a lower bound maximum of the smaller limit, so delta<=1.75. Smaller tolerance, more constraint.

why not use the standard +- notation? This would be so much easier to understand in the scope of a discussion on tolerance. A graph is also a good starting point for this discussion. y-tolerance is L+-epsilon and x-tolerance is c+-delta. Then bridge the inequality gap to show that |f(x)-L|<epsilon, and |x-c|<delta by explaining why we want to represent that as an absolute value inequality.

I need a graph here.

Why is |x-1|<epsilon/5 if both |x-1| and epsilon/5 are less than delta, and both are less than epsilon/(x^2+x-1)? How are you determining that |x-1|<epsilon/5? I can't rank |x-1|, delta, and epsilon/5 based on this information. How are you doing that? Also, you are saying that delta=epsilon/5 without stating it.

That will prove...

that

need to demonstrate

Ok, if epsilon/5 is less than delta, why do we want to set delta to less than or equal to epsilon/5?

I've never heard of scratch-work before.

so we are finding a second representation of delta other than |x-1|, and that turns out to be epsilon/5 because of our approximation of delta being 1, giving us boundary conditions of -1<x-1<1, which is 0<x<2? And then applying the same condition to x^2, and finally subtracting 1 to get boundary conditions for the equation? I've never calculated boundaries for an equation before.

approximation

approximation

approximate

approximation

assuming is wrong, it should be phrased as an approximation

approximate

This is the first time I've seen someone use an inequality to build an inequality for an equation, sort of like building boundaries for an equation based on probable deltas around |x=1|.

You should explain more about reciprocals.

How? Just because the limit is within the tolerance, why does that show that the limit exists? What if it's a removable discontinuity?

I'd like at least a graph as to why we want to do that as well as at least a sentence as to what value this has.

what are you doing? Show your work! Why didn't you just say you were adding the two previous inequalities to get the x^2+x-1 terms!!

?

So in other words, use the epsilon statement to isolate the x-2 term to identify a value for delta.

This example is quite the jump and needs explanation before asking this! You are making jumps that I cannot follow.

You need to state that |y-L|<epsilon and |x-c|<delta need to be found for x by splitting the absolute values and substituting x for any y values. We are analyzing the limit statement here.This should be standard practice.

|y-4|<epsilon comes from the equation and should be |y-L|<epsilon (|y-2|)? What are you doing here? Show your work!

pictures should always be part of this analysis.

We are talking about the general case of the limit of sqrt(x) as x approaches 4, right?

This section is really unclear. Please formalize it with some math. A couple of diagrams would help too. Stating that we want the more restrictive tolerance will help simplify this explanation.

shouldn't that be 3.9601?

So, this problem is all about using the definition.... Choose epsilon tolerances of y to determine the delta tolerances of delta as related to c, choosing the more restrictive delta to proceed with.

This is the first time you've mentioned that they are definitely symmetric.

hyphen

This could also happen on a closed interval.

Is it too much to ask you to do the math here for the derivation? I'm unfamiliar with addition and subtraction of variables with inequalities and absolute values. It seems I'm missing a learning opportunity by strictly adhering to this book, which I'm supposed to be able to do.

A graph of the tolerances would be useful here.

Explain, evaluate, or determine

explain, evaluate or determine

Explain, evaluate, or determine

estimate

calculate

calculate

What is the necessity of naming it a function?

Identifying

Explain

I thought the f(x)'s said 1000, until I realized they were 10,000.

You need to define that.

Why was this in Section 5 and now it's Section 13 also?

, ?

There's history to this even though you're not teaching it!

Graph

Graph

graph

Why show the quadrants and not explain them?

Some background information on why it's 360 degrees would be helpful here.

Would have been better off in the previous section on trigonometry.

Why not instantaneous?

tempting to say it is slowing at 68 ft/s, but the negative sign points to the downward trajectory.

s(t)

Shouldn't s'(a) be defined by x+t as t approaches 0 to be clear in our notation?

doesn't this form have a name?

?

?

!

!

!

x^(frac13)

So this is NOT differentiable, right?

?

?

So are you saying that a discontinuity is not differentialble?

I don't think this annotation belongs here.

semi-colons?

missing commas

$300? You said this was in thousands of dollars.

something is a problem '^' allowed only in math mode? You haven't defined an a and b for f(a+b)?

There's an extra -3 in the answer and the answers aren't in order. Please list them as question/answer: f(0)=0, f(1)=-2, f'(0)=0, f'(2)=-2, where f'(4)=(-2-2)/(1-(-1))=-2

Missing graph

Missing graph.

I wasn't expecting the exact same equation as the example.

It's worth saying that s(0)=6 is the initial height.

12.4.34 has an extra 4 in the denominator.

I've never seen Df(x) and df/dx

So it's a difference quotient, m_sec, AROC, derivative, and instantaneous rate of change??

Why do we have four different names for this? m_sec makes sense. Secant line is what I'm used to, ARCO is new, but f(a+h)-f(a)/h is the derivation that is necessary.

likening this to the epsilon and delta equations might be useful.

?

Condition 1= Condition 2

?

f(x)

WHAT is a component function?

!

fx??

Why aren't we identifying the type of discontinuity for all of them?

Condition 1 fails at x=5

!

in other words, does Condition 1=Condition 2

why evaluate the limit as x-->3?

So this example we're not evaluating x from the right? This section is terrible.

why are we evaluating 3+ if 3 is not part of the piecewise function for x>3? You're being very inconsistent.

Condition 2: 2-=1/2, 2+=6.5. These are not equal, so the function is not continuous.

Yes, answer the previous questions like this one!

I wish you would go through each condition and prove or disprove it for each problem. After all, this IS math.

Condition 1?

Condition 2 you mean?

?

?

There has not been enough said explicitly about these three conditions previously for me to define exactly what's being talked about. I'm unable to look up this explicit information in the previous sections to get the information I need to understand this messed up graph situation that preceded this. I can only hope these examples will help. :'(

This needs to be expanded on in each of these conditions.

!

This section is unclear. I don't understand how to identify each of these solely based on a graphical pictorial representation.

Is there a missing graph here?

?

?

?

Where are these conditions I'm supposed to be checking?

This information is useful at the beginning of learning about what a function is.

So is it -1??

Should this be written as sqrt(x)-2 in the denominator, or sqrt(x-2)?

12.2.28 missing sqrt sign

So we're not doing the math on this one?

examine

Sum of functions: xtoa

You should put this in bold print.

Didn't we cover this already?

?

I think it doesn't 'pass through' (2,f(2)), instead, I think it is the inclusive point.

So a, b, c, d are i, ii, iii, iv? Why is there only an a and b answer with a i=iv section for each?

!

so x>a, x not equal to a.

This might be a run-on sentence.

comma

Why isn't this shown as the next step in analyzing a function: "a" and "L", after determining delta's and epsilon's? Why aren't we ever told explicitly that we're analyzing graphs and functions?

That previous example didn't prepare me for this exercise. I have no idea how to proceed. Some pointers in the example answer are needed. For a, there are more than 3 gumballs that are purple?

When was this introduced before? Was this called Counting Principles?

How? I don't see any intuitive sense in doing that. And why wouldn't I just count the sample space for values less than 10 directly?

dice

dice

We should be formalizing a mathematical approach here, where's all the P(E) and P(F) and P(EUF)?

Where's the W?

Shouldn't this be P(EUF)? P(ENF)=0

P(EUF)? P(ENF)=0

The double notation is confusing, please standardize.

Why couldn't you do this with the notation you just introduced?

Wow, where's the Venn diagram and derivation of this?

Is this otherwise known as Counting Theory?