Why couldn't you start out with this equation and then describe the transformations according to this equation. Now I have constants all over the place that were c, but are now a and my notes suck.

around

This explanation does nothing for me.

Saying that x-3=0, x=3 tells me the x value when y=0.

You're missing some math here, but I see it below.

we've never called this an argument before. It's known as a variable. Inserting x-2 for x is known as a composite function.

Determine

1) ?

How so?

Determine the domain of each function.

Where's the math for this? I can determine that the maximum for f(x) occurs at f(0)=2 and the minimum for f(x) occurs at f(+-2)=0, so 0<=y<=2.

We just said that.

I'm having trouble following your math because you are skipping steps.

This is out of order

I kind of need that information in order to sketch the graph...?

trend lines, trajectories, regression lines, extrapolation, forecasting, expectations, projections, predictions....

although one can develop those equations, functions, graphs, and other mathematical tools in the field on the spot, mostly they are available by computer to compute with. I prefer to learn how to stand on my own two feet in the field like what is expected of me when I travel to other locations than where my primary hired workplace is.

f(x)

function

equation

f(x)

f(x)

Where is the derivation for this equation?

This is the first instance of us looking at +-infinity as domains and ranges.

I might say that if there is more than one point to choose from, you wouldn't usually use this equation unless you were trying to get the equation of the line in slope-intercept form, but the point-slope equation for the line will depend on your choice of points, and the slope-intercept form of the line won't.

estimating that

So, I used (x_1,y_1) like the definition said and came up with f(x0-2=.5(x+3)

determine

Why are we naming the line L and then never mentioning it again?

I appreciate the derivation.

This should have introduced this section and been the basis for which you explained everything else.

You have m for slope in the definition below, don't you want y=mx+b?

equation. Also, don't you want y=mx+b?

could this be "ordered pairs"?

no a=U?

explain

graph a function....

explain

determine

identify

where's the math for this?

x−3=.5(y−4)

x-3=-.5(y−4)

The "why's" of the math are not well explained. I'm struggling to see why there are two solutions?

I think the definition should have been before these statements.

Analyze or determine.

Given

Given

determine

determine

determine

determine

analyze

analyze

extraneous \)Thedomainis\(

Equations given are:

You should have chosen which notation to use for the domain. Based on how the others were done, even though this is simpler to look at, I would have expected (-inf,-1)U(-1,1)U(1, inf)

if, when?

not sure why this is set off with commas.

A function can have one y-intercept at most.

equation

an equation

equation

equation

equation

equations

Equations

equation

Why are we calling equations formulas?

of. It belongs to it, it's not for it.

I never remember learning set-builder notation other than recently.

I remember finding domain and range, but defining that by solving for x is new to me.

Determine the i. Doman and ii. Range of each of the following functions.

Evaluate f(1) and f(a+h) for f(x)=x^2-3x+5. Why use unnecessary commas.

Weird extra bracket stuff in answer.

Evaluate the function at

equation

identify

identify

set A in domain D.

So regarding the picture above, if x in domain {1, 2, 3, 4} equals 2, then f(2)=4 in range {2, 4, 6}.

a function of

I don't remember a lot of math about sets and elements.

called points

by formulas, are you saying that you will derive them and show how they are built so that we can learn modeling for analysis?

We defined local extrema occurring on an open interval, and theorem 3.1.1 does not state this explicitly, which does not infer an absolute maximum and absolute minimum on a closed interval...???

That's enough of this book for me.

local

This point is moot.

are defined as

local

local

local

local

local

local

local

local

local

what about the local extrema at x=0?

local

local

local

these are not synonyms: you want local

local

critical number, c is a number. x is a value of f. (c,f(c)) is a point.

undefined.

Your vocabulary and sentence structures matter.

would be nice to mention here that this is not a continuous function because of it.

Evaluate f' at each of these points.

Why use commas if you don't need to?

new word

new word

none of this was about local minimum, why didn't you include this in your analysis, and why did you say "it seems"?

Evaluate f' at each of these points.

Why use commas if you don't need to?

Why would I approximate something given? List

minimums are either local minimums or absolute minimums. The same with maximums. Minimum and maximum are too general. I have never heard this referred to as a relative minimum/maximum. Are you trying to use these definitions as related to open and closed intervals? Why not separate those sections.

How are we approximating exact values? Also, we should be using the definition to define the answer. f(c)<=f(x) for all x on I, f(c)=-27. f(c)>=f(x) for all x on I, f(c)=25.

extrema

determines

category

Extrema

random bracket

They're just called the extrema.

How many theorems do we need that repeat the same information?

Just a reminder, I am using this book to reinforce what I've learned, not just to gather information.

I'm sure you could have written this with math.

I'm not seeing more than one example. I really need more examples.

garner

?

The level of difficulty of this book is all over the place. Some things you don't cover well, others you bring up the most basic ideas that were already covered. Some things are just incomplete like the missing error analyzation of f(c+h) (because math is an analyzation, and calculus is another method of analyzation). I'm 2.7 sections into this book and am seriously thinking about dropping it for another book, but you went over the derivative at a point so well otherwise, and other books won't do that. SO...?

What's wrong with the "does not equal" sign?

Isn't f'(c)=2.582?

I'd really like to see where that's coming from, but it's not here.

Is there a chance you should substitute y=sqrt(1-x^2) and simplify?

Why is this all the way out here? It was covered in Theorem 12?

lol: apply lots of algebra!

We've never called them explicit functions until now. Maybe bring that up much earlier when we start talking about them.

what?

Again, t=(u+v)/w u: u'=ab'+a'b, where b=cos(c), and c=x^(-2) v: d=-h^2, h-sin(j), j=e^4x w: k=ln(m), m=x^2+5x^4 t'=(u'v-v'u)/w^2

much easier to say: u=v^5, v=tan(w), w=6x^3-7x u'=5v^4v', v'=sec^2(w)w', w'=18x^2-7 f'(x)=5v^4v'=5(tan(w)^4)sec^2(w)w'=5tan(6x^3-7x)sec^2(6x^3-7x)*(18x^2-7)

or uv'.....

Why not also factor out the 5x^2?

The step-by-step is missing.

I recognized that 1/e^(-x^2)=e^(x^2) and used the product rule, which I find easier. I also used the u and v notation which made everything look less complicated.

where's the step-by-step?

Isn't u=g(x)? This is confusing as written. Use the u to make things clearer.

It is strange to see the answer with a factored numerator and non-factored denominator.

Yeah, shouldn't this have been placed before Theorem 19?

You;re not using the u notation, and it makes it confusing.

what about y'=f'(u)*u'?

Does this equation assume that the tangent line is below the function, and should absolute value brackets be used?

are you going to walk us through the error between calculating it using the tangent line as a delta-h instead of the curve? There would be some df error between f(c) and f(c+h) using the tangent line instead of the function. It was difficult to determine from everything that we were, in fact, using the tangent line for our delta-h as opposed to the funciton.

A graphical explanation is required. Show f(x), tangent line f(c)+f'(c)(x-c) and/or f(c)+f'(c)×h, and maybe the secant line with (delta-y and delta-x). This will answer a lot of questions on its own.

What exactly is an f'(c)*h? Can we show it on a graph?

Clearly show in it that f(c+h)≠f(c)+f(h).

It is also good to note that h=x-c, or x=c+h. This would round out our understanding of the tangent line's notation.

This should be introduced before the examples with variables defined in each example so that I can learn what you are talking about. P'(0)h=P(h) P(c+h)=P(c)+P'(0)h=P9c)*P(h)

A graph of this would help.

So y(c+h) is the next slope h units after c for some small h? We have not defined how small "small" is for h.

the graph of the slopes of f(x)

rate of change that is a constant

(c+h)-c

expectation

This is a large jump to go from this description to the notation. It should be better explained, with a breakdown of what each is and how it all adds up.

Put that in bold print.

extrapolation?

approximate total change of...the function? the extrapolation? the trajectory? the trend line?

The explanation that needs to go along with this is that a graph of f'(x) is a graph of the slopes of f(x) for all x, and that the slope of f(c) can be found at f'(c) for x=c.

it: the function? Or maybe the derivative, it's slope?

You need more examples for these

no....? Ok, I thought you were still referring to example 54. This should be stated differently: Rewrite f.....

Now would be a great time to use u and v, since you've called the entire equation f(x) instead of the numerator.

since these are functions of x that are being evaluated, why not use the u and v notation?

so why haven't you evaluated it to finally be -x^3(ln(x))(sin(x))+x^2(cos(x))+3x^2(ln(x))(cos(x))?

y^prime?

I'm not sure why a derivation is being presented as an example.

I have problems with your choices of notation. y(x) is supposed to be for the tangent line, which is a strange notation that was never introduced. This function should have been called h(x)=f(x)g(x). Your notation for this chapter is wanting.

Is there a computation for this, or are we just eyeballing it?

Shouldn't this problem be further up? Also, it's not well-organized.

pedantic is part of learning.

rate of change that is a constant

Given

20?

?

misc bracket

misc bracket

Suppose

It is treated like a function in differential equations and the chain rule, so is it ok to leave it at this?

line tangent to f at c

Where's the derivation?

reciprocal

The equation of the line tangent to f at x=1

Suppose

cursive letter? Why not g(x)?

Why not use delta notation?

this is actually, and is not being stated, 2+h-2.

Suppose

what's going on with the brackets for a, b, and c?

Determine

for x^n and x^m....

I'd rather see these as examples.

It isn't clear with this that n and m are only the exponents being talked about. I could not interpret it for n and m as subscripts of the coefficients, which is what I initally thought you were talking about.

???

Why are you using y and f(x) interchangably?

You should introduce this section noting that vertical and horizontal asymptotes are able to be determined, with vertical asymptotes investigating f(x)--> +- infinity and horizontal asymptotes investigating x--> +- infinity.

You should introduce the vertical asymptote here.

So we're not going to use that definition we just learned?

Explanations for each step are missing. Did you just square the |x-1| term?

This choice of delta is not intuitive, please explain the reasoning.

Determine

?

\and) ?