dice

For the sake of the example, you should list what the possibilities are and note that three of them are odd numbers. You should show that n(S) is 6 and n(E) is 3, then show the P(E) math that n(E)/n(S)=3/6.

A dice?

!

Introduce the term expanded form here.

Just curious under what circumstances this is useful, really?

(11,4)=330

Derivations for these should have been included.

Notation!

This is unclear. Is there a derivation we can do for this?

What?? Where did you get 15 from? Why did it stop at 4? I don't see how we could have used the Multiplication Principle.

Maybe? But when?

A club of people? You mean 6 people in a club?

?

Is this 2!x3! or 2X3!?

no picture, just a description.

S_infinity

r=1+(.06/12)=1.005 n=(10 years)(12 mo/yr)=120 a1=200

S_120=(200(1-1.005^120))/(1-1.005) =(200(1-1.81939))/(-.005) =(200(-0.81939))/(-.005) =(-163.879346)/(-.005) =32,775.86936

?

?

what's with this notation?

This is not geometric as written

Shouldn't this say (1-r^n)?

Exercise 11.4.6A is not defined, but says it's defined.

I find r=3/2 which is greater than 1, so doesn't it diverge? That would mean that because r is not between -1 and 1 the sum is not defined?

so the series does not converge, and therefore diverges and is undefined.

Why are we not doing this for an Arithmetic Series?

b) has a formatting problem with k not showing as an exponent.

Shouldn't this have been an example to examine? Is n=n? I don't know what's going on with this problem. Is it formatting? It looks like this is supposed to be from k=1 to n=10 for a_n=5-6k.

Glad to see we'll be determining the results to this problem at the end of the chapter, unlike the previous two.

Are we neve going to do this problem then?

Maybe you could describe that situation??

missing graph

a(n-1)?

Ok, why the bold print?

I see that a5=a4+2, which makes sense, but I would have liked to see the actual derivation.

Isn't this supposed to be "arithmatic"?

Are we not doing this problem then?

Not sure where 1.026a^n came from, shouldn't this be 2.6^n? The answer would then be approximately 34,812.

Derivations are preferred over formulas. This is math, after all.

Good to know to keep the negative signs.

This 2 shouldn't be here.

Doesn't this statement belong after the derivation for r?

This should have been the first topic.

This was the introductory sequence we didn't use in an example! Frustrating organization!

We just did two problems like this, but none using the previous definition of ar, ar^2, ar^3, ...

since we're using a_2=ra_1, a_3=ra_2, etc, why not introduce that here?

a4 should be negative

I like how you're describing the situations that pertain to the study of sequence and series. It makes it more tangible.

Should the last two terms in the sequence be 5/2 and 17/6?

This is the wrong answer. How exactly do you come up with the right answer for this one? Even the sequence finders online can't find this.

Yeah, the comma conglomerate at the end is hard to decipher.

-3/2, -5/8

? notation

Wrong graph

?

[5pt] ?

Shouldn't the answer be written in terms of Ax^2+Bxy+Cy^2...?

Where's the logic that follows to this statement?

This was shown earlier with comparisons of A and C in the general equation having like or unlike signs for B=0. Opposite signs are a hyperbola, like signs are an ellipse as long as A is not equal to C and B=0.

It's unnecessary to rotate the axes to determine the conic sections, so I'm not sure about your statement.

According to the section on generalized equation identification, a positive A and C value are an ellipse. It's worth saying somewhere in here that these cases are for B not equal to 0.

...when B is not equal to zero

A and C?

This should have been mentioned the first time.

Some indicators that I'm on the right path would be nice, like sin theta and cos theta, what x' and y' =.

Where did these equations come from?

Why the comma?

i cos theta + j sin theta is new notation. are i and j vectors or scalar? Am I supposed to recognize i and j = to something specific? Why are you saying that x'i'+y'j'=xi+yj? This section needs a LOT more explanation and maybe some more graphs.

I can't make the jump, you need to explain this. u is in terms of x' and y', not x and y.

Why is u=x'i'+y'j'=xi+yj?

This, and the previous graph, need some examples.

We should have covered the axes translation in the introductory section on i and j coordinate systems and defined the unit vector there.

This needs more trigonometry and graphs to explain it. I really don't understand the cos theta projecting from the y' axis to the x axis.

Is r=1? cos(theta)=x/r, x=rcos(theta)

Are we defining the sin and cos from x' and y' or x and y, or both?

This is the first time you've used the hat notation

Why not mention that the -1 in front of the xy term is related to the slope of the axis.

so, are you saying it's actually: -Ax^2+By^2=0 and Ax^2-By^2=0?

This should be a bold print with definition

Why are the absolute values in the graph coordinates, but not the equations?

Why did I bother learning parabolas as y=ax^2? Learning it in this format isn't more helpful. Why not just state that we learned y=ax^2 and that a=4p?

I think parabola b, facing left, is supposed to have the y-axis to the right of the directix.

I'm trying to figure out why this information is useful.

This is the first time that I've heard of a latus rectum.

The vertex of y^2=4px at (0,0) is not on the axis.

?

This should have been placed before the (h,k) section.

Shouldn't x be subtracted from y?

+-9

Please include the derivation for b^2=c^2-a^2

The light cascading on the wall of a house from a lamp forms a hyperbola.

It looks like a parabola?

I take issue with the figure. Shock waves are spherical, not circular.

Isn't this the same instructions you've printed over and over?

division signs are missing.

I would've put this section following the section that had h,k centered at the origin.

This would be better served as an example. There is only one example for ellipses!

A derivation and explanation of b^2=a^2-c^2 is appropriate here too. The reasoning is not just algebraic, but geometric also, and based off of the distance formulas and right triangle created from b to O, c to O and distance a=d1=d2.

should be (0, +-c)

Only one graph shown, does not depict either description. Is actually figure 10.1.14.

First mention of this.

-6x should be -6z

This is a great observation that I didn't notice.

Much easier to read as -3R2+R1=R1

The first use of the formula should have been already covered. This section is very unorganized.

Not sure what I did wrong, a solution set would be great.

display problems for AA-1 third row matrix and A-1A 1st row, second matrix

You told me to use the formula, but then solved it without using the formula. You didn't even explain where it came from.

You didn't do any examples with the A-1 formula.

?

?

Second equation should be presented as IA

%

The annual interest in only one year?

%

%

Erroneous nesting of equation structures.

How is that the answer?? 2x+5y+8z is not equal to 1 for (1,1,1)

first time I've seen (x, y, z).

My row-echelon matrix looks a little different, which is worth mentioning, but still find that the answer is (1,-1,2). I don't understand why your R1 shows fractions when the original set for R1 should be 1, -2, 3, 9 according to the first equation. BTW, I've never seen any solution sets in here for x, y, and z expressed in parenthesis.

I learned this in high school. Why is it in preCalculus?

rows 2 and 3 in the shown row-echelon form has numbers other than zero

missing answer: "Erroneous nesting of equation structures"

This text often introduces information as if it was new even though we've been working on it. It's irritating.

15%?

Why are we calling rows both m and i, and columns both n and j?

[

Bb_1x in the first partial fraction breakdown has a large 1 and a small x...formatting

I'm not seeing it in the generalized equation expansion of P(x)/Q(x). This is just repeated information from the previous section on non-repeated linear factors.

Partial fractions are composed to form a single rational expression.

should I be defining a feasible region here?

I don't know where I may have learned this.

shouldn't this have been presented in the inequality system.

Both of these were ellipses in parabolas without considering that any of the four solutions could be invalid yielding one or no solutions.

interesting to note that this system has one solution and the line is therefore tangent to the circle.

It didn't say to graph, but I believe there should always be a picture with the story.

again, a statement about how the solutions for a single outcome and no solution will present themselves if we're not going to examine examples and do exercises on them.

You need a statement in here about how the single solution and non solution will present themselves upon examination.

Missing figure!

Examination

Examining how these inconsistent systems present themselves is necessary. It helps with recognition.

I like the geographic representation.

I would like to see more sections structured this way.

Bold print for "Gaussian elimination"

bold print?

bold print?

Organization should be taught in every problem we examine.

This should probably be after the circus ticket and meal ticket problems.

If we're just going to examine the profit of the profit-loss function, shouldn't this be an inequality? Is P(x) a parameterization?

profit-loss function

Things to cover here, C(x) and R(x) are represented by the y-axis, that's why they are annotated as y_1 and y_2. x represents the number of units (the x-axis). For the R(x)=px, there's no information on where this number would come from. If we're going to call P(x) the profit function, shouldn't we use an inequality?

Just to complete the graph and put forth the educational information, the area to the left of the break-even point is called loss.

11/2=5.5

Shouldn't the coordinates be in <> notation?

This should be annotated as v_1 and v_2, not u and v.

There is no clear reason why v/|v|=<5/sqrt(29)+2/sqrt(29)> and u/|u|=<3/sqrt(58)+7sqrt(58)> ? Should the plus signs be commas? Is there some mysterious vector u I'm not seeing here?

shouldn't these have the <> notation in the graph?

I just want to enter a note that it was not obvious that vector-v=<a,b> and <a,b> is equal to a i-vector+b j-vector.

What does this number mean? What does it represent? What is it measuring? What is it's use?

are we saying that it's time-dependent? Is it parametric?

sin theta should have a division sign in there

First time with this term.

so are the vector notations supposed to be above the i and j in the v-vector linear combination? It's unclear.

shouldn't it be y_2-y_1?

This equation is new. Maybe introduce it first since there is no arrow?

the arrows are really helpful. The bold letters are not showing up.

This is the first time I've heard it called a standard position vector.

Shouldn't the vector terminal point be labeled <2,3>?

should say v1

Not sure how you got here given the problem the example gave. 17+16=31?? This problem description doesn't even belong here. Shouldn't it be "given u= and v=, find 3u+2v="?

missing a v with the 1/2

grammar

u should be on the next line

should be 6 root 2

Missing graph

Cartesian

This has been covered, although not formally. This should have been done back at it's introduction.

Cartesian

Shouldn't we be annotating the removal of the parameter by indicating y(x)=?

none of these rotated clockwise.

Why is this in bold print with no definition?

Are you saying that Cartesian equations are not rectangular equations? Shouldn't we give credit where credit is due? Why is this in bold print with no definition?

y=1/2(ln(x))

no graph

missing graph

So, even though the instructions at the beginning of the section would have me find t=5-y and substitute into x(t) as x(5-y), it is actually going to be easier to find y(t) since the quadratic equation produced by y^2 isn't two binomials. This should be an example, not an exercise.

So we are finding coordinates x(t) and y(t) for f(t). This is a lot like drawing circles and ellipses.

So, normally m is calculated as the difference in y divided by the difference in x, which is -1/2. You're saying that since we're parameterizing, we have to find m_x dependent on t (=2) and m_y dependent on t (=-1). So the actual slope m is -1/2, it's just (m_y)/(m_x).

And then we're using the x-value and y-value of the first coordinate as the b value, which I don't understand. Just how does that play out mathematically? Are we just using the first value of (=5,3) for the b values of x(t) and y(t)? This is so unclear. Is it the first coordinate because we're adding time to it?

t column showing named as \(t)

This should have started with an example.

Why the comma?

cis is not an identity and is not useful. De Moive did not have cis in his theorem so why is it here?

cis is not an identity

cis is not an identity and is never useful.

Not sure why I can't use cos(theta)=x instead of x/r seeing as how the equation is multiplied by the radius and will cancel it out.

Not sure why you calculated r, it was given in the problem. And although the 122 is wrong, the answer is right.

Ok, so 2 cos (pi/6) is a number, why aren't we writing it that way?

This notation is not an identity. Please stop. I find it preposterous that we use a cartesian coordinate system for imaginary numbers at all. 3+4i could well be (3,4i), but the point (5,6) would only be 5x+6y, which has never been taught. There's a lot here that isn't being discussed.

I'm not using this cis notation. You can't use it to solve anything since it's a mixture of sin and cos and isn't an identity.

cis? Never heard of it before.

there's a |z| with no value in the graph.

I'm not supposed to be learning formulas in math. I'm supposed to be learning derivation and modeling.

Fill in your own math! I don't see ANYTHING about the complex conjugate in here! I don't see anything about z^2-=(x+iy)(x-iy) making |z|=sqrt(x^2+y^2)

Should be: 1<a/b<2

missing graphs

missing graphs

This whole section would be better if you included math with the +-(pi)(k)