The Tutorial Thread (School Only)

[TypeIII]

You've got questions, we've got answers.

[Mr. Bater]

"A bicyclist is riding on a path modeled by the function f(x)=.04(8x-x^2), where x and f(x) are measured in miles. Find the rate of change of elevation when x=2."

its a Calc 1 problem.

[Mickee]

.04(16-4)= .04(12)= .48 jsut my guess.. maybe...

[ShadyDave]

Derivatives with Dave, part one.

Fundementals first.

A tangent line is one that touches a curve at one and only one point. We'll call that point on the curve "u."

Since we know that u is on the tangent line, we can find the equation of the tangent line (t) if we know the slope. Remember the point-slope format? y - y1 = m ( x - x1 )

We know points y1 and x1, so now all we need is the m, the slope.

To find the slope, you need two points on that line...but we've only got one, point u.

Right now, there's no way for us to find that second point on the line and thus, our slope. But what we can do is approximate. Remember the curve (the original equation)? We created a tangent line, which touches it at one and only one point. We've got that point...now we need another. So let's pick a second point on the curve, point v. Now, imagine a line connecting point u and point v...this is a secant line (s).

So now we have a curve, two points, a secant line, and a tangent line. Let me pull up MSPaint and draw a picture.

[ShadyDave]

Okay. Now imagine a triangle connecting the points (using the secant line for a hypot).

Let's find the slope of this. As you know, slope is the change in the (blank) of the function. In this case, this is a position curve, so slope is the change in the position of two points.

m = ( y1 - y2 ) / ( x1 - x2 )

Following the graph, we see that muv = [ f(x) - f(a) ] / x - a.

Oh, sorry. Got ahead of myself. I made the coordinates of point u equal ( x, f(x) ) and the coordinates of point v equal ( a, f(a) ).

Time for a new drawing. Pardon my paint skills.

[ShadyDave]

Let's now move the second point v along the curve closer to point u.

Do you see how the slope of the secant line changes? It gets closer and closer to matching the slope of the tangent line as you take v closer and closer to u.

Now, being that v cannot equal u, it will never BE equal to u. But it can get very close. When you can't push a point any closer to another, we say that it has reached its limit, as v approaches u.

A limit can be written several ways, but this is the one we shall use:

mtan = Limv -> u muv

So, the slope of the tangent is equal to the limit of the slope of the secant, as v approaches u.

[ShadyDave]

Now, the slope of the tangent line will never truely equal the slope of the secant line...but do you see how it will get really, really close as you move v closer and closer to u?

That's good enough for right now.

Putting together our equations so far, we have this. I'm guessing this is what your prof was telling you about tangents and secants today, Shawn. It's just a problem that illustrates what a limit is.

[ShadyDave]

So how does this apply to your problem and what do we need limits for? Good question.

We've shown that the tangent line to the curve at point u is the line through u with the slope m (the equation in my last post).

There is another equation for the slope of a tangent line that comes in handy.

h = x - a

Rearrange that, and you get x = a + h.

Plugging a + h into our equation for the slope of the tangent line gives you this equation. Notice that as v -> u, x -> a.

As (a+h) approaches a, h approaches 0.

[ShadyDave]

Ooops...here's the equation.

[ShadyDave]

What is a derivative?

[ShadyDave]

That's the definition of a derivative. It's painstaking to solve anything with it, but sometimes necessary.

For most cases, there is an easier way. :)

I give you the Power Rule...

[ShadyDave]

Time to solve the problem...

f(x)=.04(8x-x^2) when x = 2.

EDIT: The reason I got to place the .04 on the left side is the constant-multiple rule...if you're multiplying the equation by a constant, you can place it outside of the differentiable function.

Also, d/dx is the same as f'(x)...it means the differentiation of the function with respect to x.

[ShadyDave]

...I think I killed it. :)

But yeah, if anyone has questions, someone here probably knows the method, or if they don't, hopefully they can at least point you in the right direction.

[k.]

You couldnt fit that in one post?

~k.

[ShadyDave]

I could have, but as of yet, I have no image hosting place. So I had to do attachments...

My apologies if it was too long. It got the point across to Shawn, he knows how to do it now, you can delete it if you like.

[k.]

I'm not going to delete it, I just wanted to know why it wasnt in one post. Thanks. :)

~k.

[Jaggorri]

Oh god my head hurts now.

[ShadyDave]

Why's that? :P

[Tadash130]

wtf is this?!?

[Fast2Seater]

Ahh....the days of calc. Glad I don't have to put up with that crap anymore. This thread may come in handy once I really start getting deep into my Database management class. Fun!

[BlueFlash]

Ya im really glad im done with Calculus 1 & 2 .. those were pretty freakin hard courses !!! oh well.. on to more hard and fun computer courses now :D

[ShadyDave]

Bah. Calc three and MV-calc were my math classes back in freshman year...and they were only the very, very beginning.

I hate math. :(

[crow]

write a function setbits(x,p,n,y) that returns x with the n bits that begin at position p set to the rightmost n bits of y, leaving the other bits unchanged.

this is in standard C, and we assume that the bit position 0 is at the right end and that n and p are sensible positive values.

[crow]

Originally posted by ShadyDave
Time to solve the problem...

f(x)=.04(8x-x^2) when x = 2.

dude, you forgot to write down the units. that's minus five billion points right there.

[Artreus]

Heres a bastard of a question.. it appears simple, but for some reason the answer eludes me:

A car rental agency rents 300 cars a day at a rate of $40 per day. For each $1 increase in rate, five fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?

edit: I know basic calculus.. however this is for a pre-calc class so i'm wondering if the prof is looking for an algebraic answer... I think i'll be able to squeeze out the points though even if i find a calculus based answer. Thanks in advance for any help on this one..

[Avram]

This is probably a bit late, but I modeled this problem in this way:

y = x*(300-5(x-40))
[300 is equivalent to a y0 value, 40 is a x0]

Where y is money earned and x is the rate chosen.

This simplifies to: y = 260x - 5x^2 (which you may notice is predictably an inverted parabola)

Using a graphing calculator to find the local maximum:

Rate of 26 dollars, netting 3380 dollars per day.

It is just as easy to find the maximum using the first derivative and setting it equal to zero:

y' = 260 - 10x
0 = 260 - 10x
10 x = 260 => x=26

[TK]

3 guys go to a hotel. There is 1 room left for $30. They each pay $10 and go to the room. The receptionist forgot the special for the weekend was $25 for the room so she gives $5 to the bellboy to take back to the guys. On his way up, he pockets $2 as a tip and gives them $1 back each. So instead of paying $10 each, they payed $9 each. But if you take the 3 guys, times $9 each, plus the $2 the bellboy kept you get......?!? Can you graph this for me please Ken?

[Mr. Bater]

you get 29 dollars. What is there to graph?

[TK]

yea, you get 29 dollars, but they started out with 30.... where's the missing dollar. And i like graphs. they're pretty

[TypeIII]

Yeah well... See when the bellboy went from the lobby to the room, he violated time-space continuum and that dollar was then split among Chuck, Toby, and Ken... So which one of them would get $0.34 instead of just $0.33?

[TK]

what, like we just hang around the continuum waiting for people to come through? Besides, everyone knows that when in the state of being in the contiuum, given that the bellboy had the ability to stop his own movement, change is never given. And if it was given, chuck and i would give you the 34 cents to contribute to your green card. Eh.

[TypeIII]

Damn it the Canadian got picked on again... Eh!

T I N A !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[Mickee]

Jimson weeds are found to have 12 chromosomes in each gamete.
(a)How many chromosomes are found in the nucleus of a megaspore mother cell for Jimson weeds?
(b)How many chromosomes are found in a nucleus of a pollen grain for Jimson weeds?
(c)How many chromatids are found in a cell at metaphase I for Jimson weeds?
(d)How many chromatids are found in a cell at metaphase II for Jimson weeds?
(e)How many chromosomes are found in a nucleus of the endosperm for Jimson weeds?

Answers:
a:24
b:12
c:12
d:6
e:12

Note: Diploid 2x and Haploid 1 ... If you guys forget...
can anyone verify if my answers are right or wrong??? Thanks....

[Mickee]

anyone.... i know CDE i have to change....

c:48
d:12
e:48

[DjRob]

Calc 1 prob:

find the limit

lim cot 2x/csc x
x->0