Multiple Calculus Questions

[generico]

Hi there. There are three problems in this post.

I browsed online and found this forum. I don't know the rules yet, but read the first thread so I'll try to keep it in line. I read that you guys don't want people to post their homework, and I'm not doing that although these are homework problems. One of which I got the right answer using an online tool, but I wasn't able to find out "how" which is more important to me. I didn't want to flood with topics, and I need a lot of help for a major exam in a few days with material I'm having a hard time understanding. I made this into one topic, although, if I need to I understand if I have to make these three separate ones. Thank you so much ahead of time for your help.

First problem:

Suppose 2 x^2+ 5 x + xy = 4 and y( 4 ) = -12. Find y'( 4 ) by implicit differentiation.

So, I know the answer is -9/4, but I don't understand how. The problem gives me a hint that "you'll have to also solve for y." Anyways, what I have so far would be:

4x + 5 + ( y + x (dy/dx) y) = 0

I'm not sure if I have that right, but I took derivation of all the parts, and used product rule for xy. I know previously I move over to one side, and get it so dy/dx = something, but not sure what I'm doing wrong after this point. I'm also not sure what putting in the y (4) = -12 impacts this.

The second problem is:

Suppose that a company needs 450000 items during a year and that in preparation for each production run costs 500 dollars. Suppose further that it costs 10 dollars to produce each item and 2 dollars to store an item for one year. Use the inventory cost model to find the number of items in each production run that will minimize the total costs of production and storage. units per production run.

My professor did one example of this, and made a mistake in setting it up but never went back to do another example. So, I don't even know how to set this up or where to start/go on from.

The third problem is like the first:

For x+e^(xy)=10, find dy/dx when y=0.

What I got to was:

d/dx (x + e^(xy) ) = d/dx 10

I know by rule the derivative of e^anything is the same.

1 + e^(xy) d/dx xy = 0

Then use the product rule for derivative of xy...

1 + e^(xy) (y + x dy/dx y) = 0

At this point I'm stuck. I have in my notes that dy/dx = ( -1 - e^ (xy) y ) / ( x e^(xy) ), but don't know how it jumped to that after my last step, and what to do next to answer this particular question.

I do have more questions after this with other problems, but wanted to wait to see how these go or if I'm in the right place/asking the right questions. Thanks again!

 

[Aladdin]

First problem:

Suppose 2 x^2+ 5 x + xy = 4 and y( 4 ) = -12. Find y'( 4 ) by implicit differentiation.

So, I know the answer is -9/4, but I don't understand how. The problem gives me a hint that "you'll have to also solve for y."

--------------------------------------

Welcome generico ~

2 x^2+ 5 x + xy = 4
xy = 4 - 2x^2 -5x

\(\displaystyle y = \frac{4 - 2x^2 - 5x }{x}\)
\(\displaystyle y = \frac{4}{x} - 2x - 5\)

Can you take it from here . . .

 

[BigGlenntheHeavy]

\(\displaystyle (3) \ x+e^{xy} \ =10, \ Note: \ When \ y \ = \ 0, \ x \ = \ 9.\)

\(\displaystyle 1+ye^{xy}+xy'e^{xy} \ = \ 0, \ \frac{d[uv]}{dx} \ = \ uv'+vu'\)

\(\displaystyle xy'e^{xy} \ = \ -1-ye^{xy}, \ y' \ = \ \frac{-1-ye^{xy}}{xe^{xy}}\)

\(\displaystyle Now, \ if \ y \ = \ 0, \ then \ y' \ = \ \frac{-1}{x} \ = \ \frac{-1}{9}\)

\(\displaystyle Is \ this \ what \ you \ wanted?\)

 

[generico]

Aladdin: I should have clarified, I know how to solve for y in terms of x in the original function, but not sure how to use that in the problem as a whole.

BigGlenn: Thanks... I need to study that a bit to see how that becomes the derivative, not sure exactly how to get to the final derivative still, but it gives me a direction to ponder and study.

For the answer, that's spot on what the system tells me. Is it safe to assume when a question is asking for dx/dy when a variable = something, I plug that in to find the opposing variable, and then use both those points in the derivative?

Like, in this instance, plug 0 for y in the original function which gives you x=9, and just plug in the points (9,0) into the derivative once I find out what the derivative is to get -1/9. I know that may seem obvious because it appears that's what you did, but I'm trying to conceptualize that for other or similar problems like that I may find on an exam.

 

[BigGlenntheHeavy]

\(\displaystyle Another \ way:\)

\(\displaystyle e^{xy} \ = \ 10-x, \ xy \ = \ ln|10-x|. \ y(x) \ = \ \frac{ln|10-x|}{x}, \ \implies \ y(9) \ = \ 0\)

\(\displaystyle y'(x) \ = \ \frac{\frac{-x}{10-x}-ln{|10-x|}}{x^{2}} \ = \ \frac{-x-(10-x)ln|10-x|}{x^{2}(10-x)}\)

\(\displaystyle y'(9) \ = \ \frac{-9-ln(1)}{81(1)} \ = \ \frac{-1}{9}\)

\(\displaystyle See \ graph, \ note, \ the \ black \ curve \ is \ the \ derivative.\)

[attachment=0:1vrfwvl6]stu.jpg[/attachment:1vrfwvl6]

 

[generico]

Thanks... That helps me a little understand the derivative. I'll need to just memorize the steps and applications of it. Does anyone know the cost-inventory model example?

 

[galactus]

The classic model for inventory involves:

K=set up cost per order. In this case, $500

D=demand. In this case, 450,000 units.

h=holding cost per unit per year. In this case, $2 each

Total Cost per Unit Time, TCU, is \(\displaystyle \frac{K}{\frac{y}{D}}+h(\frac{y}{2})\)

The optimum order is found by minimizing TCU.

\(\displaystyle \frac{d(TCU(y))}{dy}=\frac{-KD}{y^{2}}+\frac{h}{2}=0\)

\(\displaystyle y=\sqrt{\frac{2KD}{h}}=\sqrt{\frac{2(500)(450000)}{2}}=15,000\)

Order 15,000 units.

 

[generico]

Thanks galactus. That is the right answer. I have a similar question that I set up the same way but can't get right:

Suppose that a company needs 150,000 items during a year and that in preparation for each production run costs 360 dollars. Suppose further that it costs 7 dollars to produce each item and 0.75 dollars to store an item for one year. Use the inventory cost model to find the number of production runs per year that will minimize the total costs of production and storage.


Okay, so this is the exact same question but with different numbers. I used the same method. That is, I'm not understanding the derivative part, yet, but I used the final formula you used. Perhaps that's not correct if I don't understand the derivative aspect, but I got:

I took the square root of 2(150,000)(360), quantity divided by .75, and got 12,000. I was told it was wrong. I'm not sure I grasp how you set it up still, but I plugged it in; was that not the correct way in this instance?