# Tutor.com The Derivative Session

##### Jan. 15, 2013

Session Transcript - Math - Calculus, 1/8/2013 5:02PM - Tutor.comSession Date: 1/8/2013 5:02PM
Length: 35 minute(s)
Subject: Math - Calculus

System Message
[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Cindy (Customer)
[00:00:00] Hi can you pls help me w math hw? [ File > http://www.tutor.com/SharedSessionFiles/d3cfb2c3-aba7-4c6d-bc04-d5f2c8618b76_Tank.jpeg ]

Johnny M (Tutor)
[00:00:10] Hi there! And welcome to tutor.com
[00:00:17] Give me just a moment to review the problem :)

Cindy (Customer)
[00:00:30] ok

Johnny M (Tutor)
[00:02:15] Alrighty, any ideas how to approach this problem?

Cindy (Customer)
[00:02:17] it's an optimization problem and in school I have been learning about using the second deriv test
[00:03:15] is what I wrote on the board right

Johnny M (Tutor)
[00:03:29] Yes, so far so good...
[00:04:32] Good, and they tell us the volume is 500...
[00:05:36] What might we do next?
[00:07:20] What shape is the base?

Cindy (Customer)
[00:07:56] sq

Johnny M (Tutor)
[00:08:12] Very very close, remember volume is all multiplication...so we would have x * x * y = 500..
[00:09:30] Perfect, now looking at the area's...can we make an equation describing that?
[00:11:37] Looks good...now thinking of the areas of the rectangular prism...can we think of an equation describing that?
[00:13:39] Very close...first we have the area of the base, and it being a square, we have...
[00:14:07] Then the sides are 4 rectangles, with area's of xy, so we have...
[00:14:24] Make sense?

Cindy (Customer)
[00:14:35] yes

Johnny M (Tutor)
[00:14:38] Remember the top is open...so there is only one square.
[00:15:20] Great, so what might we do next?

Cindy (Customer)
[00:16:38] plug in y

Johnny M (Tutor)
[00:16:53] Perfect! That will get our area equation in terms of a single variable.
[00:17:01] So we would get...?
[00:17:55] Perfect, and next?

Cindy (Customer)
[00:19:02] i don't know

Johnny M (Tutor)
[00:19:17] Once you have the primary equation in terms of one variable, we can take our derivative.
[00:19:24] So, what would the derivative of this be...?
[00:20:20] Perfect, and once we find our derivative, we...?

Cindy (Customer)
[00:20:32] set equal to o
[00:20:39] 0

Johnny M (Tutor)
[00:20:42] Right!

Cindy (Customer)
[00:20:54] why do we use the area equation?
[00:21:05] and not the volume equation

Johnny M (Tutor)
[00:21:37] The tank is made of a metal...but, the metal does not fill up the tank, it represents what the sides are made of...
[00:21:51] So the tank with the smallest area, will use the least amount of metal.
[00:22:06] Make sense?

Cindy (Customer)
[00:22:43] sothe lowest surface area equals lowest weight?

Johnny M (Tutor)
[00:23:10] Right, because the surface area represents how much metal we would use to shape the tank.
[00:23:57] Just because it can hold 500 ft^3 of a substance, does not change the weight of it. It's empty right now, so the only thing that makes it weight more, would be using more metal to "shape" the tank.

Cindy (Customer)
[00:24:58] yes

Johnny M (Tutor)
[00:25:32] Very close, we actually don't need the second derivative. Just the first.
[00:26:56] Right, so x = 10 represents one of the dimensions, so how would we solve for y?
[00:28:08] Right!
[00:28:23] So our dimensions are 10 by 10 by 5, for our tank.
[00:28:25] Make sense?

Cindy (Customer)
[00:29:26] why did we have to take the first deriv and set eq to 0 to find X?

Johnny M (Tutor)
[00:30:03] The first derivative finds maximum and minimum, the second finds concavity and points of inflection.
[00:30:13] Since we were looking for a minimum, that's from the first derivative.

Cindy (Customer)
[00:30:55] but when you find con cavity don't younalsofind the min or max from the curve

Johnny M (Tutor)
[00:31:37] Not necessarily, what you find is that it "Is Concave Up", but it does not find the maximum point for that interval of concavity.
[00:31:40] For instance...
[00:32:13] That is concave up, but the second derivative cannot tell me the lowest point across that concavity.
[00:32:18] The first derivative can.

Cindy (Customer)
[00:32:52] ohh I understand now
[00:32:59] so the second deriv is used to justify
[00:33:08] that there is a min

Johnny M (Tutor)
[00:33:14] Right.

Cindy (Customer)
[00:33:27] thank you so much

Johnny M (Tutor)
[00:33:28] And we set it equal to 0, because the slope of a tangent line drawn at the lowest point, would be 0...
[00:33:51] My pleasure! Was there anything else I could help you with?

Cindy (Customer)
[00:34:27] no
[00:34:30] thanks

Johnny M (Tutor)
[00:34:42] Great! Be sure to fill out the survey for me, and thanks for using Tutor.com !