Tutor.com The Derivative Session

Jan. 15, 2013

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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.
[00:24:34] Follow so far?

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 !