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JohnM
Session 27: Approximation Formula Two questions about Example 1 in the PDF titled "The Tangent Approximation". 1. "Thus if |Δx|.01 andΔy|.01, we should have . . ." Where do these .01 values come from? They appear to be arbitrary. 2. "So the answer is the square with center at(1,1) given by |x −1|.01, |y −1|.01 ." I don't understand what this answer means, what it is telling me. Are we literally talking about a square with dimensions a x b? If yes, I don't understand the logic of the question asking for a "reasonable square."
Yeah this is an obtuse one here. We're looking for \[\Delta w\]with a value of .1 or less. Note the question asks for a value that will not vary by more than one tenth. We have an approximation formula that gives a proportion of how \[\Delta x\]and\[\Delta y\]affects \[\Delta w,\]\[\Delta w = 3 \Delta x + 4 \Delta y.\]We can arbitrarily pick a value to plug into delta x and delta y which will be less than or equal to one tenth. One hundredth is such a number.\[.7 \approx 3 * .01 + 4 * .01.\]The absolute values used in the pdf cause this aproximate equation to have less than or equal to signs instead of the aproximate equal symbol. Now for part two of your question. This statement: \[\left| x-1 \right| \le .01\]is a distance x - 1 less than .01. When you have an absolute value less than a number, you can remove the absolute value and get this equation: \[-.01 \le x - 1 \le .01\]What we have is just a closed interval for our reasonable square to live in. An unreasonable square might be something with a closed interval of\[-.00001 \le x - 1 \le .00001\] which is much smaller than required.
Thanks. I will study this answer. Follow-up question: "reasonable square" refers to how to frame the numbers, e.g., −.01≤x−1≤.01, yes? In this case it means a range. It doesn't literally mean a shape (e.g., a triangle or circle) that in this case is square, yes?
It is indeed a range. But, since there is a range of x values, and a range of y values, those two ranges together form a square in the x-y plane. If the range of x values was different from the y values it would form a rectangle. Later on in Real Analysis you'll study open/closed balls were a range is circular, i.e., a closed ball from a to b: \[a \le x^2 \le b\] and \[a \le y^2 \le b.\]
Thanks for your kind help. Hope you can accommodate more follow-up. 1. Real Analysis: I don't see this listed among the remainder of the Multivariate sessions nor among MIT's Open Courseware math listings. 2. Based on your most recent answer, in the case of Example 1 it seems you are telling me we want an answer that functions as both a range and literally a square. Yes? 3. #2 and the Christine Breiner Tangent Plane Approximation recitation video leads me to believe there are additional levels of understanding to the Tangent Plane Approximation that are eluding me. In trying to visually grasp the things taught in Session 27, I was trying to get my mind around the idea of finding planes and lines that are tangent to a two-variable function. Just when I thought I understood the concept (a plane tanget or roughly tangent to a point on a function), Christine introduces using Tangent Plane Approximation to find the area of a shape, in her example a square!! Now I'm having trouble wrapping my mind around the idea of using Tangent Plane Approx. to find area. How can a plane be tangent to a square? But maybe that leads back to your comments about Example 1, in which ultimately we find a square (from which we can compute area if we wanted to). So does this mean Tangent Plane Approx. can be used to find the area of functions that describe squares or rectangles (x times y functions), but not shapes that are curvy (e.g., hyperbolic)? 4. In her video Christine says we can simply calculate the area using the numbers 2 & 3 given, but she wants us to use Tangent Plane Approx. to get the answer. OK, but why are we doing this? What deep-level thing are we supposed to understand by taking a Tangent Plane Approx. rather than doing a straightforward calculation, as Christine says we can also do? Sorry for the length. I've been OK with the Multivariate course until now, but the point of this Approx. stuff is confusing me.
You're welcome. I've had the same experience where I had a grasp of a tangent plane apx. (short for approximations here on out) and then felt like I lost it, or failed to really grasp it. 1) 18.100A Introduction to Analysis. I actually watched the Harvey Mudd College videos for Real Analysis on youtube. It's easy to find. You'll probably want to get through diff equations, and linear algebra first though. 2) We are after the range. The range happens to be portion of the x-y plane that is a square. 3) I think the confusion here is that the concept of apx., is an abstract concept detached from geometric visualizations. The rectangle in the recitation video is really just an equation for area: a = x*y. We want to apx. the value of a by looking at small changes in x and y. So we have \[\Delta a = a _{x} * \Delta x + a _{y} * \Delta y,\] where\[a _{x}\]and\[a _{y}\]are the partials of "a" with respect to x and y. A rectangle which lives on a plane, wouldn't have a tangent plane, but the equation for it's area does. Apx. in and of itself is not enough to find the area of anything. But if we have an area equation, we can take a single area calculated by the equation, and apx. what a small change in one of the variables does to the value of that specific area. The deep level concept is that we want to be able to use partial derivatives to examine rates of change in a function. Now worries for the length. This stuff is important. You're right to spend the time focusing on this and getting it solid in your mind. Cheers
I'm getting it better than I did before. The best thing is to get the kind of help folks like you can provide and re-read and think hard about what the PDFs say about the tangent plane and the approx. formula. Again, thanks.