## JohnM 2 years ago 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."

1. Waynex

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.

2. JohnM

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?

3. Waynex

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 $a \le y^2 \le b.$

4. JohnM

5. Waynex

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

6. JohnM

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.