"In exercises 9 through 20, use the method of Example C to find a general expression for the slope of a tangent line to each of the indicated curves. Then find the slope for the given values of X…" OK, let's try it. I have worked out the method of Example C a couple of times on paper, so I know what I have to do. It's painstaking. I have to state all the components in ways that can lead to a clear solution for that general expression. Not only are there X's and Y's, there are the subscripts, that is the X₁ and X₂ and the Y₁ and Y₂. And Y₂ has to be stated in its relation to Y₁.

That's where the Delta comes in, the sign for "amount of change." I'm working with intervals here rather than single numbers. Eventually, in these problems, the interval dwindles down towards zero, which is how you solve it, but Delta is always there to remind me that this is not yet familiar territory for me.

So far I have not gotten any of the problems right. It's not that I don't understand the process. It's that I continue to make mistakes working out the algebra in the problems, where numbers and letters have to be multiplied and added and subtracted and canceled out. I haven't done much of this since my algebra revival days of 2001 and 2002. The negative and positive signs are especially exasperating, as I remember from those days. And if a bunch of things are inside parentheses, if you subtract them from a bunch of other things, all the signs flip over from negative to positive and vice versa.

Sometimes I copy the problem wrongly out of the book, so I fail from the start. It probably would help if I were not doing this at 3 AM after a rather pressured and frantic time at my day job at the gourmet store. We are preparing a seasonal feature with ads all over the store. I am not using calculus to find the tangents of tangerines or the polynomials of pomegranate juice. But in spare moments, I mentally recite to myself: "Y₂ = Y₁ + Delta Y. Y₁ = f(X₁)." Add and subtract and simplify. Then divide by Delta X. As Delta X approaches zero… remember the negative and positive signs! Or are they organic tomato or pumpkin butter signs? An instantaneous rate of change for point P. Amazing fall values.

Posted at 3:25 am | link