What is the derivative of a function?
It's the slope of the tangent line at any point, of course! Well, as you have probably surmised, this week's focus was derivatives. Hooray! Apparently we had learned a bit about derivatives before, especially concerning how to calculate the derivative of a function at any point (more on that later), in Honors Pre-Calc. However, instead of calling it derivatives like any normal thing would do to save confusion, the book called it the difference quotient. (Why do you do this to us, math textbooks? *shakes fists at sky*)
Our first activity with derivatives this week dealt with what I like to call the "zooming in" characteristic of derivatives. The official term is local linearity. (I like mine better, though) In this activity, we discovered that by zooming closely into a curve a point, the curve would appear to be a straight line in the graphing window. We could then calculate the slope of the line in respect to a certain point. Here we were also shown a few situations where the derivatives do not exist, a topic which was elaborated on later in the week. The derivative fails to exist if when x=a, there is a corner, cusp, vertical tangent, or discontinuity.
This activity led us to the formal definition of the derivative of a function with respect to x:
It's the slope of the tangent line at any point, of course! Well, as you have probably surmised, this week's focus was derivatives. Hooray! Apparently we had learned a bit about derivatives before, especially concerning how to calculate the derivative of a function at any point (more on that later), in Honors Pre-Calc. However, instead of calling it derivatives like any normal thing would do to save confusion, the book called it the difference quotient. (Why do you do this to us, math textbooks? *shakes fists at sky*)
Our first activity with derivatives this week dealt with what I like to call the "zooming in" characteristic of derivatives. The official term is local linearity. (I like mine better, though) In this activity, we discovered that by zooming closely into a curve a point, the curve would appear to be a straight line in the graphing window. We could then calculate the slope of the line in respect to a certain point. Here we were also shown a few situations where the derivatives do not exist, a topic which was elaborated on later in the week. The derivative fails to exist if when x=a, there is a corner, cusp, vertical tangent, or discontinuity.
This activity led us to the formal definition of the derivative of a function with respect to x:
Which we all learned in Pre-Calc as the difference quotient. See rant in first paragraph. It uses the idea of limits that we discussed last week and simplifying skills learned in Alegebra II to create a function that describes the slope of the tangent line at any point on the graph. Anyways, we used the formula to solve a few problems, but it wasn't that hard. There was just a ton of simplifying. But I have no complaints. I like to simplify expressions. No sarcasm at all. Our next topic, though, focused less on finding the derivative more on the properties of the derivative function's graph in comparison to the original function. It turns out that where the original function had a rising slope, the derivative will be above the x-axis. If it was a falling slope, the derivative would be under the x-axis. The maximums and minimums of the original function turn into the zeroes of the derivative. What fascinates me is that, although we can predict the graph of a derivative based on these properties, the inverse cannot happen. The graph of the original function cannot be obtained from the graph of its derivative because, if given only the derivative, the information is not specific enough.
The rest of the week focused on solidifying these ideas into our heads. (Or beating it into our skulls, whatever way you prefer.) I didn't find this subject particularly hard, mainly because we covered this a bit in Pre-Calc. The concepts were also relatively simple to get as well. I also found some really cool (albeit a bit cheesy) videos that NOVA created relating derivatives to surviving the zombie apocalypse. These are the videos that I've been meaning to share with Mr. Cresswell, but I keep forgetting to email him. Welp, they're here now. Now that we are getting out of the review stage of Pre-Calc and into actual Calculus, I'm getting more and more excited for this class!!
The rest of the week focused on solidifying these ideas into our heads. (Or beating it into our skulls, whatever way you prefer.) I didn't find this subject particularly hard, mainly because we covered this a bit in Pre-Calc. The concepts were also relatively simple to get as well. I also found some really cool (albeit a bit cheesy) videos that NOVA created relating derivatives to surviving the zombie apocalypse. These are the videos that I've been meaning to share with Mr. Cresswell, but I keep forgetting to email him. Welp, they're here now. Now that we are getting out of the review stage of Pre-Calc and into actual Calculus, I'm getting more and more excited for this class!!