We finished the concept of directional derivatives, introducing the notation for the gradient of a function of several variables. We proved the formula of the maximum value of the directional derivative, as well as the direction for which it is maximized. Next time we’ll begin basic optimization theory for functions of several variables.
February 14, 2007
In class on Tuesday, we stated and used the chain rule for partial derivative: For [tex]z=z(x_1, x_2, \ldots, x_n)[/tex] be a function of [tex]n[/tex] variables, such that [tex]x_i = x_i(t_1, t_2, \ldots, t_m)[/tex] is a function of [tex]m[/tex] variables for each [tex]i = 1, \ldots, n[/tex] then [tex]\displaystyle \frac{\partial z}{\partial t_j} = \sum_{i=1}^n \frac{\partial z}{\partial… [Read more…]
February 12, 2007
In Calculus IV on Thursday, February 8th, we covered the derivation of tangent planes. We also showed how the tangent plane for functions of 2 variables generalizes to functions of several variables, a process we call linearization. We also used this concept of linearization to define differentiability for functions of several variables. We also covered… [Read more…]
February 6, 2007
We have finally started taking partial derivatives in Calculus IV. This is definitely one of my favorite parts of the entire Calculus sequence. Today, we finished up talking about limits of functions of several variables and the concept of continuity. We, then, introduced the concept of holding one variable fixed and finding the derivative of… [Read more…]
February 2, 2007
We picked up where we left off prior to the Exam on Tuesday. We are discussing function of several variables. I began by recapping the techniques we use for visualizing the graphs of functions of two variables, namely, surfaces and contour maps, or level curves. We used this as a springboard to move up to… [Read more…]
January 31, 2007
During Calculus IV on Tuesday, we had Exam I. I’ve decided that I’m not generally very good at producing tests at the appropriate level for the course. Usually they are much too hard (1 or 2 problems that require a LOT of tedious calculation) or too long (too many problems). The reason is not that… [Read more…]
January 25, 2007
Having just finished a very short chapter on vector functions, we began the chapter that will cover Partial Derivatives. Before we started that lecture, I took a little class time to answer a homework question. It was one of my favorites. Having covered the concept of curvature, the students were asked to find a polynomial,… [Read more…]
February 18, 2007
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