Sections 3.5, 9.3, and 9.4

Main Points: In Section 3.5, it is stated that a periodic function has a periodic derivative. Also, the proof that dy/dx (sinx)= cosx and dy/dx (cosx)=-sin(x). Also, if a constant, k, is present in the the periodic function, the derivatives are as follows: [d/dt(sinkt) = kcoskt] and [d/dt(coskt)=-ksinkt]. In section 9.3, the book discusses partial derivatives with respect to both x and y (or a and b). The formulas for the partial derivative are, with respect to x, lim (h-> 0) [f(a+h,b) - f(a,b)]/h, and with respect to y, lim(h-> 0) [f(a,b+h) - f(a,b)]/h. Furthermore, it is proven that the partial derivative can be estimated through both tables and contour diagrams. Local linearity is introduced ( delta f is about equal to f with respect to x times delta x + f with respect to y times delta y). In 9.4, the book states that partial derivatives are able to be computed algebraically by fixing one of the variables. The second order derivative formulas are all listed (if I knew how to get the special characters, I would list them all!). Also, if fxy and fyx are continuous at point (a,b), then fxy(a,b)= fyx(a,b). This is called mixed partial derivatives.

Challenges: In section 9.4, I was a little confused on the Second Order Partial Derivatives. I'm not so sure about specifics, but I think I just need to hear them explained in real life rather than in a text.

Reflections: This information is very useful when trying to find an answer to a problem with more than one variable. Also, since we have already studied contour diagrams, getting partial derivatives using these graphs is the next and more complex step in our math education. Furthermore, partial derivatives encompass all of the knowledge we have gained from the techniques for differentiation for single-variable function and then add on another step.