-
Higher order partial derivatives: We learned how to compute higher order partial derivatives of a 2-variable function. This involved taking partial derivatives of partial derivatives. The key factor is that the higher order partial derivative of any function in this course won’t be affected by its order. However, there are some functions that have different partial derivatives when we consider different order (see Homework 2).
-
Taylor expansion: We introduced the Taylor expansion of a 2-variable function, which allowed us to approximate the value of the function at a given point using a polynomial. We give a formula and learn how to compute the first few terms of a Taylor expansion. A interesting example is that if a function $F(x,y)$ is a product of function, says $F(x,y)=f(x)g(y)$. Then, the multivariable Taylor expansion of $F$ is simply the product of Taylor expansions of $f$ and $g$. Also, if $F(x,y)= f(g(x,y))$ where $f$ is a single variable function that has a Taylor expansion at $g(0,0)$ and $g(x,y)$ is a multivariable polynomial. Then, to obtain the Taylor expansion of $F$, we simply plugin the polynomial $g(x,y)$ into the Taylor expansion of $f$, expand, and organize the rsult.
-
Examples: The Taylor expansion of $e^{(x+y)}$ at $(0,0)$ is, by formula, \(1+ x + y + \frac{1}{2!}(x^2+2xy+y^2) + \frac{1}{3!}(x^3+3x^2y+3xy^2+y^3)+\cdots.\) We observe that the Tayor expansion of $e^t$ at $0$ is \(1+t+\frac{1}{2!}t^2 + \frac{1}{3!}t^3+\cdots,\) so we regain the Taylor expansion of $e^{(x+y)}$ by replace $t$ with $x+y$. Finally, we can also gain the Taylor expansion of $e^{x+y}$ by considering the fact that $e^{x+y}=e^xe^y$. The detail is shown in the class.