Homework 7: Partial Derivatives

$\newenvironment {prompt}{}{} \newcommand {\DeclareMathOperator }[2]{\@OldDeclareMathOperator {##1}{##2}\immediate \write \myfile {\unexpanded {\DeclareMathOperator }{\unexpanded {##1}}{\unexpanded {##2}}}} \newcommand {\ungraded }[0]{} \newcommand {\reserved@a }[2]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{} \newcommand {\vnameref }[1]{\unskip ~\nameref {##1}\@vpageref [\unskip ]{##1}} \newcommand {\ref }[0]{\@ifstar \@refstar \T@ref } \newcommand {\pageref }[0]{\@ifstar \@pagerefstar \T@pageref } \newcommand {\nameref }[0]{\@ifstar \@namerefstar \T@nameref } \newcommand {\reserved@a }[2]{} \newcommand {\reserved@a }[0]{\AtBeginDocument } \newcommand {\reserved@a }[1]{} \newcommand {\reserved@a }[2]{}$

Completion Packet

Compute the partial derivatives of the function $f(x,y) = ye^{x^2}$ .

Compute the partial derivatives of the function $f(x,y) = \ln ((x+y)^2)$ .

Compute the partial derivatives of the function $f(x,y,z) = \sin (x^2yz)$ .

Consider the function $f(x,y) = 2x^2 + xy -y^2 + 3y + 1$ .

(a): Explain how you can tell that this function is differentiable at the point $(2,1)$ .
(b): Find an equation for the tangent plane to the graph of $f$ at the point $(2,1)$ .

Consider the function $f(x,y) = e^{x^2+y^2}$ .

(a): Explain how you can tell that this function is differentiable at the point $(1,2)$ .
(b): Find an equation for the tangent plane to the graph of $f$ at the point $(1,2)$ .

Suppose we have a differentiable function $f:\mathbb {R}^2\rightarrow \mathbb {R}$ such that $f(0,0) = 5$ , $f(0.1, 0) = 5.2$ , and $f(0,0.1) = 4.7$ .

(a): Estimate $f_x(0,0)$ and $f_y(0,0)$ .
(b): Give an approximate equation for the tangent plane to the graph of $f$ at the point $(0,0)$ .
(c): Using your approximation from (b), estimate the value of $f(-0.2, 0.2)$ .

Consider the function $f(x,y) = \begin {cases} \frac {x^3 + 2x^2y -xy^2 + 3y^3}{x^2+y^2} & \text { if }(x,y)\neq (0,0)\\ 0 & \text { if }(x,y) = (0,0) \end {cases}.$

(a): Compute the partial derivatives $f_x(x,y)$ and $f_y(x,y)$ for $(x,y) \neq (0,0)$ .
(b): Compute the partial derivatives $f_x(0,0)$ and $f_y(0,0)$ .

Consider the function $f(x,y) = e^{x+y}$ .

(a): Compute the partial derivatives $f_x(0,0)$ and $f_y(0,0)$ .
(b): Graph the function.
(c): Based on your graph, is $f$ differentiable at $(0,0)$ ?

Consider the function $f(x,y) = (xy)^{2/3}$ .

(a): Compute the partial derivatives $f_x(1,1)$ , $f_y(1,1)$ , $f_x(0,0)$ , and $f_y(0,0)$ .
(b): Graph the function.
(c): Based on your graph, is $f$ differentiable at $(1,1)$ ? Is $f$ differentiable at $(0,0)$ ?

Graded Problems

Consider the function $f(x,y) = x^2+y^2$ .

(a): Find an equation for the tangent plane to the graph of $f(x,y)$ at the point $(a,b)$ .
(b): At which point(s) $(a,b)$ is the tangent plane parallel to the plane $x+2y-z = 0$ ?

Find the partial derivatives of the function $f(x_1,...,x_n) = x_1x_2^2x_3^3x_4^4\cdots x_n^n.$