Homework

$1$ If $2x+3y+2z=9$ is the tangent plane to the graph of $z=f(x,y)$ at the point $(1,1,2).$

Extimate $f(1\mathrm{.}01,0\mathrm{.}98).$

$2$ Estimate ${1000}^{\frac{1}{5}}$ using linear approximation

$3$ Find $f(0\mathrm{.}01,0\mathrm{.}999)$ for $f(x,y)=cos(xy)y+sin(x+y).$

$4$ Find the linear approximation $L(x,y)$ of the function

$f(x,y)=\sqrt[\phantom{2}]{10-x^2-5y^2}$

at $(2,1)$ and use it to estimate $f(1\mathrm{.}95,1\mathrm{.}04).$

$5$ Sketch a contour map of the function

$f(x,y)=x^2+9y^2$

find the gradient vector gradf$=($:$f_x,f_y$:$)$ of $f$ at the point $(1,1).$ Draw it together with the

tangent line $ax+by=d$ to the curve at $(1,1).$