Partial Derivatives

Find the first order partial derivative with respect to y
 $f(x,y)=x^3y^2+3xe^y$  
.

•  ^{$f_y(x,y)=3x^2y^2+3e^y$}  

•  $f_y(x,y)=3x^2y^2+2x^3y+3e^y+3xe^y$  

•  $f_y(x,y)=2x^3y+3xe^y$  

•  $f_y(x,y)=6x^2+3e^y$  

The temperature at T at a location in the Northern Hemisphere depends on longitude x, latitude y, and time t, so we can write T=f(x,y,t). The time measured since January 1. What is the meaning of the partial derivatives dT/dx?

• The rate of change of temperature with respect to time

• The rate of change of temperature as longitude and time varies

• The rate of change of temperature as longitude varies with latitude and time fixed

• The rate of change of the time with respect to temperature

Find dy/dx by Implicit Differentiation
 $2xy^3-x^2y=2$  

•  $2xy-2y^3$  

•  $\frac{2y\left(x-y^2\right)}{x\left(6y^2-x\right)}$  

•  $\frac{-2y}{x}$  

•  $2$  

Suppose at the point (1,2, f(1,2)),  $f_{xx}\left(1,2\right)=-5$  , $f_{yy}\left(1,2\right)=4$  , and  $f_{xy}\left(1,2\right)=-2$  .  Then at that point there is a:  

• Saddle point

• Local Max

• Local Min

• Cannot be determined

When finding local extrema, suppose the D>0 and $$f_{xx}<0$$ at a point.  Then that point is: 

• A local max

• A local min

• A saddle point

• Cannot be determined.

Use implicit differentiation to find  $\frac{\partial z}{\partial x}$  given that  $yz^4+x^2z^3=e^{xyz}$  .

•  $\frac{yze^{xyz}-2xz^3}{4yz^3+3x^2z^2-xye^{xyz}}$  

•  $\frac{2xz^3-yze^{xyz}}{4yz^3+3x^2z^2-xye^{xyz}}$  

•  $\frac{4yz^3+3x^2z^2-xye^{xyz}}{yze^{xyz}-2xz^3}$  

•  $\frac{xye^{xyz}-4yz^3-3x^2z^2}{2xz^3-yze^{xyz}}$  

Partial derivatives graphically is define as

• A constant

• The 2D slope of the slice of a 3d graph

• The 3D derivative

• The derivative of the whole thing

Which one of the following is always true?

• $f_{xy}=f_{xx}$

• $f_{xy}=f_{yx}$

• $f_{xx}=f_{yy}$

• $f_{xy}=f_{yy}$

 $f\left(x,y\right)=x^3-2xy+3y^2$   Which of the following is true?

•  $f_x=3x^2-2y+y^3;\ f_y=6x$  

•  $f_{xx}=6x;\ f_{xy}=-2+3y^2$  

•  $f_x=-2x+6y+3xy^2;f_y=3x^2-2y+y^3$  

•  $f_{xx}=6+6xy;f_{yy}=6x$  

Find  $f_x$  and  $f_y$  from the equation  $f\left(x,y\right)=\ \ln\left(\sqrt{x^2y^3}\right)$  

•  $f_x=\frac{1}{x}\ \ \ \ \ \ \ f_y=\frac{3}{2y}$  

•  $f_x=\frac{1}{xy}\ \ \ \ \ \ \ f_y=\frac{3x}{2y}$  

•  $f_x=\frac{1}{x}\ \ \ \ \ \ \ \ f_y=\frac{3x}{2}$  

•  $f_x=\frac{x^2}{y^3}\ \ \ \ \ f_y=\frac{3x^2}{2y^3}$