Free Printable Partial Derivatives

1.

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

a)

^{$f_y(x,y)=3x^2y^2+3e^y$}

b)

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

c)

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

d)

$f_y(x,y)=6x^2+3e^y$

2.

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?

a)

The rate of change of temperature with respect to time

b)

The rate of change of temperature as longitude and time varies

c)

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

d)

The rate of change of the time with respect to temperature

3.

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

a)

$2xy-2y^3$

b)

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

c)

$\frac{-2y}{x}$

d)

$2$

4.

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:

a)

Saddle point

b)

Local Max

c)

Local Min

d)

Cannot be determined

5.

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

a)

A local max

b)

A local min

c)

A saddle point

d)

Cannot be determined.

6.

Use implicit differentiation to find

\frac{\partial z}{\partial x}

given that

yz^4+x^2z^3=e^{xyz}

.

a)

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

b)

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

c)

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

d)

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

7.

Partial derivatives graphically is define as

a)

A constant

b)

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

c)

The 3D derivative

d)

The derivative of the whole thing

8.

Which one of the following is always true?

a)

$f_{xy}=f_{xx}$

b)

$f_{xy}=f_{yx}$

c)

$f_{xx}=f_{yy}$

d)

$f_{xy}=f_{yy}$

9.

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

a)

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

b)

$f_{xx}=6x;\ f_{xy}=-2+3y^2$

c)

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

d)

$f_{xx}=6+6xy;f_{yy}=6x$

10.

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

a)

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

b)

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

c)

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

d)

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

Partial Derivatives

Total questions: 10

Worksheet time: 15mins

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