Free Printable Chapter 2 Partial Derivative

1.

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

2.

Define a two-variable functions as follows

f\left(x,y\right)=x^2+2xy+y^2,\ \frac{\partial f}{\partial x}=?

a)

$2x+2y+2x\frac{\text{d}y}{\text{d}x}+2y\frac{\text{d}y}{\text{d}x}$

b)

$2x+2y$

c)

4y

d)

$\frac{\text{d}f}{\text{d}x}+2xy$

3.

Let

f\left(a,b,c\right)=\cos\left(ab\right)+\sin\ \left(b\right)+c\ where\ f\left(\left(\frac{1}{2},\frac{\pi}{3},7\right)\right)

Evaluate

\frac{\partial f}{\partial a}\left(\frac{1}{2},\frac{\pi}{3},7\right)

a)

$\frac{\pi}{3}$

b)

$-\frac{\pi}{3}$

c)

$-\frac{\pi\sqrt{3}}{12}$

d)

$\frac{\pi}{24\sqrt{3}}$

4.

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

5.

$z=\left(2x+3y\right)^{10}$ What is the correct partial derivative?

a)

$z'=60\frac{\text{d}y}{\text{d}x}\left(2x+3y\right)^{\left(9\right)}$

b)

$f_y=20\left(2x+3y\right)^9$

c)

$f_x=20\left(2x+3y\right)^9$

d)

$f_z=30\left(2x\right)^9$

6.

Find

f_y

for the function

f\left(x,y\right)=yx^3-2x^2y^3

a)

$y-y^3$

b)

$y-6xy^2$

c)

$x^3-6x^2y^2$

d)

$yx^3-2x^2y^3$

7.

Which one of the following is always true?

a)

$f_{xy}=f_{xx}$

b)

$f_{xy}=f_{yx}$

c)

$f_{xx}=f_{yy}$

d)

$f_{xy}=f_{yy}$

8.

$f\left(x,y\right)=x^3y^2$

Find the first order partial derivative with respect to y of

a)

$f_y=3x^2y^2$

b)

$f_y=3x^2y^2+2x^2y$

c)

$f_y=6x^2y$

d)

$f_y=2x^3y$

9.

$z\left(x,y\right)=7yx-4x^3y^2$

Determine the first partial derivative with respect to x for

a)

$7y-12x^2$

b)

$7y-12x^2y^3$

c)

$7y-12x^2y^2$

d)

$7y-12x^2y^3$

10.

$f\left(x,y\right)=x^3-2xy+3y^2$ 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)

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

Chapter 2 Partial Derivative

Total questions: 10

Worksheet time: 20mins

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