Partial Derivatives Quiz

The temperature 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 is 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 the time with respect to temperature

• 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

Define a two-variable function as follows: f(x, y) = x² + 2xy + y²

∂f/dx = _______

• 2x + 2y

• 2x + 2y + 2x(dy/dx) + 2y(dy/dx)

• 4y

• df/dx + 2xy

Let f(a,b,c) = cos(ab) sin(b)+c

where f(1/2, π/3, 7)

Evaluate ∂f/∂a (1/2 ,π/3, 7)

• π/3

• -π/3

• -π(sqrt(3)) / 12

• π/24 / sqrt(3)

What can we informally define partial derivatives graphically?

• A constant always

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

• The derivative of the whole thing

• The 3D derivative

z= (2x + 3y)¹⁰

Which is the correct partial derivative?

• z' = 60 (dy/dx) (2 + 3y)⁹

• f_x= 20 (2x + 3y)⁹

• f_y= 20 (2x + 3y)⁹

• f_z= 30 (2x)⁹