# Predict sign of partial derivatives

## Homework Statement

The temperature T at a location in the Northern Hemisphere depends on the longitude x, latitude y, and time t so we can write T=f(x,y,z); time is measured in hours from the beginning of January.

Honolulu has longitude 158 degrees W, and latitude 21 degrees N. Suppose that at 9:00 AM on January 1, the wind is blowing hot air to the northeast, so the air to the west and south is warm and the air to the north and east is cooler. Would you expect fx(158, 121,9), fy(158, 121,9), and ft(158, 121,9) to be positive or negative? Explain.

## Homework Equations

dT/dx = change in temperature as longitude varies
dT/dy = change in temperature as latitude varies
dT/dt = change in temperature as time varies

## The Attempt at a Solution

As longitude varies, the temperature in the north would get cooler, so would it be a negative partial derivative?

I'm not sure how to approach this.

dynamicsolo
Homework Helper

## Homework Equations

dT/dx = change in temperature as longitude varies
dT/dy = change in temperature as latitude varies
dT/dt = change in temperature as time varies

## The Attempt at a Solution

As longitude varies, the temperature in the north would get cooler, so would it be a negative partial derivative?

There is no special trick to partial derivatives: they still express rates of change of a function with respect to a change in an independent variable -- there is just more than one variable now.

The value of (North) latitude is increasing as you go north, while the temperature is falling. So you are correct that dT/dy < 0 . (I presume you meant 'latitude' in your question, since you mentioned the temperature 'in the north'.)

Be careful about dT/dx: what longitude is being used and which way does it increase? As for dT/dt, you need to consider how the air is moving and what will happen to the temperature at Honolulu as time passes...

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try to draw a graph of temperature

if
keeping y and z constant, and x goes from a (less positive x) to b (more positive x), and Temperature decreases, then dT/dx is?

HallsofIvy