# Rate of change from one point towards another

• Punkyc7
In summary, the conversation discusses how to find the rate of change of z when moving from the origin towards the point (2,1). The function e^(xy+x-y) is used to determine the rate of change and the concept of directional derivatives is introduced. The conversation also clarifies that the starting point should be used when calculating the directional derivative. Ultimately, the answer is found to be -1/sqrt(5).
Punkyc7
Suppose z=e^(xy+x-y). How fast is z changing when we move from the origin towards the point (2,1)?

Would it just be 1/sqrt((2^2+1))=1/sqrt(5), i am asking this because does the z=e^(xy+x-y) get used at all?

The function is what entirely determines the answer.
1/sqrt(5) is how much (x,y) is changing----they're asking for how fast z is changing.

Anytime you're looking for a rate of change, you should be thinking about 'derivatives'. In this case, because they're asking about a particular direction of motion, you should be thinking about 'directional derivatives'.

So e^(xy+x-y)
<y+1(e^(xy+x-y)), x-1(e^(xy+x-y))>

would you use the starting point or the ending point?

Great start Punkyc.

Punkyc7 said:
So e^(xy+x-y)
<y+1(e^(xy+x-y)), x-1(e^(xy+x-y))>
You have the gradient there, good; that's the first part of finding the http://en.wikipedia.org/wiki/Directional_derivative" .
The key is taking the dot product of that, with a unit vector in the direction of interest.

Punkyc7 said:
would you use the starting point or the ending point?
Look at the wording of the question carefully. You're not actually moving from the origin to (1,2), but you are at the origin moving in that direction.

Last edited by a moderator:
So it woud be <1,-1> dot (1/sqrt(5)<1,2>

so we have -1/(sqrt(5))

Yup! I think that's it.
Keep in mind: even though this is almost the exact answer you guessed in the first place, that's just a coincidence, and the process is important.

## 1. What is the definition of rate of change?

The rate of change is a mathematical concept that measures the speed at which a quantity changes from one point to another. It is also known as the slope or gradient of a line.

## 2. How is rate of change calculated?

The rate of change is calculated by finding the difference in the y-coordinates (vertical change) and dividing it by the difference in the x-coordinates (horizontal change) between two points on a graph.

## 3. What does a positive rate of change indicate?

A positive rate of change indicates that the quantity is increasing as the input variable increases.

## 4. What does a negative rate of change indicate?

A negative rate of change indicates that the quantity is decreasing as the input variable increases.

## 5. How is rate of change used in real-world applications?

Rate of change is used in various fields such as physics, economics, and engineering to analyze and predict changes in quantities over time. For example, it can be used to calculate the speed of a moving object or to determine the rate of growth in a population.

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