Math CentralQuandaries & Queries


Question from Tom, a student:

I just had a quick calc question about wording that wasn't ever addressed in class. When the book says "the rate of change of y with respect to x", should it be considered how fast y is changing in
comparison to x?

I ask because the textbook says that "y is changing 3 times faster than x, so the rate of change of y with respect to x is 3." I'm use to rate being like velocity, as in units of distance per units of time. All we're told in class is that it's the slope of the tangent line, I was hoping you could clarify for me what exactly is meant by the wording of a "rate of change of something with respect to something else". More specifically, what "rate" and "with respect to" mean within this context?

Thanks for your time

Hi Tom,

You are correct, the expression "the rate of change of y with respect to x" does mean how fast y is changing in comparison to x. In your example of velocity, if y is the distance travelled in miles, and x is the time taken in hours then y/x is the average velocity in miles per hour. Velocity is the rate of change of distance with respect to time.

For another example, if you are riding a bicycle up a hill you might want to know how steep the hill is. One way to measure the steepness or grade of the hill is to measure how much your altitude changes when you go a specific distance. For example if your altitude changes 370 feet (y = 370 feet) as you go a horizontal distance of 1 mile (x = 5280 feet) then the rate of change of the altitude with respect to the horizontal distance travelled is 370/5280 = 0.07. A mathematician would say the slope is 0.07 but the sign on the road expresses it as a percentage and says "7% grade".

In both of these examples I have calculated the average rate of change of y with respect to x. I would really like to know the instantaneous rate of change, the slope of the hill at a specific point, but that means x = 0 and I can't calculate y/x. Defining the instantaneous rate of change as the slope of the tangent line at the point is the beginning of differential calculus.

I hope this helps,

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