Can someone solve this problem, please? Let fuay be a function of r. In class, w

Question

Can someone solve this problem, please?

Let fuay be a function of r. In class, we have discussed the average rate of change of y with respect to z and the instantaneous r of change of y with respect to z at the point r a. We have also discussed secant lines to the graph of f and lines tangent to the graph of f at C a. All of these concepts are related to each other and to the derivative off at a In one or two paragraphs, state precisely and discuss the explicit relation- ships between all of these concepts and describe the role of limits in these relationships. Use as few mathematical symbols as possible and answer the question in your own words

Explanation / Answer

The average rate of change of a function y = f(x) on an interval from x1 to x2 is just the ratio of the change in y to the change in x:

y/x = f(x2) f(x1)/(x2 x1)

For example, if f measures distance traveled with respect to time x, then this average rate of change is the average velocity over that interval. But that leaves us with the question of what is the in-
stantaneous velocity at some moment x0, the velocity that the speedometer in a car is claimed to give us?
The answer is in some sense quite easy to give: The instanta-neous rate of change of the function

y = f(x) at the point x0 in its domain is:
lim
xx0 y/x = lim xx0 (f(x0) f(x))/(x0 x).

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