Rate of change

[elindow]

Ok I am going over my homework that I have had graded before I take my test in a couple of days and I came across a problem that I could not figure it out so I thought that maybe someone here could help me.
Here is the problem.

A) Find the average rate of change of the area of a circle with respect to its radius r as r changes from
i) 2 to 3
ii) 2 to 2.5
iii) 2 to 2.1

B) Find the instantaneous rate of change when r = 2.
C) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle.

I would appriciate any help.

Thanks
Erik

 

[Unco]

A) Find the average rate of change of the area of a circle with respect to its radius r as r changes from
i) 2 to 3
ii) 2 to 2.5
iii) 2 to 2.1

The function for the area of a circle with radius r is, as you will know:
A(r) = pi.r^2

Geometrically, the average rate of change is just the slope of the secant line joining two points on a graph of A(r).

Just as the slope between two points (x0,y0) and (x1,y1) is given by
(y1-y0)/(x1-x0), for (i) we have
avg rate of change = [A(3) - A(2)] / (3-2) = ?

 

[Unco]

B) Find the instantaneous rate of change when r = 2.

Instantaneous rate of change is given by the derivative. Find A'(r) and substitute r=2 into this.

C) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle.

In (b) you found A'(r). Now, what is the circumference of a circle given by in terms of the radius?