The circular area A in square centimetres of a healing wound is given by
A(r)=πr^2
Where r =the radius in centimetres. Find the rate of change of the area with respect to the radius.
My work:
I'm not really sure if this is correct but I gave it a try.
Dy/dx =π*2r^2-1
= 6.283r
dA = 2 pi r dr
yes, correct
however notice that this is simply the circumference
the new area added is the circumference * dr
I'm very confused ?
Yes
dA/dr = 2 pi r = 6.28 r
correct
however think a little more
dA = 2 pi r dr = circumference * dr
If you draw a picture of a circle r and then one outside it with radius r + dr
You will see immediately that the change in area is the strip of length 2 pi r times the width dr
To find the rate of change of the area with respect to the radius, we need to take the derivative of the area function A(r) with respect to r.
The derivative of A(r) = πr^2 with respect to r can be found using the power rule of differentiation. According to the power rule, if we have a function f(x) = x^n, then the derivative f'(x) is given by f'(x) = nx^(n-1).
In this case, the function A(r) = πr^2 can be considered as f(r) = πr^2, where n = 2.
Using the power rule, we differentiate A(r) as follows:
A'(r) = 2πr^(2-1)
= 2πr
Therefore, the rate of change of the area with respect to the radius is A'(r) = 2πr.
Now, let's substitute r into this expression to find the rate of change for a particular value of r.