math problem?
The volume of a growing spherical cell is given below, where the radius is measured in micrometers (1 µm = 10^-6m).
V = 4/3pi r^3
(a) Find the average rate of change of V with respect to r when r changes from 2. Evaluate your answers numerically.
(i) 2 to 5 µm ?µm2
(ii) 2 to 3 µm ?µm2
(iii) 2 to 2.1 µm ?µm2
(b) Find the instantaneous rate of change of V with respect to r when r = 2 µm/. Evaluate your answer numerically.
V'(2) = µm^2
Comments
Find a secant line for part (a). That means find the slope between points (2, V(2)) and (5, V(5)) for problem (i).
(i) 156pi/3 = 52pi = 163.363
(ii) 76pi/3/1 = 79.587
(iii) 5.282/0.1 = 52.821
Take note that you are getting closer and closer to the actual value of the instantaneous rate of change at r=2.
Use calculus or take the limit as h goes to zero of [V(2+h)-V(2)]/h for part b.
The derivative, tangent line, or the instantaneous rate of change - all the same thing - are what you need to find.
Using calculus, V'(r) = 4*pi*r^2 = 4*pi*(2^2) = 16pi = 50.265