Find the instantaneous rate of change of V with respect to r when r=100
V(r)=.2r^2-20r+600
using limit definition.
Please show work for understanding, thank you :)
To find the instantaneous rate of change of V with respect to r when r = 100 using the limit definition, we can start by finding the difference quotient for V(r) with respect to r.
The difference quotient represents the average rate of change, and as we take the limit as the change in r approaches zero, it gives us the instantaneous rate of change.
The difference quotient formula is:
ΔV/Δr = [V(r + Δr) - V(r)] / Δr
Let's calculate it step by step:
Step 1: Calculate V(r)
V(r) = 0.2r^2 - 20r + 600
Step 2: Substitute r + Δr into the expression for V(r)
V(r + Δr) = 0.2(r + Δr)^2 - 20(r + Δr) + 600
= 0.2(r^2 + 2rΔr + Δr^2) - 20r - 20Δr + 600
= 0.2r^2 + 0.4rΔr + 0.2Δr^2 - 20r - 20Δr + 600
Step 3: Now we can substitute both V(r + Δr) and V(r) into the difference quotient formula and simplify:
ΔV/Δr = [V(r + Δr) - V(r)] / Δr
= [0.2r^2 + 0.4rΔr + 0.2Δr^2 - 20r - 20Δr + 600 - (0.2r^2 - 20r + 600)] / Δr
= [0.4rΔr + 0.2Δr^2 - 20Δr] / Δr
Step 4: Now, factor out Δr from the numerator:
ΔV/Δr = (Δr)(0.4r + 0.2Δr - 20) / Δr
Step 5: Cancel Δr from the numerator and denominator:
ΔV/Δr = 0.4r + 0.2Δr - 20
Step 6: Finally, we can take the limit as Δr approaches zero to find the instantaneous rate of change:
lim(Δr->0) [ΔV/Δr] = 0.4r - 20
Therefore, the instantaneous rate of change of V with respect to r when r = 100 using the limit definition is 0.4(100) - 20 = 40 - 20 = 20.