# Rate of Change

• May 25th 2009, 06:10 AM
lisa1984wilson
Rate of Change
I need help with this..I have 50 of these problems so if you can show me how to calculate this one I can do the rest myself...Thanks!
Let s(t)=√t
a.Find the average rate of change of s(t) with respect to t as t changes from t=1 to t=1/4
b.Use calculus to find the instantaneous rate of change of s(t) at t=1 and compare with average rate found in part(a).
• May 25th 2009, 06:21 AM
skeeter
Quote:

Originally Posted by lisa1984wilson
I need help with this..I have 50 of these problems so if you can show me how to calculate this one I can do the rest myself...Thanks!
Let s(t)=√t
a.Find the average rate of change of s(t) with respect to t as t changes from t=1 to t=1/4
b.Use calculus to find the instantaneous rate of change of s(t) at t=1 and compare with average rate found in part(a).

average rate of change ...

$\frac{s(1) - s\left(\frac{1}{4}\right)}{1 - \frac{1}{4}}$

$\frac{1 - \frac{1}{2}}{1 - \frac{1}{4}} = \frac{2}{3}$

instantaneous rate of change at t = 1 ...

$\lim_{t \to 1} \frac{s(t) - s(1)}{t - 1}$

$\lim_{t \to 1} \frac{\sqrt{t} - 1}{t - 1}$

$\lim_{t \to 1} \frac{\sqrt{t} - 1}{(\sqrt{t} - 1)(\sqrt{t} + 1)}$

$\lim_{t \to 1} \frac{1}{\sqrt{t} + 1} = \frac{1}{2}$