Rates in calculus

Question

Suppose that we are pumping water into a tank at a rate of r(t) = t/ (t^2+1) gallons/minutes. At t=0 the tank may or may not be empty.

1. Is the amount of water in the tank strictly increasing, strictly decreasing, or neither over the time interval (0, 2)? Explain your answer briefly
2. Is the rate of change of water strictly increasing, strictly decreasing, or neither over the time interval (0,2)
3. Suppose we look at time t = 10 minutes and see 5 gallons of water in the tank. How much water in the tank was there at t = 5 minutes?

Answer 1

3) r(t)= t/(t^2+1)

R(t)=integral of r(t)

= ln(t^2+1)/2 +C

R(10)=ln(101)/2 +C=5 gallons at time t=10

C= 5-ln(101)/2

R(10)=ln(101)/2+5-ln(101)/2=5

R(t)=ln(t^2+1)/t+5 -ln(101)/2

R(5)=ln(26)/2+5-ln(101)/2

=about 4.3215 gallons at time t=5

1) R(0)=5-ln(101)/2=about 5-2.308= 2.692

R(1)=ln(2)/2 +2.692= about 3.039

R(2)= ln(5)/2+2.692= about 3.497

strictly increasing from t=0 to t=2