In the tank is given by a twice-differentiable function h of time, t, measured in minutes. The table below gives the rate of change, h'(t), of the depth of the water in the tank for selected values of t over the time interval 0 less than equal to t less than equal to 15. During this interval h''(t) is greater than zero. When t=5 minutes, the depth of the water is 6 inches. Table: t 0 2 5 9 13 15 h'(t) .4 .5 .7 1.0 1.1 1.3 a) Approximate the depth of the water in the tank at t=4 minutes using the tangent line approximation at t=5. Is your estimate greater than or less than the true value? Give a reason for your answer.
b) Find the rate of change of the volume of the water in the tank at t=2 minutes. Indicate units of measure
c) Use a left Riemann sum with five subintervals indicated by the table to approximate integral 0-15 h'(t) dt. Using correct units explain the meaning of the answer in terms of the depth of the water in the tank.
d) Is the approximation in part c greater than or less than the integral value?