Water flows into a reservoir at a rate modeled by the differentiable function R, where R(t) is measured in gallons per hour and t is measured in hours since pumping began. Values of R(t) for selected values of t, 0 ≤ t≤ 12, are given in the table above.

(a) Using correct units, interpret the meaning of ∫₂¹⁰ R(t) dtin the context of this problem. Use a right Riemann sum with the three subintervals [2, 5], [5, 8], and [8, 10] to approximate the value of ∫₂¹⁰ R(t) dt.

(b) Must there exist a value of c, for 2 < c < 10, such that R′(c) = 0 ? Justify your answer.

(c) Water flows out of the reservoir at a rate modeled by W(t) = 5t e^−0.1t, where W(t) is measured in gallons per hour for 0 ≤ t≤ 12. Find the average rate of outflow of water from the reservoir over the time interval 0 ≤ t≤ 12. Show the setup for your calculations.

(d) Find the value of W′(8). Using correct units, interpret the meaning of W′(8) in the context of this problem.

A particle moves along the x-axis for 0 ≤ t ≤ 8 seconds. The velocity of the particle is given by v(t) = sin(πt/4) + cos(πt/4), where t is measured in seconds and v(t) is measured in meters per second. At time t = 0, the position of the particle is x(0) = 5 meters.

(a) Find all times t in the open interval 0 < t< 8 at which the particle changes direction. Give a reason for your answer.

(b) Find the acceleration of the particle at time t = 2 seconds. Show the setup for your calculations, and indicate units of measure. Is the particle speeding up or slowing down at time t = 2? Give a reason for your answer.

(c) Find the displacement of the particle over the time interval 0 ≤ t≤ 5. Show the setup for your calculations.

(d) Find the total distance traveled by the particle over the time interval 0 ≤ t≤ 6. Show the setup for your calculations.