**Unformatted text preview: **AF"D CALCULUS AB! CALCULUS BC
2016 SCORING GUIDELINES Question 1 mun-nu m-I-
liters 2’ hour Water is pumped into atank at a rate modeled by WU) = 2000s"!2 /20 liters per hour for 0 S t S 8, where t is
measured in hours. Water is removed from the tank at a rate modeled by RU) liters per hour, where R is differentiable and decreasing on 0 S t s 8. Selected values of RU) are shown in the table above. At time
t = 0, there are 50,000 liters ofwater in the tank. (a) Estimate R’(2). Show the work that leads to your answer. Indicate units of‘ measure. (b) Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of
water removed from the tank during the 8 hours. Is this an overestimate or an underestimate ofthe total
amount ofwater removed? Give a reason for your answer. (c) Use your answer from part (b) to find an estimate ofthe total amount ofwater in the tank, to the nearest liter,
at the end of8 hours. (d) For 0 S t S 8, is there atime I when the rate at which wateris pumped into the tank is the same as the rate
at which water is removed from the tank? Explain why or why not. . M-w- - 2 . 1=estimate
(a) R (2) =5 3—1 — 3—1 — 120 litersfhr 2. l:units
a
(b) The total amount ofwater removed is given by Io RU) dz. 1 : left Riemann sum
8 3 : l : estimate
In RU) d1 ”5 143(0) + 2‘R(1)+ 3 ‘ M3) + 2 ‘ M5) 1 : overestimate with reason
= 1(1340) + 20190) + 3(950) + 2040)
= 8050 liters This is an overestimate since R is a decreasing function. 8 (c) Total :5 50000 + In WU) d! — 8050 2 . [ l : integral
= 50000 + 7836195325 — 3050 :5 49786 liters 1 = “time
(d) W(0) — R(0) > 0, W(8) — R(8) < 0, and WU) — RU) is 2' l : considers WU)— RU)
continuous. ' l : answer with explanation Therefore, the Intermediate Value Theorem guarantees at least one
time t, 0 < t < 8, forwhich WU)— RU): 0, or WU): RU). For this value of t, the rate at which water is pumped into the tank
is the same as the rate at which water is removed from the tank. ...

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- Summer '16
- Robert Tuskey
- Calculus, Continuous function, Riemann sum