REASONING FROM TABULAR DATA

Ex. 1 Let y( t) represent the population of a town over a 20-year period. The table below shows the population recorded at selected times.

t (yrs) 0 4 10 13 20 y( t) (people) 2500 2724 3108 3697 4283

(a) Use data from the table to find an approximation for y (12) , and explain the meaning of y (12) in terms of the population of the town. Show the computations that lead to your answer, and round your answer to the nearest whole number of people.

(b) Use data from the table and a trapezoidal approximation with four subintervals to approximate the average population of the town over the 20-year period. Show the computations that lead to your answer, and round your answer to the nearest whole number of people.

Sometimes we are given data about a function and also a model for the function. When this occurs, we can compare the results given by the data and by the model to help determine whether we have made an error in our computations.

Ex. 2 The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R of time t. The table below shows the rate as selected values of t for a 12- hour period.

t (hrs) 0 2 4 6 8 10 12 R( t) (gal/hr) 12.5 13.4 13.9 14.3 14.6 14.8 14.7

12 (a) Use a midpoint Riemann sum with four subintervals to approximate R( t) dt , and 0 12 explain the meaning of R( t) dt in terms of the water flow, using correct units. 0 Show the computations that lead to your answer.

1 2 (b) A model for the rate of water flow is given by the function P( t) =750 + 24 t - t . 60 ( ) Use P( t) to approximate the average rate of water flow during the 12-hour time period. Indicate units of measure.

Ex. 3 Particle A moves along a horizontal line with a positive velocity vA ( t) , where vA ( t) is a differentiable function of t. The time t is measured in seconds, and the velocity is measured in cm/sec. The velocity of the particle at selected times is given in the table below.

t (sec) 0 2 5 7 10 v( t) (cm/sec) 17 6.8 7.4 15.6 24.9

(a) Use data from the table to approximate the distance traveled by particle A over the interval 0＃t 10 seconds by using a right Riemann sum with four subintervals. Show the computations that lead to your answer, and indicate units of measure.

2 (b) Particle B moves along the same line with a velocity of vB ( t) = t -7 t + 16 cm/sec,

where t is measured in seconds and vB ( t) is measured in cm/sec. Which particle traveled a greater distance over the interval 0＃t 10 seconds? Explain your answer. Answers to Examples Ex. 1 (a) Since t = 12 is not one of the times given in the table, we should approximate the derivative by using a difference quotient with the best (closest) data available. Because 12 lies between 10 and 13, the best approximation for y (12) is found by y(13) - y ( 10) 3697- 3108 y (12) � = 196.333 196 people per year. 13- 10 3 When t = 12 years, the population of the town is increasing at a rate of approximately 196 people per year.

Note: It is not necessary to simplify answers on the AP Free Response questions, but the computations must be shown.

1 20 (b) The average value of y( t) from t = 0 to t = 20 years is given by y( t) dt 20 0 1骣 1 1 1 1 � 琪 (4)( 2500+ 2724) +( 6)( 2724 + 3108) +( 3)( 3108 + 3697) + ( 7)( 3697 4283) 20桫 2 2 2 2 = 3304.075 3304 The average population over the 20-year period is approximately 3304 people.

Ex. 2 12 (a) R( t) dt�4 R( 2) + R( 6) = R ( 20) + 4( 13.4 + 14.3 14.8) = 170 gallons. 0 ( ) Approximately 170 gallons of water flowed into the tank between t = 0 and t = 12 hours.

112 1 (b) Average rate of water flow = P( t) dt =(169.2) = 14.1 gal/hr. 120 12 12 (Notice that the value of P( t) dt is 169.2 gallons which is very close to the answer to (a).) 0

Ex. 3 (a) Distance �(2)( 6.8) +( 3)( 7.4) +( 2)( 15.6) = ( 3)( 24.9) 141.7 cm

10 (b) Distance traveled by particle B = t2 -7 t + 16 dt = 143.333cm , while 0 ( ) particle A traveled 141.7 cm. Therefore particle B traveled the greater distance.