Question

lt34: i can calculate average rate of change of a function over a specified interval

1. a family is travelling from their home to a vacation resort hotel. the table below shows their distance from home as a function of time.

time (hrs) 0 2 5 7
distance (mi) 0 140 375 490
a) determine the average rate of change for $0 \leq h \leq 2$
b) determine the average rate of change for $2 \leq h \leq 7$

1. the table below represents the height of bird above the ground during flight, with $p(t)$ representing height in feet and $t$ representing time in seconds.

t $p(t)$
0 6.70
2 6.28
4 6
9 2.41
calculate the average rate of change from for $0 \leq t \leq 4$ seconds, in feet per second. round to the nearest tenth.

1. the table below shows the average diameter of a pupil in a persons eye as he or she grows older.

determine the average pupil diameter in mm per year for each age group, rounded to the nearest hundredth:
age (years) average pupil diameter (mm)
20 4.7
30 4.3
40 3.9
50 3.5
60 3.1
70 2.7
80 2.3
a) age 20 to 50
b) age 60 to 70
c) age 30 to 80

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $f(x)$ over $[a,b]$ is $\frac{f(b)-f(a)}{b-a}$.

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Problem 1a

Step1: Identify values for interval $0\leq h\leq2$

$t_1=0, d_1=0$; $t_2=2, d_2=140$

Step2: Compute average rate of change

$\frac{140-0}{2-0} = \frac{140}{2} = 70$

Problem 1b

Step1: Identify values for interval $2\leq h\leq7$

$t_1=2, d_1=140$; $t_2=7, d_2=490$

Step2: Compute average rate of change

$\frac{490-140}{7-2} = \frac{350}{5} = 70$

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Problem 2

Step1: Identify values for interval $0\leq t\leq4$

$t_1=0, P_1=6.71$; $t_2=4, P_2=6$

Step2: Compute average rate of change

$\frac{6-6.71}{4-0} = \frac{-0.71}{4} = -0.1775$

Step3: Round to nearest tenth

$-0.1775 \approx -0.2$

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Problem 3a

Step1: Identify values for age 20 to 50

$a_1=20, d_1=4.7$; $a_2=50, d_2=3.5$

Step2: Compute average rate of change

$\frac{3.5-4.7}{50-20} = \frac{-1.2}{30} = -0.04$

Problem 3b

Step1: Identify values for age 60 to 70

$a_1=60, d_1=3.1$; $a_2=70, d_2=2.7$

Step2: Compute average rate of change

$\frac{2.7-3.1}{70-60} = \frac{-0.4}{10} = -0.04$

Problem 3c

Step1: Identify values for age 30 to 80

$a_1=30, d_1=4.3$; $a_2=80, d_2=2.3$

Step2: Compute average rate of change

$\frac{2.3-4.3}{80-30} = \frac{-2}{50} = -0.04$

Answer:

1a) 70 miles per hour
1b) 70 miles per hour

1. -0.2 feet per second

3a) -0.04 mm per year
3b) -0.04 mm per year
3c) -0.04 mm per year