Question

question 7
the following graph shows the number of viewers who attended a live stream during the first 10 minutes after its broadcast started. which statement is correct?
a. the average rate of change from 1 to 2 minutes is less than the average rate of change from 3 to 4 minutes.
b. the average rate of change from 1 to 2 minutes is greater than the average rate of change from 7 to 8 minutes.
c. the instantaneous rate of change from 5 to 6 minutes is greater than the instantaneous rate of change from 2 to 3 minutes.
d. the instantaneous rate of change from 1 to 2 minutes is less than the instantaneous rate of change from 2 to 3 minutes.
graph: x - axis labeled minutes (0–10), y - axis labeled viewers (100–900); plotted points (approximate): (0, 300), (1, ~375), (2, 500), (3, 600), (4, 400), (5, ~375), (6, 700), (7, ~775), (8, 800), (9, ~725), (10, ~775)

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{f(b)-f(a)}{b - a} \). For instantaneous rate of change (slope of the tangent, but in a linear segment, it's the slope of the segment) between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Analyze Option A

- From 1 to 2 minutes: Let's find the number of viewers. At \( x = 1 \), \( y = 375 \) (approx, looking at the graph: 300 at 0, 375 at 1, 500 at 2). So \( \frac{500 - 375}{2 - 1}=\frac{125}{1}=125 \).
- From 3 to 4 minutes: At \( x = 3 \), \( y = 600 \); at \( x = 4 \), \( y = 400 \). So \( \frac{400 - 600}{4 - 3}=\frac{- 200}{1}=-200 \). The average rate of change from 1 - 2 is 125, from 3 - 4 is -200. So 125 > -200. So A is incorrect.

Step3: Analyze Option B

- From 1 to 2 minutes: As above, \( \frac{500 - 375}{2 - 1}=125 \).
- From 7 to 8 minutes: At \( x = 7 \), \( y = 775 \); at \( x = 8 \), \( y = 800 \). So \( \frac{800 - 775}{8 - 7}=\frac{25}{1}=25 \). Since 125 > 25, the average rate of change from 1 - 2 is greater than from 7 - 8. So B is correct? Wait, let's check other options to be sure.

Step4: Analyze Option C

- From 5 to 6 minutes: At \( x = 5 \), \( y = 375 \); at \( x = 6 \), \( y = 700 \). The slope (instantaneous rate, since it's a linear segment) is \( \frac{700 - 375}{6 - 5}=325 \).
- From 2 to 3 minutes: At \( x = 2 \), \( y = 500 \); at \( x = 3 \), \( y = 600 \). Slope is \( \frac{600 - 500}{3 - 2}=100 \). 325 > 100, but wait, the option says "greater than", but let's check the graph again. Wait, at x=5, the viewers are 375? Wait, the graph: at 0:300, 1: ~375, 2:500, 3:600, 4:400, 5:375, 6:700, 7:775, 8:800, 9:725, 10:775. So from 5 to 6: 700 - 375 = 325 over 1 minute. From 2 to 3: 600 - 500 = 100 over 1 minute. But the option C says "instantaneous rate of change from 5 to 6 is greater than from 2 to 3". But let's check option D.

Step5: Analyze Option D

- From 1 to 2 minutes: slope is \( \frac{500 - 375}{2 - 1}=125 \).
- From 2 to 3 minutes: slope is \( \frac{600 - 500}{3 - 2}=100 \). So 125 > 100, so "instantaneous rate from 1 - 2 is less than from 2 - 3" is false.

Wait, but earlier in option B: from 1 - 2, average rate is 125; from 7 - 8, average rate is 25. So 125 > 25, so B is correct? Wait, but let's re - check the graph values more accurately.

Let's get exact values from the graph:

- At \( t = 0 \): 300 viewers.
- \( t = 1 \): 375 (since between 300 and 400, mid - way? Wait, no, the y - axis: 300, 400, 500, 600, 700, 800, 900. At \( t = 1 \), the point is at 375? Wait, no, looking at the graph, at \( t = 1 \), the y - value is 375? Wait, the first segment: 0 (300) to 1 (let's say 375), 1 to 2 (500), 2 to 3 (600), 3 to 4 (400), 4 to 5 (375), 5 to 6 (700), 6 to 7 (775), 7 to 8 (800), 8 to 9 (725), 9 to 10 (775).

So:

- Option A: 1 - 2: \( \frac{500 - 375}{1}=125 \); 3 - 4: \( \frac{400 - 600}{1}=-200 \). 125 > - 200, so A is wrong.
- Option B: 1 - 2: 125; 7 - 8: \( \frac{800 - 775}{1}=25 \). 125 > 25, so B is correct.
- Option C: 5 - 6: \( \frac{700 - 375}{1}=325 \); 2 - 3: \( \frac{600 - 500}{1}=100 \). 325 > 100, but the option says "greater than", but let's check the option's wording: "The instantaneous rate of change from 5 to 6 minutes is greater than the instantaneous rate of change from 2 to 3 minutes." But is this correct? Wait, but we need to check all options. Wait, maybe I made a mistake in the graph reading. Wait, at \( t = 5 \), the viewers are 375? Wait, the point at \( t = 5 \) is at 375? The graph shows at \( t = 5 \), it's 375 (between 300 and 400, closer to 400? Wait, no, the y - axis: each grid is 100? Wait, 300, 400, 500, 600, 700, 800, 900. So at \( t = 5 \), the point is at 375? Wait, maybe the y - value at \( t = 5 \) is 375, \( t = 6 \) is 700, \( t = 2 \) is 500, \( t = 3 \) is 600.

But let's check option B again. From 1 - 2: change in viewers is 500 - 375 = 125, time change 1 minute, rate 125. From 7 - 8: change is 800 - 775 = 25, time change 1 minute, rate 25. So 125 > 25, so B is correct.

Answer:

B. The average rate of change from 1 to 2 minutes is greater than the average rate of change from 7 to 8 minutes.