Estimate the average rate of change of the functions f(x)=2x−8−−−−−−√3 and g(x)=2x−8−−−−−−√

Question

over the given intervals. Select the interval over which f(x)
has an estimated average rate of change greater than that of g(x)
.(1 point)
Responses

[4,4.7]
left bracket 4 comma 4 point 7 right bracket

[4,4.3]
left bracket 4 comma 4 point 3 right bracket

[4,4.9]
left bracket 4 comma 4 point 9 right bracket

[4,4.5]

Answer 1

To find the average rate of change of the functions \( f(x) = \sqrt{2x - 8} \) and \( g(x) = 2x - 8 \) over the specified intervals, we need to use the formula for the average rate of change, which is:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

where \( [a, b] \) is the interval.

1. Calculate the average rate of change for \( f(x) \)

Let's calculate \( f(x) \) for the given intervals:

• Interval [4, 4.7]:

  • \( f(4) = \sqrt{2(4) - 8} = \sqrt{0} = 0 \)
  • \( f(4.7) = \sqrt{2(4.7) - 8} = \sqrt{1.4} \)

  The average rate of change over this interval is: \[ \frac{f(4.7) - f(4)}{4.7 - 4} = \frac{\sqrt{1.4} - 0}{0.7} = \frac{\sqrt{1.4}}{0.7} \]

• Interval [4, 4.3]:

  • \( f(4.3) = \sqrt{2(4.3) - 8} = \sqrt{0.6} \)

  Average rate of change: \[ \frac{\sqrt{0.6} - 0}{0.3} = \frac{\sqrt{0.6}}{0.3} \]

• Interval [4, 4.9]:

  • \( f(4.9) = \sqrt{2(4.9) - 8} = \sqrt{1.8} \)

  Average rate of change: \[ \frac{\sqrt{1.8} - 0}{0.9} = \frac{\sqrt{1.8}}{0.9} \]

• Interval [4, 4.5]:

  • \( f(4.5) = \sqrt{2(4.5) - 8} = \sqrt{1} = 1 \)

  Average rate of change: \[ \frac{1 - 0}{0.5} = 2 \]

2. Calculate the average rate of change for \( g(x) \)

Now for \( g(x) = 2x - 8 \):

• Interval [4, 4.7]:

  • \( g(4.7) = 2(4.7) - 8 = 1.4 \)

  Average rate of change: \[ \frac{1.4 - 0}{0.7} = 2 \]

• Interval [4, 4.3]:

  • \( g(4.3) = 2(4.3) - 8 = 0.6 \)

  Average rate of change: \[ \frac{0.6 - 0}{0.3} = 2 \]

• Interval [4, 4.9]:

  • \( g(4.9) = 2(4.9) - 8 = 1.8 \)

  Average rate of change: \[ \frac{1.8 - 0}{0.9} = 2 \]

• Interval [4, 4.5]:

  • \( g(4.5) = 2(4.5) - 8 = 1 \)

  Average rate of change: \[ \frac{1 - 0}{0.5} = 2 \]

Summary

Now, let's summarize the average rates of change:

• For \( f(x) \):

  • [4, 4.7]: \( \frac{\sqrt{1.4}}{0.7} \approx 1.414/0.7 \approx 2.02 \)
  • [4, 4.3]: \( \frac{\sqrt{0.6}}{0.3} \approx 1.225/0.3 \approx 4.08 \)
  • [4, 4.9]: \( \frac{\sqrt{1.8}}{0.9} \approx 1.341/0.9 \approx 1.49 \)
  • [4, 4.5]: \( 2 \)

• For \( g(x) \):

  • Each interval: \( 2 \)

Now we can compare these values. The average rate of change for \( f(x) \) over the interval [4, 4.3] is greater than that for \( g(x) \):

Conclusion: The interval over which \( f(x) \) has an estimated average rate of change greater than that of \( g(x) \) is [4, 4.3].