Question

Question

Use the table to answer the question.

x p(x)=x−−√−5
q(x)=5x−1−−−−√3
0 −5
−5
1 −4
0
9 −2
10
Compare the estimated average rates of change for the functions p(x)=x−−√−5
and q(x)=5x−1−−−−√3
over the interval [0.1,8.9]
.

(1 point)
Responses

The estimated average rate of change of q(x)
is greater than the estimated average rate of change of p(x)
over [0.1,8.9]
.
The estimated average rate of change of q times x is greater than the estimated average rate of change of p times x over left bracket 0 point 1 comma 8 point 9 right bracket .

The estimated average rates of change of p(x)
and q(x)
are both 13
over [0.1,8.9]
.
The estimated average rates of change of p times x and q times x are both 1 third over left bracket 0 point 1 comma 8 point 9 right bracket .

The estimated average rates of change of p(x)
and q(x)
are both 53
over [0.1,8.9]
.
The estimated average rates of change of p times x and q times x are both 5 thirds over left bracket 0 point 1 comma 8 point 9 right bracket .

The estimated average rate of change of q(x)
is less than the estimated average rate of change of p(x)
over [0.1,8.9]
.
The estimated average rate of change of q times x is less than the estimated average rate of change of p times x over left bracket 0 point 1 comma 8 point 9 right bracket .
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Answer 1

To compare the estimated average rates of change for the functions \( p(x) = \sqrt{x} - 5 \) and \( q(x) = \frac{\sqrt{5x - 1}}{3} \) over the interval \([0.1, 8.9]\), we first need to calculate the values of \( p \) and \( q \) at the endpoints of the interval.

1. Calculate \( p(0.1) \) and \( p(8.9) \):

   • \( p(0.1) = \sqrt{0.1} - 5 \approx 0.316 - 5 \approx -4.684 \)
   • \( p(8.9) = \sqrt{8.9} - 5 \approx 2.983 - 5 \approx -2.017 \)

2. Calculate the average rate of change of \( p \) over \([0.1, 8.9]\): \[ \text{Average rate of change of } p(x) = \frac{p(8.9) - p(0.1)}{8.9 - 0.1} = \frac{-2.017 - (-4.684)}{8.9 - 0.1} \approx \frac{2.667}{8.8} \approx 0.303 \]

3. Calculate \( q(0.1) \) and \( q(8.9) \):

   • \( q(0.1) = \frac{\sqrt{5(0.1) - 1}}{3} = \frac{\sqrt{0.5 - 1}}{3} \) which is not defined as the value under the square root is negative.
   • \( q(8.9) = \frac{\sqrt{5(8.9) - 1}}{3} = \frac{\sqrt{44.5 - 1}}{3} = \frac{\sqrt{43.5}}{3} \approx \frac{6.6}{3} \approx 2.2 \)

Since \( q(0.1) \) is not defined or negative and \( q(8.9) \) is positive, we might consider the average rate of change with only \( q(8.9) \) that can lead to discrepancy in calculation.

4. Thus, for the interval where both are defined: This situation presents challenges as \( q(x) \) does not produce a real value at the lower bound.

Hence, overall, comparing \( p(x) \) and \( q(x) \), we find that the estimated values lead \( p(x) \) values from negative to less negative; while \( q(x) \) values emerge from undefined to positive.

Concluding statement:

Since we cannot evaluate \( q(x) \) properly at the start, we could summarize that the average change seems undefined whereas \( p(x) \) continuously grows, hence the estimated average rate of change of \( p(x) \) is greater than that of \( q(x) \) over [0.1, 8.9]. However, since \(\text{q(0.1)}\) is negative (undefined), further analyzing remains moot.

Generally taken:

• The correct comparison does therefore imply analysis leaning towards \( p(x) \) dominating trends but specifics of answers should be further validated. Since options do outlay specifically a defined 53 misconception consideration beforehand illustrates rates where \( p, q \) both estimate intervals, ideally neither surpassing average output.