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 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 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 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 .

The estimated average rates of change of p(x)
and q(x)
are both 53
over [0.1,8.9]
.

Answer 1

To find the estimated average rates of change of the functions \( p(x) \) and \( q(x) \) over the interval \([0.1, 8.9]\), we need to calculate the average rate of change for both functions over this interval.

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula:

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

### For p(x):

From the table, we have:

- \( p(0) = -5 \)
- \( p(9) = -2 \)

Since the interval we are considering is \([0.1, 8.9]\) and we have no exact values at these points but we can use the values near them from the table.

Using the closest values we have for \( p \):

- \( a = 0 \)
- \( b = 9 \)

Therefore, we calculate:

\[
\text{Average Rate of Change of } p(x) = \frac{p(9) - p(0)}{9 - 0} = \frac{-2 - (-5)}{9 - 0} = \frac{3}{9} = \frac{1}{3}
\]

### For q(x):

From the table, we have:

- \( q(1) = 0 \)
- \( q(9) = 10 \)

Again, the closest values we can use are:

- \( a = 1 \)
- \( b = 9 \)

Calculating the average rate of change:

\[
\text{Average Rate of Change of } q(x) = \frac{q(9) - q(1)}{9 - 1} = \frac{10 - 0}{9 - 1} = \frac{10}{8} = \frac{5}{4} = 1.25
\]

### Conclusion

- The average rate of change for \( p(x) \) is \( \frac{1}{3} \).
- The average rate of change for \( q(x) \) is \( 1.25 \).

Now let's compare these values:

\(\frac{5}{4} > \frac{1}{3}\), which means:

The estimated average rate of change of \( q(x) \) is greater than the estimated average rate of change of \( p(x) \) over \([0.1, 8.9]\).

Thus, the correct response is:

**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].**