Find the average rate of change of 

g

(

x

)

=

–

19

x

 over the interval 

[

19

,

20

]

.

To find the average rate of change of the function \( g(x) = \frac{-19}{x} \) over the interval \([19, 20]\), we use the formula:

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

Where \( a = 19 \) and \( b = 20 \).

1. First, calculate \( g(a) = g(19) \):

\[
g(19) = \frac{-19}{19} = -1
\]

2. Then, calculate \( g(b) = g(20) \):

\[
g(20) = \frac{-19}{20} = -\frac{19}{20}
\]

3. Now, apply the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{-\frac{19}{20} - (-1)}{20 - 19}
\]

4. Simplify the expression:

\[
\text{Average Rate of Change} = \frac{-\frac{19}{20} + 1}{1}
\]

5. Find a common denominator to combine the fractions:

\[
-\frac{19}{20} + 1 = -\frac{19}{20} + \frac{20}{20} = \frac{1}{20}
\]

6. So, the average rate of change is:

\[
\text{Average Rate of Change} = \frac{1}{20}
\]

Thus, the average rate of change of \( g(x) \) over the interval \([19, 20]\) is \(\frac{1}{20}\).

Would you like further details or have any questions?

---

Here are some related questions you might find interesting:
1. What would be the average rate of change over the interval \([20, 21]\)?
2. How does the value of the average rate of change compare with the derivative of \( g(x) \) at \( x = 19.5 \)?
3. What does the small value of \(\frac{1}{20}\) indicate about the change in the function over this interval?
4. How would the average rate of change differ if the interval was much larger, say \([10, 30]\)?
5. How does the average rate of change help in approximating the function's behavior between the two points?

**Tip:** The closer the two points are on the interval, the more the average rate of change approximates the instantaneous rate of change (derivative) at a point in between.

Learn how to calculate the average rate of change for the function g(x) = -19/x over the interval [19, 20]. Follow a step-by-step solution using the Average Rate of Change formula. Suitable for Grades 11-12 students and those studying calculus concepts.

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