Math Problem Statement

Find the average rate of change of 

g

(

x

)

=

19

x

 over the interval 

[

19

,

20

]

.

Solution

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

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

Where a=19a = 19 and b=20b = 20.

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

g(19)=1919=1g(19) = \frac{-19}{19} = -1

  1. Then, calculate g(b)=g(20)g(b) = g(20):

g(20)=1920=1920g(20) = \frac{-19}{20} = -\frac{19}{20}

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

Average Rate of Change=1920(1)2019\text{Average Rate of Change} = \frac{-\frac{19}{20} - (-1)}{20 - 19}

  1. Simplify the expression:

Average Rate of Change=1920+11\text{Average Rate of Change} = \frac{-\frac{19}{20} + 1}{1}

  1. Find a common denominator to combine the fractions:

1920+1=1920+2020=120-\frac{19}{20} + 1 = -\frac{19}{20} + \frac{20}{20} = \frac{1}{20}

  1. So, the average rate of change is:

Average Rate of Change=120\text{Average Rate of Change} = \frac{1}{20}

Thus, the average rate of change of g(x)g(x) over the interval [19,20][19, 20] is 120\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][20, 21]?
  2. How does the value of the average rate of change compare with the derivative of g(x)g(x) at x=19.5x = 19.5?
  3. What does the small value of 120\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][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.

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Math Problem Analysis

Mathematical Concepts

Average Rate of Change
Functions

Formulas

Average Rate of Change formula: \( \frac{g(b) - g(a)}{b - a} \)

Theorems

-

Suitable Grade Level

Grades 11-12