Select the correct answer.
Consider function [tex]g[/tex].
[tex]g(x)=\frac{5}{x-1}+2[/tex]
What is the average rate of change of function [tex]g[/tex] over the interval [-4,3] ?
A. [tex]-\frac{1}{2}[/tex]
B. [tex]-\frac{7}{2}[/tex]
C. [tex]\frac{1}{2}[/tex]
D. 2
Answer (1)
Calculate g ( 3 ) by substituting x = 3 into the function: g ( 3 ) = 3 − 1 5 + 2 = 2 9 .
Calculate g ( − 4 ) by substituting x = − 4 into the function: g ( − 4 ) = − 4 − 1 5 + 2 = 1 .
Apply the average rate of change formula: 3 − ( − 4 ) g ( 3 ) − g ( − 4 ) = 7 2 9 − 1 .
Simplify the expression to find the average rate of change: 7 2 7 = 2 1 .
Explanation
Understanding the Problem We are asked to find the average rate of change of the function g ( x ) = x − 1 5 + 2 over the interval [ − 4 , 3 ] . The average rate of change is given by the formula: 3 − ( − 4 ) g ( 3 ) − g ( − 4 )
Calculating g(3) and g(-4) First, we need to find the values of g ( 3 ) and g ( − 4 ) .
g ( 3 ) = 3 − 1 5 + 2 = 2 5 + 2 = 2 5 + 2 4 = 2 9 g ( − 4 ) = − 4 − 1 5 + 2 = − 5 5 + 2 = − 1 + 2 = 1
Calculating the Average Rate of Change Now, we can plug these values into the formula for the average rate of change: 3 − ( − 4 ) g ( 3 ) − g ( − 4 ) = 3 + 4 2 9 − 1 = 7 2 9 − 2 2 = 7 2 7 = 2 7 × 7 1 = 2 1
Final Answer The average rate of change of the function g ( x ) over the interval [ − 4 , 3 ] is 2 1 .
Examples
The average rate of change is a fundamental concept in calculus with applications in various fields. For example, in physics, it can represent the average velocity of an object over a time interval. If g ( x ) represents the position of an object at time x , then the average rate of change over the interval [ − 4 , 3 ] would be the average velocity of the object between time -4 and time 3. Similarly, in economics, if g ( x ) represents the cost of producing x units of a product, then the average rate of change over an interval would represent the average cost per unit over that production range. Understanding average rates of change helps in making informed decisions and predictions in these real-world scenarios.
