Find the net change in the value of the function between the given inputs.
[tex]g(t)=1-t^2 ; \quad \text { from }-4 \text { to } 9[/tex]
Answer (1)
Calculate the value of the function at t = − 4 : g ( − 4 ) = 1 − ( − 4 ) 2 = − 15 .
Calculate the value of the function at t = 9 : g ( 9 ) = 1 − ( 9 ) 2 = − 80 .
Find the net change by subtracting the initial value from the final value: g ( 9 ) − g ( − 4 ) = − 80 − ( − 15 ) = − 65 .
The net change in the value of the function from t = − 4 to t = 9 is − 65 .
Explanation
Understanding the Problem We are given the function g ( t ) = 1 − t 2 and asked to find the net change in the value of the function from t = − 4 to t = 9 . The net change is the difference between the function's value at the final input and its value at the initial input.
Calculating g(-4) First, we need to find the value of the function at t = − 4 . We substitute − 4 into the function: g ( − 4 ) = 1 − ( − 4 ) 2 = 1 − 16 = − 15
Calculating g(9) Next, we need to find the value of the function at t = 9 . We substitute 9 into the function: g ( 9 ) = 1 − ( 9 ) 2 = 1 − 81 = − 80
Finding the Net Change Now, we find the net change by subtracting the initial value g ( − 4 ) from the final value g ( 9 ) :
g ( 9 ) − g ( − 4 ) = − 80 − ( − 15 ) = − 80 + 15 = − 65
Examples
Imagine you're tracking the temperature change in a room over a period. The function g ( t ) = 1 − t 2 could represent the temperature, where t is the time. Finding the net change from t = − 4 to t = 9 tells you the overall temperature difference during that time. This concept is useful in many real-world scenarios, such as analyzing stock prices, population growth, or the trajectory of a moving object. Understanding how functions change over intervals helps us make predictions and informed decisions.
