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 .
Calculate the net change: g ( 9 ) − g ( − 4 ) = − 80 − ( − 15 ) = − 65 .
The net change in the value of the function 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 as t varies from − 4 to 9 . The net change is the difference between the function's value at the final input and its value at the initial input.
Calculate g(-4) First, we need to calculate the value of the function at t = − 4 . g ( − 4 ) = 1 − ( − 4 ) 2 = 1 − 16 = − 15
Calculate g(9) Next, we calculate the value of the function at t = 9 . g ( 9 ) = 1 − ( 9 ) 2 = 1 − 81 = − 80
Calculate the Net Change Now, we find the net change by subtracting the initial value g ( − 4 ) from the final value g ( 9 ) . Net change = g ( 9 ) − g ( − 4 ) = − 80 − ( − 15 ) = − 80 + 15 = − 65
Final Answer Therefore, the net change in the value of the function g ( t ) = 1 − t 2 from t = − 4 to t = 9 is − 65 .
Examples
Understanding net change is useful in many real-world scenarios. For example, if you're tracking the temperature change throughout the day, the net change tells you the overall increase or decrease in temperature from the beginning to the end of the day. Similarly, in finance, the net change in a stock's price indicates the overall gain or loss in value over a specific period. This concept helps in making informed decisions based on overall trends.
