Find the net change in the value of the function between the given inputs.
[tex]g(t)=1-t^2[/tex]; from [tex] -4 [/tex] to [tex]9[/tex]
Answer (1)
Calculate g ( − 4 ) : g ( − 4 ) = 1 − ( − 4 ) 2 = − 15 .
Calculate g ( 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 asked to find the net change in the value of the function g ( t ) = 1 − t 2 as t varies from − 4 to 9 . The net change is the difference between the value of the function at the final value of t (which is 9 ) and the value of the function at the initial value of t (which is − 4 ). In other words, we need to compute g ( 9 ) − g ( − 4 ) .
Calculating g(-4) First, let's calculate g ( − 4 ) . We substitute t = − 4 into the expression for g ( t ) : g ( − 4 ) = 1 − ( − 4 ) 2 = 1 − 16 = − 15.
Calculating g(9) Next, let's calculate g ( 9 ) . We substitute t = 9 into the expression for g ( t ) : g ( 9 ) = 1 − ( 9 ) 2 = 1 − 81 = − 80.
Calculating the Net Change Now, we can find the net change by subtracting g ( − 4 ) from g ( 9 ) : g ( 9 ) − g ( − 4 ) = − 80 − ( − 15 ) = − 80 + 15 = − 65.
Final Answer Therefore, the net change in the value of the function g ( t ) from t = − 4 to t = 9 is − 65 .
Examples
Understanding net change is crucial in various real-world scenarios. For instance, in finance, it helps determine the profit or loss of an investment over a specific period. If an investment's value changes according to the function g ( t ) = 1 − t 2 , where t represents time, calculating the net change from t = − 4 to t = 9 would reveal the overall gain or loss during that period. This concept is also applicable in physics to calculate displacement and in economics to analyze changes in market trends.
