Given the function [tex]\( f(t) = 4t - 12 \)[/tex], we need to evaluate the area function [tex]\( A(x) = \int_3^x f(t) \, dt \)[/tex].
### (a) Evaluation of [tex]\( A(4) \)[/tex] and [tex]\( A(5) \)[/tex]
1. Evaluate [tex]\( A(4) \)[/tex]:
[tex]\[ A(4) = \int_3^4 (4t - 12) \, dt \][/tex]
Let's find the antiderivative of [tex]\( f(t) \)[/tex]:
[tex]\[ \int (4t - 12) \, dt = 2t^2 - 12t \][/tex]
Now, evaluate this antiderivative from 3 to 4:
[tex]\[ A(4) = \left[ 2t^2 - 12t \right]_3^4 \][/tex]
Calculate the values at the upper and lower limits:
Upper limit (when [tex]\( t = 4 \)[/tex]):
[tex]\[ 2(4)^2 - 12(4) = 2(16) - 48 = 32 - 48 = -16 \][/tex]
Lower limit (when [tex]\( t = 3 \)[/tex]):
[tex]\[ 2(3)^2 - 12(3) = 2(9) - 36 = 18 - 36 = -18 \][/tex]
Now, find [tex]\( A(4) \)[/tex]:
[tex]\[ A(4) = (-16) - (-18) = -16 + 18 = 2 \][/tex]
Thus, [tex]\( A(4) = 2 \)[/tex].
2. Evaluate [tex]\( A(5) \)[/tex]:
[tex]\[ A(5) = \int_3^5 (4t - 12) \, dt \][/tex]
Again, using the antiderivative [tex]\( 2t^2 - 12t \)[/tex]:
[tex]\[ A(5) = \left[ 2t^2 - 12t \right]_3^5 \][/tex]
Calculate the values at the upper and lower limits:
Upper limit (when [tex]\( t = 5 \)[/tex]):
[tex]\[ 2(5)^2 - 12(5) = 2(25) - 60 = 50 - 60 = -10 \][/tex]
Lower limit (when [tex]\( t = 3 \)[/tex]):
As before,
[tex]\[ 2(3)^2 - 12(3) = -18 \][/tex]
Now, find [tex]\( A(5) \)[/tex]:
[tex]\[ A(5) = (-10) - (-18) = -10 + 18 = 8 \][/tex]
Thus, [tex]\( A(5) = 8 \)[/tex].
3. Expression for [tex]\( A(x) \)[/tex] for all [tex]\( x \geq 3 \)[/tex]:
The general expression for [tex]\( A(x) \)[/tex] can be found by evaluating the definite integral from 3 to [tex]\( x \)[/tex]:
[tex]\[ A(x) = \int_3^x (4t - 12) \, dt \][/tex]
Using the antiderivative [tex]\( 2t^2 - 12t \)[/tex]:
[tex]\[ A(x) = \left[ 2t^2 - 12t \right]_3^x \][/tex]
Evaluate at the upper and lower limits:
Upper limit (when [tex]\( t = x \)[/tex]):
[tex]\[ 2x^2 - 12x \][/tex]
Lower limit (when [tex]\( t = 3 \)[/tex]):
[tex]\[ 2(3)^2 - 12(3) = 18 - 36 = -18 \][/tex]
So,
[tex]\[ A(x) = (2x^2 - 12x) - (-18) = 2x^2 - 12x + 18 \][/tex]
Thus, the expression for [tex]\( A(x) \)[/tex] for all [tex]\( x \geq 3 \)[/tex] is:
[tex]\[ A(x) = 2x^2 - 12x + 18 \][/tex]
### Summary
- [tex]\( A(4) = 2 \)[/tex]
- [tex]\( A(5) = 8 \)[/tex]
- The expression for [tex]\( A(x) \)[/tex] for all [tex]\( x \geq 3 \)[/tex] is [tex]\( A(x) = 2x^2 - 12x + 18 \)[/tex].
