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Sagot :
Let's solve the question step by step.
Given: [tex]\( f(t) = 4t - 12 \)[/tex]
We are given two area functions:
1. [tex]\( A(x) = \int_3^x f(t) \, dt \)[/tex]
2. [tex]\( F(x) = \int_6^x f(t) \, dt \)[/tex]
### Part (a)
First, let's evaluate [tex]\( A(x) \)[/tex].
Evaluate [tex]\( A(4) \)[/tex] and [tex]\( A(5) \)[/tex]:
[tex]\[ A(4) = \int_3^4 (4t - 12) \, dt \][/tex]
Let's compute the above integral:
[tex]\[ \int_3^4 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^4 \][/tex]
[tex]\[ = \left( 2(4)^2 - 12(4) \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 16 - 48 \right) - \left( 2 \cdot 9 - 36 \right) \][/tex]
[tex]\[ = (32 - 48) - (18 - 36) \][/tex]
[tex]\[ = -16 + 18 \][/tex]
[tex]\[ = 2 \][/tex]
Thus, [tex]\( A(4) = 2 \)[/tex].
Next, evaluate [tex]\( A(5) \)[/tex]:
[tex]\[ A(5) = \int_3^5 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^5 \][/tex]
[tex]\[ = \left( 2(5)^2 - 12(5) \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 25 - 60 \right) - \left( 2 \cdot 9 - 36 \right) \][/tex]
[tex]\[ = (50 - 60) - (18 - 36) \][/tex]
[tex]\[ = -10 + 18 \][/tex]
[tex]\[ = 8 \][/tex]
Thus, [tex]\( A(5) = 8 \)[/tex].
General expression for [tex]\( A(x) \)[/tex]:
[tex]\[ A(x) = \int_3^x (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^x \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - (18 - 36) \][/tex]
[tex]\[ = 2x^2 - 12x + 18 \][/tex]
Thus, the expression for [tex]\( A(x) \)[/tex] when [tex]\( x \geq 3 \)[/tex] is [tex]\( 2x^2 - 12x + 18 \)[/tex].
### Part (b)
Now, let's evaluate [tex]\( F(7) \)[/tex] and [tex]\( F(8) \)[/tex], and find the general expression for [tex]\( F(x) \)[/tex].
Evaluate [tex]\( F(7) \)[/tex] and [tex]\( F(8) \)[/tex]:
[tex]\[ F(7) = \int_6^7 (4t - 12) \, dt \][/tex]
Let's compute the above integral:
[tex]\[ \int_6^7 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^7 \][/tex]
[tex]\[ = \left( 2(7)^2 - 12(7) \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 49 - 84 \right) - \left( 2 \cdot 36 - 72 \right) \][/tex]
[tex]\[ = (98 - 84) - (72 - 72) \][/tex]
[tex]\[ = 14 - 0 \][/tex]
[tex]\[ = 14 \][/tex]
Thus, [tex]\( F(7) = 14 \)[/tex].
Next, evaluate [tex]\( F(8) \)[/tex]:
[tex]\[ F(8) = \int_6^8 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^8 \][/tex]
[tex]\[ = \left( 2(8)^2 - 12(8) \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 64 - 96 \right) - \left( 2 \cdot 36 - 72 \right) \][/tex]
[tex]\[ = (128 - 96) - (72 - 72) \][/tex]
[tex]\[ = 32 - 0 \][/tex]
[tex]\[ = 32 \][/tex]
Thus, [tex]\( F(8) = 32 \)[/tex].
General expression for [tex]\( F(x) \)[/tex]:
[tex]\[ F(x) = \int_6^x (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^x \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - (72 - 72) \][/tex]
[tex]\[ = 2x^2 - 12x \][/tex]
Thus, the expression for [tex]\( F(x) \)[/tex] when [tex]\( x \geq 6 \)[/tex] is [tex]\( 2x^2 - 12x \)[/tex].
### Summary:
1. [tex]\( A(4) = 2 \)[/tex]
2. [tex]\( A(5) = 8 \)[/tex]
3. [tex]\( A(x) = 2x^2 - 12x + 18 \)[/tex]
4. [tex]\( F(7) = 14 \)[/tex]
5. [tex]\( F(8) = 32 \)[/tex]
6. [tex]\( F(x) = 2x^2 - 12x \)[/tex] when [tex]\( x \geq 6 \)[/tex]
Given: [tex]\( f(t) = 4t - 12 \)[/tex]
We are given two area functions:
1. [tex]\( A(x) = \int_3^x f(t) \, dt \)[/tex]
2. [tex]\( F(x) = \int_6^x f(t) \, dt \)[/tex]
### Part (a)
First, let's evaluate [tex]\( A(x) \)[/tex].
Evaluate [tex]\( A(4) \)[/tex] and [tex]\( A(5) \)[/tex]:
[tex]\[ A(4) = \int_3^4 (4t - 12) \, dt \][/tex]
Let's compute the above integral:
[tex]\[ \int_3^4 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^4 \][/tex]
[tex]\[ = \left( 2(4)^2 - 12(4) \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 16 - 48 \right) - \left( 2 \cdot 9 - 36 \right) \][/tex]
[tex]\[ = (32 - 48) - (18 - 36) \][/tex]
[tex]\[ = -16 + 18 \][/tex]
[tex]\[ = 2 \][/tex]
Thus, [tex]\( A(4) = 2 \)[/tex].
Next, evaluate [tex]\( A(5) \)[/tex]:
[tex]\[ A(5) = \int_3^5 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^5 \][/tex]
[tex]\[ = \left( 2(5)^2 - 12(5) \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 25 - 60 \right) - \left( 2 \cdot 9 - 36 \right) \][/tex]
[tex]\[ = (50 - 60) - (18 - 36) \][/tex]
[tex]\[ = -10 + 18 \][/tex]
[tex]\[ = 8 \][/tex]
Thus, [tex]\( A(5) = 8 \)[/tex].
General expression for [tex]\( A(x) \)[/tex]:
[tex]\[ A(x) = \int_3^x (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_3^x \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - \left( 2(3)^2 - 12(3) \right) \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - (18 - 36) \][/tex]
[tex]\[ = 2x^2 - 12x + 18 \][/tex]
Thus, the expression for [tex]\( A(x) \)[/tex] when [tex]\( x \geq 3 \)[/tex] is [tex]\( 2x^2 - 12x + 18 \)[/tex].
### Part (b)
Now, let's evaluate [tex]\( F(7) \)[/tex] and [tex]\( F(8) \)[/tex], and find the general expression for [tex]\( F(x) \)[/tex].
Evaluate [tex]\( F(7) \)[/tex] and [tex]\( F(8) \)[/tex]:
[tex]\[ F(7) = \int_6^7 (4t - 12) \, dt \][/tex]
Let's compute the above integral:
[tex]\[ \int_6^7 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^7 \][/tex]
[tex]\[ = \left( 2(7)^2 - 12(7) \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 49 - 84 \right) - \left( 2 \cdot 36 - 72 \right) \][/tex]
[tex]\[ = (98 - 84) - (72 - 72) \][/tex]
[tex]\[ = 14 - 0 \][/tex]
[tex]\[ = 14 \][/tex]
Thus, [tex]\( F(7) = 14 \)[/tex].
Next, evaluate [tex]\( F(8) \)[/tex]:
[tex]\[ F(8) = \int_6^8 (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^8 \][/tex]
[tex]\[ = \left( 2(8)^2 - 12(8) \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2 \cdot 64 - 96 \right) - \left( 2 \cdot 36 - 72 \right) \][/tex]
[tex]\[ = (128 - 96) - (72 - 72) \][/tex]
[tex]\[ = 32 - 0 \][/tex]
[tex]\[ = 32 \][/tex]
Thus, [tex]\( F(8) = 32 \)[/tex].
General expression for [tex]\( F(x) \)[/tex]:
[tex]\[ F(x) = \int_6^x (4t - 12) \, dt \][/tex]
[tex]\[ = \left[ 2t^2 - 12t \right]_6^x \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - \left( 2(6)^2 - 12(6) \right) \][/tex]
[tex]\[ = \left( 2x^2 - 12x \right) - (72 - 72) \][/tex]
[tex]\[ = 2x^2 - 12x \][/tex]
Thus, the expression for [tex]\( F(x) \)[/tex] when [tex]\( x \geq 6 \)[/tex] is [tex]\( 2x^2 - 12x \)[/tex].
### Summary:
1. [tex]\( A(4) = 2 \)[/tex]
2. [tex]\( A(5) = 8 \)[/tex]
3. [tex]\( A(x) = 2x^2 - 12x + 18 \)[/tex]
4. [tex]\( F(7) = 14 \)[/tex]
5. [tex]\( F(8) = 32 \)[/tex]
6. [tex]\( F(x) = 2x^2 - 12x \)[/tex] when [tex]\( x \geq 6 \)[/tex]
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