Answered

Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Let [tex]$f(t)=4t-12$[/tex] and consider the two area functions [tex]$A(x)=\int_3^x f(t) \, dt$[/tex] and [tex][tex]$F(x)=\int_6^x f(t) \, dt$[/tex][/tex]. Complete parts (a)-(c).

a. Evaluate [tex]A(4)[/tex] and [tex]A(5)[/tex]. Then use geometry to find an expression for [tex]A(x)[/tex] for all [tex]x \geq 3[/tex].

The value of [tex]A(4)[/tex] is [tex]\square[/tex]. (Simplify your answer.)

Sagot :

GinnyAnswer
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].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.