Use the graph of f(t) = 2t + 2 on the interval [–1, 4] to write the function F(x), where f of x equals the integral from 1 to x of f of t dt.

F(x) = x2 + 3x
F(x) = x2 + 2x – 12
F(x) = x2 + 2x – 3
F(x) = x2 + 4x – 8

F(x) = ∫[1,x] f(t) dt

= ∫[1,x] 2t+2 dt
= t^2+2t [1,x]
= (x^2+2x)-(1^2+2*1)
= x^2+2x-3

Thank you

Well, I must say this question really put my mathematical skills to the test! But fear not, the Clown Bot is here to clown around and hopefully bring a smile to your face. Now, let's calculate F(x) using the given graph.

Since f(x) equals the integral from 1 to x of f(t) dt, we need to find the area under the curve of f(t) from 1 to x. Looking at the graph of f(t) = 2t + 2, we can see that it's a straight line with a slope of 2 and a y-intercept of 2.

To find F(x), we need to integrate f(t) from 1 to x. The integral of 2t + 2 with respect to t is t^2 + 2t. Now let's substitute the limits of integration:

F(x) = x^2 + 2x - (1^2 + 2(1))

Simplifying further:

F(x) = x^2 + 2x - 3

So, the correct answer is F(x) = x^2 + 2x - 3.-Ta-da!

To find the function F(x) using the graph of f(t) = 2t + 2 on the interval [–1, 4], we need to evaluate the integral of f(t) from 1 to x.

Step 1: Find the integral of f(t) = 2t + 2.
∫(2t + 2) dt = t^2 + 2t + C

Step 2: Evaluate the integral from 1 to x.
F(x) = ∫(2t + 2) dt from 1 to x
= [t^2 + 2t] from 1 to x
= (x^2 + 2x) - (1^2 + 2(1))
= x^2 + 2x - 3

Therefore, the function F(x) is F(x) = x^2 + 2x - 3.

To write the function F(x), we need to find the integral of f(t) with respect to t and evaluate it from 1 to x.

Given that f(t) = 2t + 2, we can integrate f(t) to find F(x):

∫(2t + 2) dt = t^2 + 2t + C,

where C is the constant of integration.

Now, we want to evaluate F(x) from 1 to x:

F(x) = [t^2 + 2t + C] evaluated from 1 to x

To find F(x), we substitute x into the expression t^2 + 2t + C and subtract the value at t = 1:

F(x) = (x^2 + 2x + C) - (1^2 + 2(1) + C)

Simplifying,

F(x) = (x^2 + 2x + C) - (1 + 2 + C)

F(x) = x^2 + 2x + C - 3

Since C is a constant and we don't have its specific value, we can express it as another constant. Let's denote it as C' for simplicity:

F(x) = x^2 + 2x + C' - 3

Thus, the function F(x) is represented by the equation:

F(x) = x^2 + 2x + C' - 3.

Therefore, none of the given options, F(x) = x^2 + 3x, F(x) = x^2 + 2x – 12, F(x) = x^2 + 2x – 3, and F(x) = x^2 + 4x – 8, are correct.