Area functions and the Fundamental Theorem Consider the functi...

Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t if ―2 ≤ t < 0

t²/2 if 0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

fig

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

Accepted Answer

Hello. In this video, we are given the piecewise function whose graph is provided below. We are going to let F of X be defined as the definite integral from -2 to X of little F of TDT. We want to find an expression for F F of X between -4 and 0 by using the fundamental theorem of calculus. OK. In order to approach this problem, let's just go ahead and write down the given capital F function. We are given F of X, and that is defined as the definite integral from -2 to X of little F of T D T. Now, in order to define the integrand, we want to go ahead and define. Little FFT on the boundary from -4 to 0. Now, in this case here, we know that little T is defined between -4 and 0 as T + 1. So we can rewrite the capital F of X function as T + 1 since we want a derivative between -4 and 0. So we have capital F of X equal to the definite integral from -2 to X of T + 1 DT. Now, since we are trying to find F of X, this means that we need to take a derivative of F of X with respect to X. And in order to do this, we are going to use the fundamental theorem of calculus, part one. Now, the fundamental theorem of calculus part one states the following. If we have an area function, let's say G of X, which is defined as some constant to FXX of H of TDT, If we take the derivative with respect to X, Then all we are doing is we are plugging in the upper bound, which is a function of X into the integrand, and that will give us H of F of X, then we need to multiply by the derivative of that upper bound. So In this step, if we apply the fundamental theorem of calculus part one and take the derivative with respect to X, then we are going to plug in this upper bound of X into the integrand. That is going to give us X + 1. Then we need to multiply by the derivative of that function of X. And the derivative of X is just going to be 1. So that leaves us with X + 1 multiplied by 1, which is going to just leave us with X + 1. And this is going to be the derivative of the capital F of X function by the fundamental theorem of calculus part 1. And with that being said, the overall solution is going to be C. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Fundamental Theorem of Calculus

Definite Integral

Piecewise Function

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