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.

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

Accepted Answer

Hello. In this video, we are given a piecewise function whose graph is provided below. We are going to let capital F of X be defined as the definite integral from -2 to X a little F of TDT. We want to find an expression for F of X for X between 0 and 2 using the fundamental theorem of calculus. So, 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 was defined as the definite integral from 2 to X of little F of TDT. Now, we want to replace little FFT with the function that is between 0 and 2. Now, in this case here, T is defined between 0 and 2 as T2 + 1. So we are going to rewrite the integrand as T2 + 1. That is going to give us F of X, which is defined as -2 to X of T2 + 1DT. Now, in order to find F, this term F tells us that we need to take a derivative with respect to X. And in order to take a derivative of a definite integral, we need to use the fundamental theorem of calculus, part one. The fundamental theorem of calculus 1 is defined as the following. Let's suppose that we have a function G of X, and that is defined as the definite integral from some constant to a function of X of the integrand with respect to another variable, let's say H of TDT. In order to take the derivative of this kind of integral. All we are doing is we are going to plug in this function of X into the integrand. And afterward, we will need to multiply by the derivative of that function of X. So in this step here, if we take the derivative with respect to X by the fundamental theorem of calculus part one, we are going to define F of X. By plugging in our upper function of X, which is just X, into the integrand. That is going to give us X2 + 1, then we need to multiply by the derivative of that function of X, which again is just X. Now, the derivative of X with respect to X is going to be 1, and that is going to leave us with X2 + 1 multiplied by 1, which will simplify to be X2 + 1. And this is going to be the derivative of the capital F of X function. And the overall solution to this problem is going to be B. 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 Functions

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