Question

In Exercises \(73-78,\) use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{-2}^{x}\left(t^{2}-2 t\right) d t $$

Short answer

The derivative \( F'(x) \) of the given function is \( x^{2} - 2x \).

Step-by-step solution

Identity The Function

First, identify the function inside the integral which is \( t^{2} - 2t\). Here, \( t \) is the variable of integration.

Apply The Second Fundamental Theorem of Calculus

The next step is to apply the Second Fundamental Theorem of Calculus. We take the function inside the integral and replace \( t \) with \( x \). This yields the expression \( x^{2} - 2x \).

Final Answer

The derivative of \( F(x) \) is the equivalent of the function inside the integral, therefore \( F'(x) = x^{2} - 2x \).

Key concepts

Indefinite Integrals

Before diving into the Second Fundamental Theorem of Calculus, it's essential to understand indefinite integrals. An indefinite integral, sometimes known as an antiderivative, is essentially the reverse process of taking a derivative. When you integrate a function without definite limits, you're finding all the possible original functions which could have been differentiated to your given function.

For example, when you have a function like \( f(t) = t^2 - 2t \), taking the indefinite integral would mean finding a function \( F(x) \) such that \( F'(x) = f(x) \). This is represented mathematically as \( \int f(t) \, dt = F(x) + C \), where \( C \) is the constant of integration, accounting for the fact that derivatives remove constants. One of the challenges here is recognizing that there are infinitely many antiderivatives for any given function, each differing by a constant.

Fundamental Theorems of Calculus

The Fundamental Theorems of Calculus bridge the gap between differentiation and integration, two principal concepts in calculus. The First Fundamental Theorem states that if a function \( f \) is continuous on an interval \( [a,b] \) and \( F \) is an antiderivative of \( f \) on that interval, then \( \int_a^b f(x) \, dx = F(b) - F(a) \). This theorem shows how you can calculate the definite integral of a function over an interval if you know its antiderivative.

The Second Fundamental Theorem of Calculus is particularly useful for solving problems like the one in this exercise. It tells us that if \( F \) is defined by an integral with an upper limit of \( x \), then \( F \) is continuous on the interval and differentiable where \( f \) is continuous, and \( F'(x) = f(x) \). This theorem allows us to directly find the derivative of a function defined by an integral, simply by evaluating the integrand at the upper limit of integration, as demonstrated in the provided solution steps.

Calculus Derivative

The derivative represents the rate at which a function changes at any given point and is a fundamental concept in calculus. When calculating a derivative, you're looking for the instantaneous rate of change or the slope of the tangent line at any point on a function's curve.

In the context of the exercise, the process of finding \( F'(x) \) means to determine how the function \( F(x) \) changes as \( x \) changes. We used the Second Fundamental Theorem of Calculus to directly find the derivative of our integral-defined function. In more straightforward terms, the act of differentiating 'undoes' the integration on the function \( t^2 - 2t \) within the given limits of integration. Thus, our final answer, \( F'(x) = x^2 - 2x \), represents the rate of change of the accumulated area under the curve \( t^2 - 2t \) from \( -2 \) to \( x \). In practical scenarios, understanding derivatives is crucial as they're used in various fields like physics, engineering, economics, and more for modeling change and understanding dynamic systems.