Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(G^{\prime}(x)\). $$ G(x)=\int_{1}^{x} 2 t d t $$

Step-by-Step Solution

Answer

\( G'(x) = 2x \)

Step 1: Identify the Function to Differentiate

We are given a function in the form of an integral: \( G(x) = \int_{1}^{x} 2t \, dt \). The task is to find the derivative \( G'(x) \). Since this is a definite integral with a variable upper limit, we should apply the Fundamental Theorem of Calculus.

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(t) \) is an antiderivative of \( f(t) \), then \( \frac{d}{dx} \left[ \int_{a}^{x} f(t) \, dt \right] = f(x) \). In our problem, \( f(t) = 2t \), so directly applying the theorem, we get \( G'(x) = 2x \).

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve within a specified interval on the x-axis. In our case, the integral is from 1 to \( x \). It is mathematically expressed as \( \int_{a}^{b} f(t) \, dt \). This involves finding the antiderivative and evaluating it at the limits, but thanks to the Fundamental Theorem of Calculus, we have a shortcut. The theorem allows us to use differentiation to find the integral's rate of change as its upper limit varies. Here, we are interested in finding what happens when the upper limit is not fixed but varies, say \( x \), which connects us to differentiation through the fundamental theorem.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us the rate at which a function's value changes as its input changes. In this exercise, we apply differentiation to an integral. Specifically, we're finding \( G'(x) \), the derivative of \( G(x) \). Rather than calculate the integral first and then differentiate, we use the Fundamental Theorem of Calculus. This theorem tells us that for a function \( f(t) \), the derivative of the integral from a constant \( a \) to a variable \( x \) is simply \( f(x) \). Thus, with \( f(t) = 2t \), differentiation reveals the instantaneous rate of change at \( x \), giving us \( G'(x) = 2x \).

Antiderivative

An antiderivative is a function whose derivative is the original function we are integrating. For instance, if you know \( F'(t) = f(t) \), then \( F(t) \) can be considered an antiderivative of \( f(t) \). When dealing with the Fundamental Theorem of Calculus, knowing the antiderivative helps compute definite integrals. Here, it helps link differentiation with integration. For the function \( f(t) = 2t \), its antiderivative would be \( F(t) = t^2 + C \), where \( C \) is a constant of integration. Normally, we'd evaluate this antiderivative at the bounds to find a definite integral. But because we are differentiating, we skip directly to finding \( G'(x) \) through the theorem. The antiderivative hints at this process because it implicitly contains the information about how the function accumulates.