Problem 18 - Calculus

Question

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

Step-by-Step Solution

Answer

The derivative \( G'(x) = -2x \).

Step 1: Identify the Fundamental Theorem of Calculus

The problem asks us to find the derivative of a function defined by an integral. This is a perfect scenario to apply the Fundamental Theorem of Calculus, which states that if \( F(t) \) is an antiderivative of \( f(t) \), then \( \frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x) \).

Step 2: Recognize the Situation

We need to find \( G'(x) \) where \( G(x) = \int_{x}^{1} 2t \, dt \). However, notice that the integral has the upper limit \( 1 \) and lower limit \( x \). This isn't in the standard form, so we need to address this by manipulating the integral bounds.

Step 3: Change the Integration Limits

Using the property of definite integrals, you can switch the limits. This will give: \( G(x) = -\int_{1}^{x} 2t \, dt \).

Step 4: Apply the Fundamental Theorem of Calculus

Now the lower limit is a constant and the upper limit is \( x \). According to the Fundamental Theorem of Calculus, the derivative of \( \int_a^x f(t) \, dt \) with respect to \( x \) is simply \( f(x) \). Thus, the derivative of \( -\int_{1}^{x} 2t \, dt \) is \( -2x \).

Step 5: Write the Derivative

Therefore, the derivative \( G'(x) \) is \( -2x \).

Key Concepts

definite integrals

Definite integrals are a fundamental concept in calculus, important for calculating the area under a curve between two points. Imagine you want to find out how much space is taken by a shape bounded by a function and two specific points on the x-axis.
Definite integrals help us

• Calculate areas under curves.
• Assess the total accumulation, such as distance when speed changes continuously.

In our exercise, we deal with the definite integral \( \int_{x}^{1} 2t \, dt \). Typically, definite integrals use a lower and an upper limit to signify the interval over which the integration is performed. Here, the integral's limits are from \( x \) to 1, indicating we initially evaluate from variable \( x \) to a fixed point 1. Understanding the properties of definite integrals, like switching the integration limits, is crucial. If we swap the limits, it changes the sign of the integral, hence introducing a minus sign in our calculations.

antiderivative

An antiderivative gives you a function whose derivative is the original function you started with. Think of it as reversing differentiation. Antiderivatives tell us about accumulation, much like integration.
For any function \( f(t) \), finding its antiderivative \( F(t) \) means that \( F'(t) = f(t) \). In simpler terms, if you differentiate \( F(t) \), you arrive back at \( f(t) \).
The exercise is a good fit for recognizing the antiderivative concept. When you see an integral like \( \int 2t \, dt \), the antiderivative of \( 2t \) would be a function \( F(t) = t^2 \). Without constant limits complicating it, determining this helps answer the bigger picture of what the original function looks like before differentiation. Antiderivatives are broadly useful for solving many calculus problems involving integrals.

derivative of integrals

Derivatives and integrals have a special relationship defined by the Fundamental Theorem of Calculus. This theorem connects differentiation and integration, showing how one operation can reverse the other.
The exercise illustrates how we can take the derivative of an integral function. Initially, the task was to find \( G'(x) \) for \( G(x) = \int_{x}^{1} 2t \, dt \). However, by changing the limits, our integral becomes \( -\int_{1}^{x} 2t \, dt \). Based on the theorem, when the upper limit of the integral is a variable, the derivative of this integral with respect to the upper variable is simply the integrand evaluated at that variable.
Therefore, the result \( -2x \) is straightforwardly derived by acknowledging the link between integrating and differentiating, a hallmark example of the Fundamental Theorem of Calculus. Understanding that the derivative of \( \int_a^x f(t) \, dt \) is \( f(x) \) helps solve such exercises more intuitively.