Problem 52 - Calculus

Question

Find \(f(x)\) if \(\int_{0}^{x} f(t) d t=x^{2}\)

Step-by-Step Solution

Answer

\(f(x) = 2x\)

Step 1: Identify the Problem Type

We need to find the function \( f(x) \) given the integral equation \( \int_{0}^{x} f(t) \, dt = x^2 \). This is an application of the Fundamental Theorem of Calculus, which can be used to relate differentiation and integration.

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \int_{a}^{x} f(t) \, dt = F(x) - F(a) \). Here, \( F(x) = x^2 \), so \( F'(x) = f(x) \).

Step 3: Differentiate the Given Equation

Differentiate both sides of the equation \( \int_{0}^{x} f(t) \, dt = x^2 \) with respect to \( x \). On the right side, the derivative of \( x^2 \) is \( 2x \). Therefore, \( f(x) = (x^2)' = 2x \).

Step 4: Write the Solution

After differentiating, we find that \( f(x) = 2x \). This satisfies the condition given by the integral equation, as integrating \( 2x \) from \( 0 \) to \( x \) gives \( x^2 \).

Key Concepts

Antiderivative

In calculus, an antiderivative of a function is another function whose derivative gives us back the original function. This is an essential concept when we explore the Fundamental Theorem of Calculus. The theorem links the process of differentiation with integration—two core operations in calculus.

- If you have a function, say, \( f(x) \), and you "undo" the derivative of \( f(x) \), you find its antiderivative. For instance, if the derivative of \( F(x) \) is \( f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
- Antiderivatives are unique up to a constant, meaning if \( F(x) \) is an antiderivative of \( f(x) \), then \( F(x) + C \) is also an antiderivative, where \( C \) is a constant.

In our original exercise, using the concept of the antiderivative, we determined that \( F(x) = x^2 \), since its derivative \( F'(x) \) matches \( f(x) = 2x \). This understanding revealed the function \( f(x) \) originally given by the integral equation.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative measures how a function changes as its input changes—the rate of change. In mathematical terms, if \( f(x) \) is our function, then its derivative \( f'(x) \) represents how \( f(x) \) changes with each infinitesimal movement in \( x \).

- Differentiation is a tool to find instantaneous rates of change. Say, for example, in physics, it helps us find the velocity of an object if we know its position over time.
- A derivative at a point gives the slope of the tangent line to the function at that point, helpful for understanding function behavior locally.

In our step-by-step solution, we used differentiation to find \( f(x) \). By differentiating the integral \( \int_{0}^{x} f(t) \, dt = x^2 \), we applied the Fundamental Theorem of Calculus. Differentiation of the right-hand side, \( x^2 \), yielded \( 2x \), showing that \( f(x) = 2x \). This connected the given integral with a function defined explicitly using differentiation.

Integration

Integration is the process of finding the integral of a function, which can be thought of as the reverse process of differentiation. When we integrate a function, we are essentially finding the accumulated total or area under the curve the function describes.

- There are two types of integration: indefinite and definite. Indefinite integrals provide a general form of antiderivatives, whereas definite integrals calculate the net area under a curve between two points.
- Integration has applications in various fields such as physics, engineering, and statistics, helping in solving problems related to area, volume, growth rates, constraints, etc.

In the context of our problem, we considered the definite integral \( \int_{0}^{x} f(t) \, dt \) and equated it to \( x^2 \). By reversing the process of differentiation—through finding antiderivatives—we confirmed the solution \( f(x) = 2x \) using crucial integration concepts. Thus, integration was pivotal in formulating the connection between \( f(x) \) and its integral representation.