Problem 51 - Calculus

Question

Find \(f(x)\) if \(\int_{1}^{x} f(t) d t=2 x-2\)

Step-by-Step Solution

Answer

The function \( f(x) = 2 \).

Step 1: Analyze the Given Integral Equation

We are given the equation \( \int_{1}^{x} f(t) \, dt = 2x - 2 \). The left-hand side of the equation represents the definite integral of the function \( f(t) \) from 1 to \( x \). The right-hand side simplifies to a linear expression in terms of \( x \).

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(x) \) is the antiderivative of \( f(t) \), then \( \frac{d}{dx} \int_{1}^{x} f(t) \, dt = f(x) \). This means \( f(x) = \frac{d}{dx} (2x - 2) \).

Step 3: Differentiate the Right-Hand Side

Differentiate the expression \( 2x - 2 \) with respect to \( x \). The derivative is \( 2 \), since the derivative of \( 2x \) is 2 and the derivative of a constant \(-2\) is 0.

Step 4: Conclude the Result

Since \( f(x) \) corresponds to the derivative of the expression \( 2x - 2 \), we have \( f(x) = 2 \). This means the function \( f(x) \) is a constant function equal to 2.

Key Concepts

Definite Integral

The definite integral is a concept in calculus used to compute the accumulation of quantities. Simply put, it's a way of adding up a continuous range of values, such as areas under curves. In our problem, the definite integral is from 1 to \( x \), and it is written as:

• \( \int_{1}^{x} f(t) \, dt \)

This notation signifies that we integrate the function \( f(t) \) over the interval \([1, x]\). The result tells us the total accumulated change of the function between those two points. In this case, the integral is equal to \( 2x - 2 \), which is a linear function of \( x \). This means that the accumulated changes follow a straight line pattern, directly connected to how \( x \) changes.

Antiderivatives

Antiderivatives are closely related to the process of differentiation, but they work in reverse. If differentiating gives you the rate of change of a function (derivative), finding the antiderivative gives you the original function whose derivative was taken.

• If \( F(x) \) represents the antiderivative of \( f(t) \) over \( [1, x] \), then according to the Fundamental Theorem of Calculus: \( F(x) \) is such that \( F'(x) = f(x) \).

In the given exercise, since the definite integral \( \int_{1}^{x} f(t) \, dt = 2x - 2 \), we are told that the derivative of this linear expression must be \( f(x) \). The antiderivative here is the accumulated integral itself, which resolves into identifying "\( f(x) \) as the function whose rate of change yields \( 2x - 2 \)".

Differentiation

Differentiation is the process of calculating the derivative, which is the rate at which a function is changing at any given point. It's a core tool in calculus for understanding how functions behave.

• The derivative of \( 2x - 2 \) was calculated in the step-by-step solution to find \( f(x) \).
• The process involves applying the rule that the derivative of \( ax \) is \( a \), and the derivative of a constant is 0. Thus, for \( 2x - 2 \), the derivative is simply \( 2 \).

This means \( f(x) \), the function needed to make the integral equation hold true, is a constant function with a value of \( 2 \). It highlights the straightforward nature of differentiation, revealing changes directly tied to modifications in \( x \). In our problem, since \( 2x - 2 \) is linear, its rate of change or slope is constant.