Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given that the integral of a function \( f(t) \) from \( 1 \) to \( x \) is equal to \( 2x - 2 \). We need to find the function \( f(x) \).

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 \( \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x) \). According to this theorem, the derivative of \( 2x - 2 \) with respect to \( x \) can be used to find \( f(x) \).

Step 3: Differentiate the Given Equation

Differentiate \( 2x - 2 \) with respect to \( x \). The derivative of \( 2x \) is \( 2 \) and the derivative of \( -2 \) is \( 0 \). Thus, \( \frac{d}{dx}(2x - 2) = 2 \).

Step 4: Conclude the Solution

Based on the differentiation, we find that \( f(x) = 2 \). This is the function whose integral from \( 1 \) to \( x \) results in \( 2x - 2 \).

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that measures the accumulation of quantities, such as area under a curve. It is represented in the form \( \int_{a}^{b} f(t) \, dt \), where \( f(t) \) is a continuous function and \( a \) and \( b \) are the limits of integration. In this context, the integral gives the total change in a quantity from \( a \) to \( b \).

For example, in the given exercise, the integral \( \int_{1}^{x} f(t) \, dt = 2x - 2 \) represents the area under the function \( f(t) \) from \( t = 1 \) to \( t = x \). This integral is equal to a function of \( x \), specifically \( 2x - 2 \), which means the accumulation of \( f(t) \) over this interval behaves linearly with \( x \).

• The lower limit of integration 1 is where the accumulation starts.
• The upper limit \( x \) indicates that the end point varies, making it a variable limit integral.

Understanding definite integrals as the net change or the total accumulated value over an interval is crucial for solving calculus problems.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which a function changes with respect to a variable. It links directly to how functions increase or decrease. In simpler terms, it is the calculation of the slope of the function at any point.

In the original exercise, differentiation is used to find \( f(x) \) from the integral equation \( \int_{1}^{x} f(t) \, dt = 2x - 2 \). By the Fundamental Theorem of Calculus, the derivative of this integral with respect to \( x \) gives \( f(x) \).

• When we differentiate \( 2x - 2 \), we find its derivative to be \( 2 \).
• This tells us the rate of change of the linear function is constant, which then identifies the function \( f(x) \) as \( 2 \).

Differentiation simplifies to finding how one function stacks up against its ongoing rate of change.

Antiderivative

An antiderivative is essentially the reverse of differentiation. If you have a derivative, the antiderivative returns the original function. It is also known as an indefinite integral.

Antiderivatives are important in calculus because they help in finding the exact functions behind rates of change described by derivatives. Utilizing the Fundamental Theorem of Calculus, which states that integration (finding the area under the curve) is inversely related to differentiation (finding the slope).

• In our problem, identifying the function \( f(x) \) from \( 2x-2 \) showcases this relationship.
• Given that \( \frac{d}{dx} \int_{a}^{x} f(t) dt = f(x) \), we find that the antiderivative of \( f(x) \) when differentiated gives \( 2 \); thus, \( f(x) \) must be a constant as its antiderivative simply increments linearly with \( x \).

Understanding antiderivatives deepens the comprehension of how mathematical operations invert each other to form complete functions.