Finding Functions from Derivatives

In Exercises 31–36, find all possible functions with the given derivative.

a. y′ = x

검증된 단계별 안내

To find the original function from its derivative, we need to perform integration. The given derivative is \( y' = x \).

Integrate the derivative \( y' = x \) with respect to \( x \). This means we need to find \( \int x \, dx \).

The integral of \( x \) with respect to \( x \) is \( \frac{x^2}{2} \). This is because the power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

After integrating, we have \( y = \frac{x^2}{2} + C \), where \( C \) is an arbitrary constant. This represents the family of functions whose derivative is \( x \).

The constant \( C \) can be any real number, which means there are infinitely many functions that satisfy the given derivative, each differing by a constant.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. To find a function from its derivative, you need to determine its antiderivative. For example, if y' = x, the antiderivative of x is (1/2)x^2 plus a constant C, representing all possible functions with the given derivative.

Integration

Integration is the process of finding the antiderivative of a function. It involves calculating the integral of the function, which can be indefinite or definite. In this context, finding the indefinite integral of y' = x will yield the general form of the function y = (1/2)x^2 + C, where C is an arbitrary constant.

Constant of Integration

The constant of integration, denoted as C, arises when computing indefinite integrals. It represents an infinite number of possible functions that differ by a constant. When finding functions from derivatives, this constant accounts for all vertical shifts of the antiderivative, ensuring the solution encompasses all possible functions with the given derivative.

Finding Functions from DerivativesIn Exercises 31–36, find all possible functions with the given derivative.a. y′ = x