Finding Functions from Derivatives

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

b. y′ = x²

검증된 단계별 안내

To find the original function from its derivative, we need to perform integration. The given derivative is \( y' = x^2 \).

The process of finding the original function is called integration. We need to integrate \( x^2 \) with respect to \( x \).

The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Here, \( n = 2 \).

Apply the power rule for integration: \( \int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \).

Thus, the original function \( y \) is \( y = \frac{x^3}{3} + C \), where \( C \) is any constant, representing the family of functions with the given derivative.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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, we perform the process of integration, which is essentially finding the antiderivative. For example, if y' = x², the antiderivative is y = (1/3)x³ + C, where C is the constant of integration.

Indefinite Integration

Indefinite integration is the process of finding the antiderivative of a function. It involves integrating the function without specific limits, resulting in a general form that includes a constant of integration, C. This constant accounts for all possible vertical shifts of the antiderivative, reflecting the family of functions that share the same derivative.

Constant of Integration

The constant of integration, denoted as C, arises when performing indefinite integration. It represents an arbitrary constant added to the antiderivative, accounting for the fact that multiple functions can have the same derivative. For example, if y' = x², the antiderivative is y = (1/3)x³ + C, where C can be any real number.

Finding Functions from DerivativesIn Exercises 31–36, find all possible functions with the given derivative.b. y′ = x²