Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

f'(x) = 2x − 1, P(0,0)

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To find the original function f(x) from its derivative f'(x) = 2x - 1, we need to perform integration. The process of finding a function from its derivative is called antidifferentiation or integration.

Integrate the derivative f'(x) = 2x - 1 with respect to x. This means we need to find the indefinite integral of 2x - 1. The integral of 2x is x^2, and the integral of -1 is -x. Therefore, the integral of f'(x) is: ∫(2x - 1) dx = x^2 - x + C, where C is the constant of integration.

The constant C represents an unknown value that can be determined using the given point P(0,0). Substitute x = 0 and f(x) = 0 into the equation f(x) = x^2 - x + C to find C.

Substituting the point P(0,0) into the equation gives: 0 = 0^2 - 0 + C, which simplifies to C = 0.

Now that we have determined C, the function f(x) is x^2 - x. Therefore, the function whose derivative is 2x - 1 and passes through the point P(0,0) is f(x) = x^2 - x.

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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, we perform integration, which is the reverse process of differentiation. In this context, finding the antiderivative of f'(x) = 2x - 1 will yield the original function f(x).

Integration

Integration is the process of finding the antiderivative of a function. It involves determining a function whose derivative is the given function. For f'(x) = 2x - 1, integrating with respect to x will provide the general form of f(x), which includes an arbitrary constant C.

Initial Conditions

Initial conditions are used to find the specific solution to a differential equation by determining the constant of integration. Given the point P(0,0), we substitute x = 0 and f(x) = 0 into the integrated function to solve for C, ensuring the function passes through the specified point.

Finding Functions from DerivativesIn Exercises 37–40, find the function with the given derivative whose graph passes through the point P.f'(x) = 2x − 1, P(0,0)