Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

b. f(1) = 0

검증된 단계별 안내

To find the function f(x) from its derivative f'(x) = 2x, we need to integrate the derivative. The integral of f'(x) = 2x with respect to x is f(x) = ∫2x dx.

Perform the integration: ∫2x dx = x^2 + C, where C is the constant of integration. This gives us the general form of the function: f(x) = x^2 + C.

We are given the initial condition f(1) = 0. Use this information to find the constant C. Substitute x = 1 into the function: f(1) = 1^2 + C = 0.

Solve for C: 1 + C = 0, which implies C = -1.

Now that we have determined C, the specific function is f(x) = x^2 - 1. To find f(2), substitute x = 2 into this function: f(2) = 2^2 - 1.

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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. In this context, finding the antiderivative of f'(x) = 2x involves determining a function f(x) such that its derivative is 2x. The general form of the antiderivative of 2x is x^2 + C, where C is a constant.

Initial Conditions

Initial conditions are specific values given for a function at a particular point, which help determine the constant of integration when finding an antiderivative. Here, the initial condition f(1) = 0 allows us to solve for the constant C in the antiderivative f(x) = x^2 + C, ensuring the function satisfies this condition.

Definite Evaluation

Definite evaluation involves using the determined function, including the constant of integration, to find the value of the function at a specific point. After finding the antiderivative and constant, we use f(x) = x^2 - 1 to evaluate f(2), which involves substituting x = 2 into the function to find the specific value of f(2).

Finding Functions from DerivativesSuppose that f'(x) = 2x for all x. Find f(2) ifb. f(1) = 0