Let be a continuous function on

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let ff be a continuous function on R\mathbb R, and let yy satisfy dydx=f(x)\dfrac{dy}{dx}=f(x) with the initial condition y(5)=−3y(5)=-3. Which of the following statements best justifies that y(x)=−3+∫5xf(t) dty(x)=-3+\displaystyle\int_{5}^{x}f(t)\,dt for all real xx? (a) Because ff is continuous, any antiderivative of ff must have the form −3+∫x5f(t) dt-3+\displaystyle\int_{x}^{5}f(t)\,dt. (b) Because the Fundamental Theorem of Calculus states that the derivative of the integral from a constant to xx of f(t)f(t) is f(x)f(x), differentiating −3+∫5xf(t) dt-3+\displaystyle\int_{5}^{x}f(t)\,dt reproduces f(x)f(x). (c) Because integrating dydx=f(x)\dfrac{dy}{dx}=f(x) gives y(x)=∫f(x) dx+Cy(x)=\int f(x)\,dx+C, and applying y(5)=−3y(5)=-3 makes C=−3C=-3, yielding y(x)=−3+∫f(x) dxy(x)=-3+\displaystyle\int f(x)\,dx. (d) Because the Mean Value Theorem for Integrals guarantees ∫5xf(t) dt=f(c)(x−5)\displaystyle\int_{5}^{x}f(t)\,dt = f(c)(x-5) for some cc between 55 and xx, so yy has slope f(x)f(x) everywhere.

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