Two students integrate and obtain and

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

Problem

Two students integrate

G'(x)=\cos x

and obtain

G_{1}(x)=\sin x+C_{1}

and

G_{2}(x)=\sin x+C_{2}

. Given the condition

G(0)=2

, which statement best explains why

C_{1}

and

C_{2}

must both equal

2

? (a) Because

\sin 0 = 1

, the only way to obtain

G(0)=2

is to set

C=1

. (b) The constant of integration is arbitrary, so

C_{1}

and

C_{2}

never have to equal

2

. (c) Evaluating

\sin x + C

x = 0

gives

C = 2

, so any correct antiderivative must satisfy this value of

C

. (d) Because

G'(0)=\cos 0 = 1

, the Mean Value Theorem forces the function through

(0,1)

, implying

C=1