EXAMPLE 6 A particle moves in a straight line and has acceleration given by a(t) = 12t+4. Its initial velocity is v(0) = -4 cm/s and its initial displacement is s(0) = 8 cm. Find its position function, s(t). SOLUTION Since v'(t) = a(t) = 12t+4, antidifferentiation gives v(t) = Note that v(0) = C. But we are given that v(0) = -4, so C = v(t) = Since v(t) = s'(t), s is the antiderivative of v: |) + 4( [ s(t) = = + 4t + C = ( s(t) = + D. This gives s(0) = D. We are given that s(0) = 8, so D = is + C. - 4t+ D and and the required position function

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EXAMPLE 6 A particle moves in a straight line and has acceleration given by

a (t) = 12 t + 4

. Its initial velocity is

v (0) = ? 4 cm / s

and its initial displacement is

s (0) = 8 cm

. Find its position function,

s (t)

. SOLUTION Since

v^{?} (t) = a (t) = 12 t + 4

, antidifferentiation gives

v (t) = + 4 t + c = + C .

Note that

v (0) = C

. But we are given that

v (0) = ? 4

, so

C =

and

v (t) =

Since

v (t) = s^{?} (t)

,

s

is the antiderivative of

v

:

s (t) = 6 () + 4 () ? 4 t + D = + D .

This gives

s (0) = D

. We are given that

s (0) = 8

, so

D =

and the required position function is

s (t) =

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