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Chapter 3 | Derivatives
Find f ″( x ) for f ( x ) = x 2 .
3.9
Example 3.16 Finding Acceleration
The position of a particle along a coordinate axis at time t (in seconds) is given by s ( t ) =3 t 2 −4 t +1 (in meters). Find the function that describes its acceleration at time t .
Solution Since v ( t ) = s ′( t ) and a ( t ) = v ′( t ) = s ″( t ), we begin by finding the derivative of s ( t ) : s ′( t ) = lim h →0 s ( t + h )− s ( t ) h = lim h →0 3( t + h ) 2 −4( t + h )+1− ⎛ ⎝ 3 t 2 −4 t +1 ⎞ ⎠ h
=6 t −4.
Next,
s ′( t + h )− s ′( t ) h
s ″( t ) = lim h →0
6( t + h )−4−(6 t −4) h
= lim
h →0
=6.
Thus, a =6m/s 2 .
For s ( t ) = t 3 , find a ( t ).
3.10
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