284
Chapter 3 | Derivatives
Solution We can see right away that for the 74th derivative of sin x , 74 = 4(18) + 2, so d 74 dx 74 (sin x ) = d 72+2 dx 72+2 (sin x ) = d 2 dx 2 (sin x ) =−sin x .
3.32
For y = sin x , find d 59 dx 59
(sin x ).
Example 3.47 An Application to Acceleration
A particle moves along a coordinate axis in such a way that its position at time t is given by s ( t ) =2−sin t . Find v ( π /4) and a ( π /4). Compare these values and decide whether the particle is speeding up or slowing down.
Solution First find v ( t ) = s ′( t ):
v ( t ) = s ′( t ) =−cos t .
Thus,
v ⎛ ⎞ ⎠ = − 1 2 . Next, find a ( t ) = v ′( t ). Thus, a ( t ) = v ′( t ) = sin t and we have a ⎛ ⎝ π 4 ⎞ ⎠ = 1 2 . Since v ⎛ ⎝ π 4 ⎞ ⎠ = − 1 2 <0 and a ⎛ ⎝ π 4 ⎞ ⎠ = 1 2 ⎝ π 4
>0, we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is travelling. Consequently, the particle is slowing down.
3.33 A block attached to a spring is moving vertically. Its position at time t is given by s ( t ) =2sin t . Find v ⎛ ⎝ 5 π 6 ⎞ ⎠ and a ⎛ ⎝ 5 π 6 ⎞ ⎠ . Compare these values and decide whether the block is speeding up or slowing down.
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