calculus pending

Q1. The position of an object is given by s(t) = sin⁡(3t) − 2t + 4.

Determine where in the interval [0,3] the object is moving to the

right and moving to the left.

Given arccos(2/3) = 0.841 radians.

and

2.pie = 6.283 radians.
S1.

s'(t) = 3cos(3t) - 2


First find critical points:

3cos(3t) - 2 = 0 => cos(3t) = 2/3

=> Since cos(3t) is positive, there are 2 solutions (1st and 4th quadrant).

3t = acos(2/3) + 2.pie.k = 0.841 + 2.pie.k

and

3t = [2.pie - acos(2/3)] + 2.pie.k

= 5.442 + 2.pie.k


How many valid solutions are there for 3t in [0,9] since t in [0,3].

Apart from the 2 above, there is only 1 more when 3t = .841 + 6.283 = 7.124


So these 3 are our critical points.

Let's evaluate at each of them.

Case 1: 3t between [0,.841)

s'(0) = 1 => +ive => moving right

Case 2: 3t between (.841, 5.442), try 3t = 3(t = 1) => 3cos(3t) - 2 = 3cos3 - 2

cos3 is close to cos(3.14), i.e. -1 so value is negative, moving left.

Case 3: 3t between (5.442,7.124), try 3t = 6.283 = 2.pi => val = 1 => positive => right

Case 4: 3t between (7.124,9], try 3t = 9 which is close to 3.pie but less => s'(t) = -ive => left


Q2.

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