ASAP --> AB CALC REVIEW QUESTION (Answer parts a b c )

Question

A particle travels up and down in a wave pattern so that its position from its origin point, in millimeters, is given by

s(t)=4sin(t)-3cos(t), where t is time, in seconds

a) Is the particle moving up, down, or neither at t= 2pi/3 seconds? Justify workout

b) Is the particle speeding up, slowing down, or neither at t= 2pi/3 seconds? Explain workout

c) At any time when the particle is at rest, what is the value for tan(t)?

Answer 1

We first need to make an assumption that positive movement is up and negative movement is down.

If you want to know if the particle is moving up, down, or neither, you could look at its velocity. If the velocity is positive, we know it's moving up. The velocity is the derivative of position so s '(t) = 4cos(t)+3sin(t). Plugging in 2π/3: s '(2π/3) = 4cos(2π/3)+3sin(2π/3) = 4(-1/≈)+3(√3/2) ≈ 0.6 so since the velocity is positive, the particle is moving up.

If you want to know if the particle is speeding up or slowing down, you would look at the acceleration. If acceleration is positive, then the particle is speeding up. acceleration is the second derivative of position or the derivative of velocity. So, take the derivative of the velocity function: v '(t) = -4sin(t)+3cos(t) and v '(2π/3) = -4sin(2π/3)+3cos(2π/3) = -4(√3/2)+3(1/2) ≈ -5 so since the acceleration is negative, the particle is slowing down.

If you want to know when the particle is at rest, you want to know when the velocity is zero. So, set the velocity function equal to zero. v(t) = s '(t) = 4cos(t)+3sin(t)

4cos(t) + 3sin(t) = 0

4cos(t) = -3sin(t)

divide both sides by cost(t) to get:

4 = -3sin(t)/cos(t)

divide both side by -3 to get:

-4/3 = sin(t)/cos(t) but tan(t) = sin(t)/cos(t) so:

tan(t) = -4/3