Question

In Exercises 44 and \(45,\) a spring is bobbing up and down on the end of a spring according to \(s(t)=-3 \sin t .\) True or False The spring is traveling upward at \(t=3 \pi / 4\) Justify your answer.

Short answer

The spring is not moving upward at \(t=3 \pi / 4\), thus the statement is false.

Step-by-step solution

Calculating the Derivative

First, calculate the derivative of \(s(t)=-3 \sin t\) using the chain rule. The derivative of \(s(t)\) with respect to \(t\) will be \(s'(t)\).

Insert t = \(3 \pi / 4\)

Then, substitute \(t=3 \pi / 4\) in to our derivative function and calculate the resulting value.

Determining the Movement

If \(s'(3 \pi / 4)\) > 0, the spring is moving upward at \(t=3 \pi / 4\), otherwise it is moving downwards.