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An object moves along the x-axis so that its position at any time 𝘵 > 0 is given by x(𝘵) = s(𝘵). Find the velocity of the object as a function of 𝘵.

To find the velocity of the object as a function of \( t \), we need to compute the derivative of the position function \( s(t) \) with respect to time \( t \). The position function \( s(t) \) is given by:

\[
s(t) = \frac{4}{3\pi} \sin(3t) + \frac{4}{5\pi} \cos(5t)
\]

### Step 1: Differentiate the first term

\[
\frac{d}{dt} \left( \frac{4}{3\pi} \sin(3t) \right) = \frac{4}{3\pi} \cdot \frac{d}{dt} \left( \sin(3t) \right) = \frac{4}{3\pi} \cdot 3 \cos(3t) = \frac{12}{3\pi} \cos(3t) = \frac{4}{\pi} \cos(3t)
\]

### Step 2: Differentiate the second term

\[
\frac{d}{dt} \left( \frac{4}{5\pi} \cos(5t) \right) = \frac{4}{5\pi} \cdot \frac{d}{dt} \left( \cos(5t) \right) = \frac{4}{5\pi} \cdot (-5) \sin(5t) = \frac{-20}{5\pi} \sin(5t) = \frac{-4}{\pi} \sin(5t)
\]

### Step 3: Combine the derivatives

The velocity function \( v(t) \), which is the derivative of \( s(t) \), is:

\[
v(t) = \frac{4}{\pi} \cos(3t) - \frac{4}{\pi} \sin(5t)
\]

Thus, the velocity of the object as a function of \( t \) is:

\[
v(t) = \frac{4}{\pi} \left( \cos(3t) - \sin(5t) \right)
\]

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Do you want details on any of the steps or have any further questions?

Here are some related questions to further explore this topic:
1. How do you find the acceleration of the object as a function of \( t \)?
2. How would the velocity change if the position function had a different frequency for the sine and cosine terms?
3. What are the units of velocity if the position \( s(t) \) is given in meters and \( t \) is in seconds?
4. How can you determine when the object is momentarily at rest using the velocity function?
5. What effect do the coefficients in the trigonometric terms have on the velocity and acceleration?

**Tip:** Remember that differentiation of trigonometric functions often involves the chain rule, especially when dealing with functions of the form \( \sin(k t) \) or \( \cos(k t) \).

An object moves along the x-axis so that its position at any time t > 0 is given by x(t) = s(t). Find the velocity of the object as a function of t.

This problem explains how to differentiate a trigonometric position function to find the velocity of an object moving along the x-axis. It involves applying the chain rule to functions involving sine and cosine with different frequencies. Suitable for students in Grades 11-12.

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