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The position function defines the location of a particle along a defined path at any given time. For our particle moving along the x-axis, its position as a function of time is given by: \[ x(t) = 3 \cos t + 4 \sin t. \] This function tells you exactly where the particle is situated on the x-axis when you plug in a specific value of time, \( t \). Here's how it works in our example:

• At \( t=0 \), you substitute zero into the position function: \[ x = 3 \cos(0) + 4 \sin(0) = 3 \times 1 + 4 \times 0 = 3 \text{ feet}. \]
• At \( t=\pi/2 \), substitute \( \pi/2 \): \[ x = 3 \cos(\pi/2) + 4 \sin(\pi/2) = 3 \times 0 + 4 \times 1 = 4 \text{ feet}. \]
• At \( t=\pi \), use \( \pi \) in the equation: \[ x = 3 \cos(\pi) + 4 \sin(\pi) = 3 \times (-1) + 4 \times 0 = -3 \text{ feet}. \]

Understanding how to substitute these values can really help you see how the particle changes position over time.

Velocity describes how fast and in what direction a particle is moving at a particular point in time. To find the velocity function, take the derivative of the position function with respect to \( t \). This gives you the rate at which position changes.Starting with the equation:\[ x(t) = 3 \cos t + 4 \sin t, \]we differentiate each term:\( \frac{d}{dt}[3 \cos t] = -3 \sin t \)and\( \frac{d}{dt}[4 \sin t] = 4 \cos t \).Thus, the velocity function is:\[ v(t) = -3 \sin t + 4 \cos t. \]Using this function, you can find the particle's velocity at any point:

• At \( t=0 \), \[ v(0) = -3 \sin(0) + 4 \cos(0) = 0 + 4 = 4 \text{ feet per second}. \]
• At \( t=\pi/2 \), \[ v(\pi/2) = -3 \sin(\pi/2) + 4 \cos(\pi/2) = -3 + 0 = -3 \text{ feet per second}. \]
• At \( t=\pi \), \[ v(\pi) = -3 \sin(\pi) + 4 \cos(\pi) = 0 - 4 = -4 \text{ feet per second}. \]

Calculating velocity this way gives you a clear picture of how the particle's speed and direction change over time.

Trigonometric derivatives are a set of standard rules that help us differentiate trigonometric functions such as sine and cosine. These derivatives are incredibly useful when dealing with motion, vibrations, and waves, as many real-world phenomena are described using trigonometric functions.The main derivatives you should be familiar with are:

• The derivative of \( \cos t \) is \( -\sin t \).
• The derivative of \( \sin t \) is \( \cos t \).

In our velocity function, for instance, we found \( \frac{d}{dt}[3 \cos t] = -3 \sin t \) and \( \frac{d}{dt}[4 \sin t] = 4 \cos t \). These derivatives allowed us to transform the position function into a velocity function, a crucial step in understanding particle motion.Knowing these trigonometric derivatives is a fundamental skill in calculus.

Simple Harmonic Motion (SHM) describes the motion of particles or objects that move back and forth about a central point. This type of motion is periodic and can be mathematically represented by sine or cosine functions, similar to our particle's position function.In SHM scenarios, the velocity and acceleration of an object are directly related to its position:

• The position function of an SHM can be noted as \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase shift.
• Our position function \( x(t) = 3 \cos t + 4 \sin t \) is a variation of standard SHM, indicating the particle's oscillatory motion along the x-axis without additional angular frequency or phase shift.

Understanding how these cycles of motion behave helps in modeling real-world phenomena like sound waves, electromagnetic waves, and even the swing of a pendulum. This knowledge is foundational in physics and engineering, making SHM a key topic in both calculus and real-world applications.