Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
a. Graph the position function.
$f\left(t\right)=6t^3+36t^2-54t;0\le t\le 4$

Verified step by step guidance

1. Step 1: Understand the position function given by $f(t)=6t^3+36t^2-54t$ for $0\le t\le 4$. This function describes the position of an object moving along a line over time.
2. Step 2: Identify the key features of the function to graph it. These include finding the intercepts, critical points, and inflection points. Start by finding the y-intercept by evaluating $f(0)$.
3. Step 3: Find the critical points by taking the derivative of the position function to get the velocity function $f^{\prime }(t)=18t^2+72t-54$. Set $f^{\prime }(t)=0$ and solve for $t$ to find the critical points.
4. Step 4: Determine the nature of the critical points (whether they are maxima, minima, or points of inflection) by using the second derivative test. Compute the second derivative $f^{″}(t)=36t+72$ and evaluate it at the critical points.
5. Step 5: Use the information from the intercepts, critical points, and the behavior of the function as $t\to 0$ and $t\to 4$ to sketch the graph of the position function $f(t)$ over the interval $0\le t\le 4$.

Key Concepts

Position Function

The position function, denoted as s = f(t), describes the location of an object at any given time t. In this context, the function f(t) = 6t³ + 36t² - 54t represents the position of the object in feet as a function of time in seconds. Understanding this function is crucial for analyzing the object's movement along a line.

Graphing Functions

Graphing the position function involves plotting the values of f(t) against time t on a coordinate system. This visual representation helps in understanding the behavior of the object over the specified interval (0 ≤ t ≤ 4). It allows one to observe key features such as the object's position at specific times, trends in movement, and any changes in direction.

Calculus and Motion

Calculus plays a vital role in analyzing motion through concepts like velocity and acceleration, which are derived from the position function. The velocity is the first derivative of the position function, indicating how fast the position changes over time, while acceleration is the second derivative, showing how the velocity changes. These concepts are essential for understanding the dynamics of the object's movement.