To find the total distance traveled by the particle along the x-axis from $t = 0$ to $t = 4$, we need to calculate the particle's position function, find its velocity, and determine where the particle changes direction.

Step 1: Position and Velocity Functions

The position of the particle is given by:

$x(t) = t^4 - 18t^2 - 17.$

To find the velocity function, we differentiate $x(t)$ with respect to $t$:

$v(t) = \frac{d}{dt} (t^4 - 18t^2 - 17) = 4t^3 - 36t.$

Step 2: Find Critical Points (Where the Velocity is Zero)

We set the velocity function to zero to find the critical points:

$4t^3 - 36t = 0.$

Factoring out the common term:

$4t(t^2 - 9) = 0.$

This gives us:

$t = 0, \quad t^2 - 9 = 0 \implies t = 3 \text{ and } t = -3.$

Since $t \geq 0$, we only consider $t = 0$ and $t = 3$.

Step 3: Evaluate Position at Critical Points and Endpoints

Next, we evaluate the position at the critical points and the endpoints $t = 0$ and $t = 4$:

1. At $t = 0$:

   $x(0) = 0^4 - 18(0)^2 - 17 = -17.$

2. At $t = 3$:

   $x(3) = 3^4 - 18(3)^2 - 17 = 81 - 162 - 17 = -98.$

3. At $t = 4$:

   $x(4) = 4^4 - 18(4)^2 - 17 = 256 - 288 - 17 = -49.$

Step 4: Calculate Total Distance Traveled

To find the total distance traveled, we add the absolute values of the changes in position between these points:

1. Distance from $t = 0$ to $t = 3$:

   $|x(3) - x(0)| = |-98 - (-17)| = |-98 + 17| = 81.$

2. Distance from $t = 3$ to $t = 4$:

   $|x(4) - x(3)| = |-49 - (-98)| = |-49 + 98| = 49.$

Thus, the total distance traveled by the particle is:

$\text{Total distance} = 81 + 49 = 130.$

Final Answer

$\boxed{130}$

This is the total distance traveled by the particle from $t = 0$ to $t = 4$.

Would you like more details or have any questions?

Here are five related questions that might interest you:

1. How do you find the speed of the particle at a given time?
2. What is the significance of velocity turning points in particle motion?
3. How do you determine the direction of motion from the velocity function?
4. What happens when the particle’s velocity is negative?
5. How would the motion change if the position function had a different form?

Tip: To understand particle motion, always analyze both the velocity and the position functions to find when and where the direction changes.