Total distance

To find the total distance, you need to add up all the individual distances traveled.

For example, if you traveled 10 miles, then 15 miles, and finally 20 miles, the total distance would be:

10 + 15 + 20 = 45 miles

So, the total distance traveled in this case would be 45 miles.

Problem:

Find the total distance traveled by a particle moving along a straight line with the given position function.

Solution:

Step 1: Define the Distance Function

The distance function measures the total distance traveled by the particle, regardless of direction. The distance function is defined as the integral of the absolute value of the velocity function.

```
d(t) = ∫|v(t)|dt
```

where:

- d(t) is the distance function
- v(t) is the velocity function

Step 2: Find the Velocity Function

The velocity function is the derivative of the position function.

```
v(t) = d/dt[x(t)]
```

where:

- x(t) is the position function

Step 3: Determine the Intervals of Motion

Find the critical points (zeroes) of the velocity function to determine the intervals of motion. The particle changes direction at critical points.

Step 4: Calculate the Distance on Each Interval

- For each interval where the velocity is positive, the distance is equal to the integral of the velocity function:
```
d_positive = ∫|v(t)|dt = ∫v(t)dt
```

- For each interval where the velocity is negative, the distance is equal to the integral of the negative velocity function:
```
d_negative = ∫|v(t)|dt = ∫(-v(t))dt
```

Step 5: Sum the Distance

The total distance traveled is the sum of the distances on all the intervals:

```
d(t) = Σ(d_positive + d_negative)
```

Example:

Suppose a particle moves along a straight line with the position function:

```
x(t) = t^3 - 3t^2 + 2t + 1
```

Solution:

Step 1: Define the Distance Function
```
d(t) = ∫|v(t)|dt
```

Step 2: Find the Velocity Function
```
v(t) = d/dt[t^3 - 3t^2 + 2t + 1] = 3t^2 - 6t + 2
```

Step 3: Determine the Intervals of Motion
The critical points of v(t) are t = 1 and t = 1/3. The particle changes direction at these points.

Step 4: Calculate the Distance on Each Interval
- From t = 0 to t = 1, v(t) > 0, so
```
d_positive = ∫[0,1] (3t^2 - 6t + 2)dt = [t^3 - 3t^2 + 2t]_0^1 = 0
```

- From t = 1 to t = 1/3, v(t) < 0, so
```
d_negative = ∫[1,1/3] (-3t^2 + 6t - 2)dt = [-t^3 + 3t^2 - 2t]_1^1/3 = -1/3
```

- From t = 1/3 to t = ∞, v(t) > 0, so
```
d_positive = ∫[1/3,∞] (3t^2 - 6t + 2)dt = [t^3 - 3t^2 + 2t]_1/3^∞ = ∞
```

Step 5: Sum the Distance
The total distance traveled is:

```
d(t) = d_positive + d_negative = 0 + (-1/3) + ∞ = ∞
```

Therefore, the particle travels an infinite distance.