To find the instantaneous velocity of an object, we need to differentiate the given position function $s(t)$ with respect to time $t$, which gives us the velocity function $v(t)$.

Step 1: Position function

The given position function is:

$s(t) = 4t^2 - 2t$

Step 2: Differentiate the position function

We differentiate $s(t)$ with respect to $t$ to find the velocity function $v(t)$:

$v(t) = \frac{d}{dt} \left( 4t^2 - 2t \right)$

Using the power rule for differentiation:

$v(t) = \frac{d}{dt}(4t^2) - \frac{d}{dt}(2t)$

$v(t) = 8t - 2$

Step 3: Calculate the instantaneous velocity at $t = 1$

Now that we have the velocity function $v(t) = 8t - 2$, substitute $t = 1$ into the equation:

$v(1) = 8(1) - 2 = 8 - 2 = 6 \, \text{ft/s}$

Conclusion:

The instantaneous velocity of the object at $t = 1$ is $6 \, \text{ft/s}$.

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Related Questions:

1. How is instantaneous velocity different from average velocity?
2. What is the velocity at $t = 2$ for this motion?
3. How would the result change if the function involved a cubic term?
4. What does it mean if the velocity is negative?
5. How is acceleration related to velocity in this context?

Tip: Instantaneous velocity is simply the derivative of the position function with respect to time, reflecting how fast the object is moving at a specific instant.