Use the definition to find an expression for the instantaneous velocity of an object moving with rectilinear motion according to the given function relating s​ (in ft) and t​ (in s). Then calculate the instantaneous velocity for the given value of t.
sequals4 t squared minus 2 t​; tequals1

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 \( 6 \, \text{ft/s} \).

Would you like more details or have any questions?

### 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.

Use the definition to find an expression for the instantaneous velocity of an object moving with rectilinear motion according to the given function relating s​ (in ft) and t​ (in s). Then calculate the instantaneous velocity for the given value of t. s equals 4t^2 - 2t; t equals 1

Learn how to calculate the instantaneous velocity of an object in rectilinear motion using the function s(t) = 4t^2 - 2t. This detailed guide walks you through the process of differentiating the position function and calculating velocity at a given time t = 1. Suitable for high school students in Grades 10-12.

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