The position of a hummingbird flying along a straight line in t seconds is given by s(t) = 3t3 − 7t meters.
Determine the velocity of the bird at t =1 sec.

The position of the hummingbird is given by the function \( s(t) = 3t^3 - 7t \) (in meters), where \( t \) is time in seconds.

To find the velocity of the bird, we need to take the derivative of the position function \( s(t) \) with respect to time \( t \). The derivative of the position function with respect to time is the velocity function \( v(t) \).

### Step 1: Find the derivative of \( s(t) \).
The derivative of \( s(t) = 3t^3 - 7t \) is:

\[
v(t) = \frac{d}{dt}(3t^3 - 7t)
\]

Using basic differentiation rules:
- The derivative of \( 3t^3 \) is \( 9t^2 \),
- The derivative of \( -7t \) is \( -7 \).

Thus, the velocity function is:
\[
v(t) = 9t^2 - 7
\]

### Step 2: Evaluate the velocity at \( t = 1 \) second.
Substitute \( t = 1 \) into the velocity function:

\[
v(1) = 9(1)^2 - 7 = 9 - 7 = 2 \, \text{m/s}
\]

### Conclusion:
The velocity of the hummingbird at \( t = 1 \) second is \( 2 \, \text{m/s} \).

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:
1. What is the velocity at \( t = 2 \) seconds?
2. How would you find the acceleration of the bird at any time \( t \)?
3. What does a negative velocity imply about the bird's motion?
4. Can the bird's velocity ever be zero? If so, at what time(s)?
5. How would the bird's position change if the velocity function were constant?

**Tip:** The velocity is the rate of change of position, and acceleration is the rate of change of velocity.

The position of a hummingbird flying along a straight line in t seconds is given by s(t) = 3t^3 − 7t meters. Determine the velocity of the bird at t =1 sec.

Learn how to determine the velocity of a hummingbird flying along a straight line by differentiating its position function, s(t) = 3t^3 − 7t. This calculus problem focuses on applying differentiation rules to find the velocity function, then evaluating it at t = 1 second to get the velocity of 2 m/s. Suitable for high school students in Grades 11-12.

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