The position of a hummingbird flying along a straight line in t seconds is given by s(t) = 3t^3 − 7t meters. leave your answer in a simplified form that may include square
roots of numbers.

determine the acceleration of the bird at t=1 sec

The position of the hummingbird as a function of time is given by:

\[
s(t) = 3t^3 - 7t
\]

### Step 1: Find the velocity function
The velocity \( v(t) \) is the derivative of the position function with respect to time \( t \):

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

Differentiating:

\[
v(t) = 9t^2 - 7
\]

### Step 2: Find the acceleration function
The acceleration \( a(t) \) is the derivative of the velocity function:

\[
a(t) = \frac{d}{dt} [9t^2 - 7]
\]

Differentiating:

\[
a(t) = 18t
\]

### Step 3: Evaluate the acceleration at \( t = 1 \) second
Now, substitute \( t = 1 \) into the acceleration function:

\[
a(1) = 18(1) = 18
\]

### Final Answer:
The acceleration of the hummingbird at \( t = 1 \) second is:

\[
a(1) = 18 \, \text{m/s}^2
\]

Would you like any further details or clarification on this? 

Here are 5 related questions:
1. How do you find the velocity from the acceleration function?
2. What would the position be at \( t = 1 \) second?
3. How does acceleration affect the velocity of the bird?
4. Can you find the velocity at \( t = 1 \) second for this problem?
5. What is the physical significance of a negative velocity or acceleration?

**Tip:** When differentiating polynomial functions, pay attention to the power rule \( \frac{d}{dt}(t^n) = n t^{n-1} \).

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

Find the acceleration of a hummingbird at t = 1 second using calculus. The problem involves differentiating the position function s(t) = 3t^3 − 7t to find the velocity and acceleration functions. Key concepts include derivatives and the power rule. Suitable for Grades 11-12 or Calculus I.

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