Estimate the instantaneous rate of change of  h(t)= 3/(t+3) at the point t=-2

To find the instantaneous rate of change of \( h(t) = \frac{3}{t+3} \) at the point \( t = -2 \), we need to calculate the derivative of \( h(t) \) with respect to \( t \) and evaluate it at \( t = -2 \).

1. **Define the function:**
   \[
   h(t) = \frac{3}{t + 3}
   \]

2. **Differentiate \( h(t) \) with respect to \( t \):**
   Using the quotient rule, where \( h(t) = \frac{f(t)}{g(t)} \) with \( f(t) = 3 \) and \( g(t) = t + 3 \), the derivative \( h'(t) \) is:
   \[
   h'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{[g(t)]^2}
   \]
   Since \( f(t) = 3 \) (a constant), \( f'(t) = 0 \), and \( g(t) = t + 3 \) so \( g'(t) = 1 \), we have:
   \[
   h'(t) = \frac{0 \cdot (t + 3) - 3 \cdot 1}{(t + 3)^2} = \frac{-3}{(t + 3)^2}
   \]

3. **Evaluate \( h'(t) \) at \( t = -2 \):**
   Substitute \( t = -2 \) into the derivative:
   \[
   h'(-2) = \frac{-3}{(-2 + 3)^2} = \frac{-3}{1} = -3
   \]

Thus, the instantaneous rate of change of \( h(t) \) at \( t = -2 \) is \( \mathbf{-3} \).

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

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Here are some related questions you might explore:
1. How would you interpret the meaning of this rate of change?
2. What is the instantaneous rate of change at other points for \( h(t) \)?
3. How does the rate of change of \( h(t) \) compare to a linear function?
4. Could you use limits to verify the instantaneous rate of change?
5. How would you find the instantaneous rate of change at any arbitrary point \( t = a \)?

**Tip:** For rational functions like \( \frac{3}{t+3} \), the quotient rule can streamline finding the derivative efficiently!

Estimate the instantaneous rate of change of h(t) = 3/(t+3) at the point t = -2

Learn how to find the instantaneous rate of change of the function h(t) = 3/(t+3) at t = -2 using calculus. This solution demonstrates the application of the quotient rule for derivatives, providing a step-by-step approach for calculating the derivative and evaluating it at a specific point. Suitable for high school students in Grades 11-12.

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