The document contains several applied calculus problems. Here's a summary and breakdown of each question:

1. Compute derivatives using the first principle method:

   • $y = 2x^3 - 3x + 5$
   • $f(x) = \sin(3x - 2)$

2. Calculate the derivative of $y = \frac{x+1}{3 - \frac{x}{x^2 - 1}}$ with respect to $x$.

3. Population growth problem:

   • A population grows according to $p(t) = 250 \left( 1 + \frac{2t}{30 + t^2} \right)$.
   • Find the growth rate ($\frac{dp}{dt}$) at $t = 2$.

4. Find the equation of the tangent line:

   • Curve: $x^2 - xy + y^2 = 3$.
   • Point: $(1, 1)$.

5. Open box maximum volume problem:

   • Material dimensions: Square piece, 24 inches on each side.
   • Volume function $V(x)$ to be derived based on the cut-out square size $x$.
   • Use calculus to find the maximum volume.

Would you like detailed solutions to any of these problems? Let me know which one to start with!

Here are some related questions to think about:

1. What does the "first principle method" reveal about the nature of derivatives?
2. How is the chain rule applied in trigonometric functions like $\sin(3x - 2)$?
3. How does differentiation help analyze growth in real-life scenarios, such as bacterial populations?
4. Why is finding the tangent line important in understanding a curve?
5. What are the steps to solve optimization problems in calculus?

Tip: For optimization problems, always confirm that your solution corresponds to a maximum or minimum using the second derivative test.