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Let's work through the problems shown in the image:

### Problem 3 (Cobb-Douglas production function)
The Cobb-Douglas production function is given as:

\[
Q = 4K^{\frac{1}{2}} L^{\frac{1}{2}}
\]

Inputs \( K \) and \( L \) vary with time \( t \) and interest rate \( r \), via the expressions:

\[
K(t, r) = \frac{10^2}{r} \quad \text{and} \quad L(t, r) = 6t^2 + 250r
\]

You are asked to find the rate of change of \( Q \) with respect to \( t \) when \( t = 10 \) and \( r = 0.1 \).

Steps:
1. First, substitute the values for \( K(t, r) \) and \( L(t, r) \) into the Cobb-Douglas production function.
2. Compute \( K(10, 0.1) \) and \( L(10, 0.1) \).
3. Use the product and chain rule to differentiate \( Q \) with respect to \( t \).

I'll calculate this after summarizing the rest of the problems.

---

### Problem 4 (Indefinite Integral)

The integral is:

\[
\int \left( 4x^3 - 5x + 5 + \frac{1}{x} + \frac{3}{x^2} \right) dx
\]

You can integrate each term separately:

- \( \int 4x^3 \, dx = x^4 \)
- \( \int -5x \, dx = -\frac{5x^2}{2} \)
- \( \int 5 \, dx = 5x \)
- \( \int \frac{1}{x} \, dx = \ln|x| \)
- \( \int \frac{3}{x^2} \, dx = -\frac{3}{x} \)

Summing these up, the result is:

\[
x^4 - \frac{5x^2}{2} + 5x + \ln|x| - \frac{3}{x} + C
\]

This matches the third option (answer choice 3).

---

### Problem 5 (Marginal Rate of Substitution)

Given the utility function:

\[
U(x, y) = \ln(x) + \ln(y)
\]

The Marginal Rate of Substitution (MRS) is the negative ratio of the marginal utilities of \( x \) and \( y \):

\[
MRS = -\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}}
\]

Calculating the partial derivatives:

- \( \frac{\partial U}{\partial x} = \frac{1}{x} \)
- \( \frac{\partial U}{\partial y} = \frac{1}{y} \)

So:

\[
MRS = -\frac{1/x}{1/y} = -\frac{y}{x}
\]

The correct answer is \( \frac{x}{y} \) (answer choice 4).

---

Now I'll calculate the rate of change for Problem 3.For Problem 3, the rate of change of \( Q \) with respect to \( t \) when \( t = 10 \) and \( r = 0.1 \) is approximately \( 3035.79 \).

Since this value is closest to **3000**, the correct answer is **option 2**.

---

Would you like further details on any part of the solution? Here are some additional related questions for you to explore:

1. How do you apply the chain rule to more complex functions?
2. Can you explain the steps to differentiate a production function?
3. How does the Cobb-Douglas production function relate to economic theory?
4. How do you verify indefinite integrals using differentiation?
5. What are the economic implications of the marginal rate of substitution (MRS)?

**Tip:** Always double-check the differentiation and integration steps to avoid small errors in handling constants or powers!

Consider the Cobb-Douglas production function Q = 4K^(1/2) L^(1/2). Inputs K and L vary with time t and interest rate r, via the expressions K(t, r) = 102/r and L(t, r) = 6t^2 + 250r. Find the rate of change of Q with respect to t when t=10 and r=0.1.

Solve a calculus-based economics problem by finding the rate of change of the Cobb-Douglas production function Q = 4K^(1/2) L^(1/2), where K and L vary with time and interest rate. Detailed step-by-step solution using the product and chain rule, with applications to economic theory.

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