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!