Question 1

Which of the following is the correct antiderivative for the function $f(x) = x^2$?

Accepted Answer

$\frac{1}{3}x^3 + C$

Question 2

Evaluate the definite integral $\int_1^3 x^2 dx$ using the Fundamental Theorem of Calculus.

Accepted Answer

$26$

Question 3

Given $F(x) = x^3 - 4x^2 + 5x$, what is $\int F(x) dx$?

Accepted Answer

$\frac{1}{4}x^4 - 2x^3 + \frac{5}{2}x^2 + C$

Question 4

Suppose the cost function for a firm is $C(Q) = \int_0^Q MC(q) dq + K$, where $MC(q)$ is the marginal cost and $K$ is a fixed cost. If $MC(q) = 200$ and $K = 500$, what is $C(Q)$?

Accepted Answer

$200Q + 500$

Question 5

If $Q = 2K^{0.5}L^{0.5}$, what is the partial derivative $\frac{\partial Q}{\partial K}$?

Accepted Answer

$K^{-0.5}L^{0.5}$

Question 6

A manufacturer finds that the marginal cost (MC) of producing $Q$ units is $MC = 200 + 5Q$. What is the total cost function $C(Q)$ if the fixed cost is $K = 800$?

Accepted Answer

$C(Q) = 200Q + \frac{5}{2}Q^2 + 800$

Question 7

Given the demand function for labor $D_L(p) = \frac{200}{p}$, what is the demand for labor when the price $p = 10$?

Accepted Answer

$20$

Question 8

If $Q = 2K^{0.5}L^{0.5}$, and $K = 100$, $L = 25$, what is the marginal productivity of labor $\frac{\partial Q}{\partial L}$?

Accepted Answer

$0.6325$

Question 9

Suppose the marginal cost for a product is $MC = 50 + 3Q$. What is the increase in total cost when output increases from $Q = 10$ to $Q = 20$?

Accepted Answer

$350$

Question 10

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It states that the definite integral of a function over $[a, b]$ is equal to the difference of its antiderivative evaluated at $b$ and $a$.

Question 11

A cost function is given by $C(Q) = Q^3 - 5Q^2 + 8Q + 222$. What is the marginal cost $MC(Q)$?

Accepted Answer

$3Q^2 - 10Q + 8$

Question 12

If $f(x) = x^2 + 3x$, what is $\int_0^2 f(x) dx$?

Accepted Answer

$14$

Question 13

Suppose $Q = 2K^{0.5}L^{0.5}$, $K = 400$, $L = 100$. What is the output $Q$?

Accepted Answer

$200$

Question 14

If the marginal cost $MC(Q) = 10Q$, what is the total cost function $C(Q)$ if the fixed cost is $K$?

Accepted Answer

$5Q^2 + K$

Question 15

A firm’s production function is $Q = 2K^{0.5}L^{0.5}$. If $K$ is held constant, what does $\frac{\partial Q}{\partial L}$ represent?

Accepted Answer

The marginal productivity of labor.

Question 16

Given $C(Q) = Q^3 - 5Q^2 + 8Q + 222$, what is the cost of producing the first unit ($Q=1$)?

Accepted Answer

$226$

Question 17

If $MC(Q) = 200$ for all $Q$, what does this imply about the cost function $C(Q)$?

Accepted Answer

It is a linear function of $Q$.

Question 18

Suppose the marginal cost for a product is $MC = 50 + 3Q$. What is the total cost function $C(Q)$ if the fixed cost is $K$?

Accepted Answer

$50Q + \frac{3}{2}Q^2 + K$

Question 19

If $Q = 2K^{0.5}L^{0.5}$, what is the marginal productivity of capital $\frac{\partial Q}{\partial K}$ when $K = 25$ and $L = 16$?

Accepted Answer

$0.64$

Integration and Partial Derivatives in Business Calculus

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