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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