Question 1

Suppose the demand function for a product is given by $$q = 240 - 2p$$, where $$q is $$the quantity demanded and $$p is $$the price. What is the price elasticity of demand when $$p = 100$$?

Accepted Answer

$$-0.71$$

Question 2

A manufacturer’s marginal cost function is $$C'(q) = 3q^2$ - 6q + 400. If $$the total cost of producing the first 2 units is $$900$$, what is the total cost of producing the first 5 units?

Accepted Answer

$$1,587$$

Question 3

If the supply function is $$x^2 - 2x $$rac{dy}{dt} - $$p^2 = 31$, and at p = 9$ the price is increasing at 0.20$ per week, how fast is the supply changing?

Accepted Answer

$$206$$ units/week

Question 4

A manager determines that daily demand for a dish is $$q = 300 - p^2$, where p$ is the price. For what values of p$ is the demand elastic?

Accepted Answer

$$0 \u003c p \u003c 10$$

Question 5

Given the total cost function $$C(q) = 0.8q^2$ + 3q + 40$$, at what level of production does marginal cost equal average cost?

Accepted Answer

$$q = 10$$

Question 6

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$, where $$F is $$any antiderivative of $$f$$

Question 7

A retailer receives a shipment of $$10,000 kg of $$rice to be used over $$5$$ months at a constant rate of $$2,000 kg/$$month. If storage costs are $$0.01$$ per kg per month, how much will the retailer pay in storage charges over the $$5$$ months?

Accepted Answer

$$250$$

Question 8

Suppose the consumer demand function for a commodity is $$D(q) = 4(25 - q^2$)$$ dollars per unit. What is the total amount of money consumers are willing to spend to get $$3$$ units?

Accepted Answer

$$264$$

Question 9

If the marginal cost for producing products is $$3(q - 4)^2$ dollars per unit, how much would the production cost increase if production is increased from 6$ to 10$ units?

Accepted Answer

$$208$$

Question 10

Jane is deciding between two investments. Investment 1 costs $$1,000$$ and generates $$3,000e$$^{0.02t}$$ per year for 5$ years. Investment 2 costs 4$$,000$$ and generates 4$$,$$000e^{0.015t}$$ per year for $$5$$ years. If the interest rate is $$5\%$$ compounded continuously, which investment is better?

Accepted Answer

Investment 2

Question 11

The area under the marginal cost curve $$C'(x) = 180$$ from $$x = 5 to$$ $$x = 10$$ represents:

Accepted Answer

The increase in total cost as production increases from $$5 to$$ $$10$$ units

Question 12

If the maintenance cost rate is $$M'(x) = 90x^2$ + 5,000$$, what is the total maintenance cost from year $$3 to $$year $$7$$?

Accepted Answer

$$29,480$$

Question 13

The consumers’ surplus (CS) at equilibrium price $$p_0 is $$given by:

Accepted Answer

$$CS = \$$int_0$$^{q_0$} [f(q) - p_0$] dq$$

Question 14

Suppose the profit function is $$P(q) = -1.6q^2$ + 19.2q + 40. At $$what production level is profit maximized?

Accepted Answer

$$q = 6$$

Question 15

If $$f(x) = 3x^2$ - 2e^x + \frac{3}{x}$$, what is the definite integral $$\$$int_1$$^3 f(x) dx$$?

Accepted Answer

$$-5.44$$

Question 16

If the equilibrium price and quantity in a market are $$p_0$$ and $$q_0$$, what does the intersection of the supply and demand curves represent?

Accepted Answer

The point of market equilibrium

Question 17

If $$\int_a^b f(x) dx = F(b) - F(a)$$, what is $$\$$int_2$$^5 3 dx$$?

Accepted Answer

$$9$$

Question 18

Suppose a manufacturer’s marginal cost function is $$C'(q) = 0.6q + 2. $$How much would it cost to increase production from $$80 to$$ $$100$$ units per week?

Accepted Answer

$$11,120$$

Question 19

If the demand function is $$q = 240 - 2p$$, at what price is the elasticity of demand $$-1$$?

Accepted Answer

$$p = 60$$

Question 20

If $$\$$int_0$$^a f(x) dx = 0$$ for all $$a$$, what can you conclude about $$f(x)$$?

Accepted Answer

$$f(x) = 0$$ for all $$x$$

Business Applications of Derivatives and Integrals – Calculus Study Notes

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