Business Calculus Quiz: Net Change And Total Change

The marginal cost for a company to produce $q$ units of a product is given by $C'(q) = 0.03q^2 - 1.2q + 15$ dollars per unit. What is the total increase in cost if the company raises its production level from $q=20$ to $q=40$ units?

A new technology startup has a continuous revenue stream with a rate of flow modeled by $R'(t) = 150e^{-0.1t}$ thousands of dollars per month, where $t$ is in months. The project had initial start-up costs, resulting in a total revenue of -$50$ thousand (a loss) at time $t=0$. What is the total accumulated revenue, in thousands of dollars, after the first 10 months?

$V(t)$ represents the value of a commercial property in thousands of dollars, $t$ years after its purchase. Financial analysts determine that $\int_5^{10} V'(t) dt = -150$. Which of the following statements is the correct interpretation of this result?

A company's cash reserves change based on its rate of income, $I'(t) = 25e^{-0.05t}$, and its rate of expenditure, $E'(t) = 10 + 0.2t$, both in thousands of dollars per month. What is the net change in the company's cash reserves over the first six months (from $t=0$ to $t=6$)?

A company's marginal profit function is $P'(q)$ dollars per unit, where $q$ is the number of units produced. Which of the following is the best interpretation of the statement $\int_{500}^{600} P'(q) dq = 4000$?

The population of subscribers to a streaming service is growing at a rate of $N'(t) = \frac{1000}{\sqrt{t+1}}$ subscribers per month, where $t$ is the number of months since the service launched. Three months after launch ($t=3$), the service had 2500 subscribers. How many subscribers, to the nearest whole number, did the service have after one year ($t=12$)?

The rate of change of items in a warehouse is given by $I'(t) = 3t^2 - 18t + 15$ items per hour for an 8-hour workday, $0 \le t \le 8$. A positive rate indicates items are being added, and a negative rate indicates items are being removed. What is the total number of items moved (either in or out of the warehouse) during the first 4 hours?

A company's rate of profit is modeled by $P'(t) = 1.2 - 0.6t$ million dollars per year, where $t$ is the number of years from the company's launch. The profit rate is positive for the first two years and then becomes negative. What is the net change in the company's profit during the first 4 years of operation?

A manufacturer's inventory of a specific smartphone model changes based on a constant production rate of 1000 units per week and a sales rate of $S'(t) = 20t^2 - 120t + 1180$ units per week, for $0 \le t \le 6$. If the initial inventory at $t=0$ was 500 units, what is the inventory level at the end of week 6?

The value of a piece of industrial equipment, $V(t)$, is decreasing at a rate given by $V'(t) = -1000(t+2)^{-2}$ dollars per year, where $t$ is the number of years since its purchase. What is the total loss in value of the equipment during its third year of operation?

Let $A(t)$ be the amount of oil, in barrels, in a storage tank at time $t$ in hours. Oil flows into the tank at a rate of $I(t)$ barrels per hour and flows out at a rate of $O(t)$ barrels per hour. Which expression represents the net change in the amount of oil in the tank from $t=2$ to $t=5$?

The rate of production at a factory is given by $P(t)$ units per hour. The selling price for each unit produced at time $t$ is given by $S(t)$ dollars per unit. Which of the following expressions represents the total revenue generated by the factory from time $t=a$ to $t=b$?

The rate of change of profit for Company A is $P_A'(t) = 10t + 5$ and for Company B is $P_B'(t) = t^2 + 8$, where profit is in thousands of dollars and $t$ is the number of years from now ($t \ge 0$). Over the time interval from $t=0$ to $t=6$, by how much does the net change in profit for Company A exceed the net change in profit for Company B?

The net rate of change of the number of employees at a corporation is modeled by $E'(t) = 20 - 4t$ people per month, where $t$ is the number of months since the start of the year. When $E'(t) > 0$, there is net hiring, and when $E'(t) < 0$, net departures exceed hires. What was the total number of employees who were either hired or who departed during the first 8 months of the year ($0 \le t \le 8$)?

The rate of change of inventory in a warehouse is modeled by the function $I'(t)$ in items per day, where $t$ is the number of days since the beginning of the month. The function is positive when new stock arrives and negative when orders are shipped out. What is the correct business interpretation of the quantity $\int_{0}^{30} |I'(t)| \,dt$?

The rate at which water is draining from a tank is given by $R(t)$ in liters per minute, where $R(t) > 0$ for all $t \ge 0$. Which statement provides the most accurate interpretation of the value calculated from $\int_{5}^{10} R(t) \,dt$?

A company's rate of revenue is $R'(t)$ and its rate of cost is $C'(t)$, both in thousands of dollars per quarter, where $t$ is the number of quarters since the beginning of the fiscal year. The company's profit at the beginning of the year ($t=0$) was $P_0$ thousand dollars. Which expression represents the company's total profit, in thousands of dollars, at the end of the second quarter ($t=2$)?