Business Calculus Quiz: Higher Order Derivatives

Let $P(t)$ represent the profit of a company, in millions of dollars, $t$ months after a new product launch. At month $t=6$, an analyst determines that $P'(6) > 0$ and $P''(6) < 0$. Which of the following statements best describes the company's profit at this time?

The total cost, in dollars, to produce $x$ units of a certain electronic component is given by $C(x) = 0.002x^3 - 0.9x^2 + 150x + 5000$. The marginal cost is the rate of change of the total cost. For what range of production levels is the marginal cost increasing?

The number of new users for a mobile app $t$ weeks after a promotional campaign begins is modeled by $N(t) = 500t e^{-0.2t}$ for $t \ge 0$. The rate of growth in new users initially accelerates and then begins to slow down. At what time $t$ does this rate of growth start to slow down?

The revenue $R(t)$ of a company is changing over time $t$ (in years). The rate of change of the rate of change of revenue is constant, with $R''(t) = -8$ (in millions of dollars per year squared). At time $t=3$ years, the revenue was growing at a rate of $12$ million dollars per year. At what time $t$ does the company's revenue reach its peak?

The total cost to produce $q$ units of a product is given by $C(q) = 2q^3 + bq^2 + 192q + 4000$, where $b$ is a constant. The marginal cost is minimized at a production level of $q=12$ units. What is the value of the constant $b$?

For the function $f(x) = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x$, if $f^{(4)}(x)$ represents the fourth derivative, what can be concluded about $f^{(5)}(x)$ and $f^{(6)}(x)$?

The temperature of a chemical reaction follows $T(t) = 25 + 40e^{-0.2t}$ degrees Celsius, where $t$ is time in minutes. If safety protocols require monitoring when the rate of temperature change is itself changing most rapidly, at what time should this occur?

Let $C(q)$ be a company's cost function for producing a quantity $q$ of a product. At a production level of $q_0 = 500$ units, it is known that $C''(500) = 0$ and $C'''(500) < 0$. What does this imply about the marginal cost, $C'(q)$, at this production level?

The productivity $P(x)$ of a company, in thousands of units per month, is given by the function $P(x) = -x^3 + 30x^2 + 10x + 500$, where $x$ is the amount spent on employee training in thousands of dollars. The company experiences diminishing returns when the rate of change of productivity begins to decrease. At what level of training expenditure $x$ do diminishing returns set in?

A marketing director states, "Our market share is still increasing, but the gains we're making each month are smaller than the gains from the month before." If $M(t)$ is the market share at time $t$ in months, which pair of conditions must be true for the current time $t_0$?

The efficiency $E(h)$ of an assembly line worker, measured in units assembled per hour, is modeled by $E(h) = -0.02h^3 + 0.9h^2 + 12h$ after $h$ hours of training, for $0 \le h \le 30$. The worker's rate of improvement is the rate of change of their efficiency. After an initial period, this rate of improvement begins to decline. At how many hours of training, $h$, does this decline begin?

The profit $P(x)$ from producing $x$ items is such that the marginal profit at a production level of $x=1000$ is zero. The second derivative of the profit function is given by $P''(x) = 0.04x - 30$. Based on this information, what can be concluded about the profit at the production level of $x=1000$?

The total cost $C(q)$ to produce $q$ items has a marginal cost $C'(q)$ that is observed to be decreasing when production is less than 200 units ($q < 200$) and increasing when production is greater than 200 units ($q > 200$). Assuming marginal cost is always positive, which statement accurately describes the total cost function at a production level of $q=250$?

A company's revenue function is $R(t) = 100\ln(t+1) + 50t$ thousand dollars, where $t$ is time in years since startup. What does $R''(2)$ represent, and what is its value?

An economist models consumer demand as $D(p) = \frac{1000}{p^2 + 1}$, where $D$ is quantity demanded and $p$ is price. The second derivative $D''(p)$ indicates how the rate of change of demand responds to price changes. For what price does $D''(p) = 0$?

A population model is given by $N(t) = 1000e^{0.05t}$, where $N(t)$ is the population after $t$ years. If $N'(t)$ represents the growth rate and $N''(t)$ represents the acceleration of growth, what is the relationship between these quantities?

For a production function $Q(L) = 20L^{2/3}$, where $Q$ is output and $L$ is labor input, economists define the marginal product as $Q'(L)$ and the rate of change of marginal product as $Q''(L)$. If $Q''(L) = -\frac{80}{9}L^{-5/3}$, what can be concluded about the production process?

The velocity of a particle is given by $v(t) = 6t^2 - 24t + 18$ meters per second. At $t = 1$ second, what is the rate of change of the acceleration?