Business Calculus Quiz: Riemann Sums

The marginal cost function for producing a certain electronic component, $C'(q)$, is known to be a continuous and strictly decreasing function for $q > 0$. Let $A$ be the actual cost of increasing production from $q=50$ to $q=100$. Let $R_5$ be the right Riemann sum approximation of $\int_{50}^{100} C'(q) dq$ with 5 subintervals of equal width. Which of the following statements correctly compares $R_5$ to $A$?

The marginal cost function for producing $x$ widgets is $C'(x) = 15 + 0.02x^2$ dollars per widget. If the current production level changes from 10 to 20 widgets, and you must choose between a left Riemann sum with 5 subintervals or a right Riemann sum with 5 subintervals to estimate the change in total cost, which approach gives a more conservative (lower) estimate?

The marginal profit function $P'(x)$ for a product is positive and strictly increasing for $x \ge 0$. To estimate the total profit gained from selling the 101st unit through the 200th unit, which of the following Riemann sum approximations would provide the most accurate, guaranteed underestimate?

The rate of appreciation of an investment is given by a differentiable function $V'(t)$, where $t$ is time in years. It is known that the rate of appreciation is slowing down, meaning the graph of $V'(t)$ is concave down for $t \ge 0$. An approximation for the total appreciation over the first 5 years, $\int_0^5 V'(t) dt$, is calculated using the Trapezoidal Rule with $n=10$ subintervals. How does this approximation, $T_{10}$, compare to the actual appreciation, $A$?

The rate of oil consumption by a factory is modeled by $f(t) = 50 + 10\sin(\pi t/12)$ gallons per hour, where $t$ is hours after midnight. If a left Riemann sum with 3 subintervals is used to estimate oil consumption from $t = 0$ to $t = 6$ hours, which expression represents this approximation?

A chemical spill is causing a pond to be contaminated. The rate at which the pollutant enters the pond is given by $P(t) = 15e^{0.1t}$ liters per hour, where $t$ is the number of hours since the spill began. What is the approximate total volume of pollutant that enters the pond during the second hour (from $t=1$ to $t=2$), using a midpoint Riemann sum with 4 subintervals?

The marginal revenue from selling $x$ units of a product is $R'(x)$ dollars per unit. The total revenue from selling units 21 through 40 is approximated using a left Riemann sum with $n=4$ subintervals. What is the economic interpretation of the second term in this sum?

Let $C'(t)$ be the rate of growth of a company's profit, in millions of dollars per year. The function $C'(t)$ is known to be strictly increasing and concave up on the interval $[0, 5]$. Let $I = \int_0^5 C'(t) dt$ be the actual total increase in profit over 5 years. Let $L_n$, $R_n$, and $T_n$ be the left Riemann sum, right Riemann sum, and trapezoidal sum approximations of $I$ with $n$ subintervals, respectively. Which of the following inequalities correctly orders these values?

A financial analyst models the total accumulated value of an asset over 3 years using a definite integral. The following right Riemann sum is used to approximate this value: $\sum_{k=1}^{60} 500(1.02)^{k/20} \cdot \frac{1}{20}$ Which of the following definite integrals does this sum approximate?

The marginal cost $C'(q)$ for producing $q$ items is a continuous function that is strictly increasing for $q \ge 0$. Let $\Delta C$ be the true change in cost from $q=10$ to $q=50$. Let $L$ be the left Riemann sum approximation and $R$ be the right Riemann sum approximation for $\int_{10}^{50} C'(q) \,dq$, both using $n=10$ subintervals of equal width. Which of the following correctly orders these three values?

A storage tank is leaking oil at a rate of $r(t) = 15e^{-0.1t}$ liters per hour, where $t$ is the number of hours since the leak began. Use a midpoint Riemann sum with four subintervals of equal width to approximate the total amount of oil that leaks out during the first 8 hours.

The expression $S = \sum_{k=1}^{40} \left(1 + \frac{2k}{40}\right)^3 \cdot \frac{1}{20}$ is a right Riemann sum approximation for $\int_a^b f(x) \,dx$. Which of the following are the correct function $f(x)$ and interval $[a, b]$?

The rate of return on an investment, $r(t)$, is a continuous function that is known to be both decreasing and concave up over the interval $0 \le t \le 5$ years. Four estimates are calculated for the total return $\int_0^5 r(t) \,dt$ using $n=10$ subintervals: Left Riemann Sum ($L$), Right Riemann Sum ($R$), Midpoint Riemann Sum ($M$), and Trapezoidal Rule ($T$). Which of the following correctly orders these four estimates from smallest to largest?

The rate of visitors, $v(t)$, entering a theme park is measured in hundreds of people per hour, where $t$ is hours after opening at 9 a.m. Park management observes that the rate of entry is always increasing between 9 a.m. ($t=0$) and 1 p.m. ($t=4$). A left Riemann sum, $L_4 = \sum_{i=0}^{3} v(t_i) \Delta t$, is calculated with $\Delta t=1$ to estimate the total number of visitors who entered during this period. Which statement best describes this estimate?

A water tank is being filled at a variable rate described by $r(t) = 8 + 2\cos(\pi t/6)$ gallons per minute, where $t$ is time in minutes. The difference between the right Riemann sum and left Riemann sum approximations (right minus left) using 6 subintervals over $[0, 12]$ is:

The velocity of a particle is $v(t) = t^2 - 4t + 5$ ft/sec. When comparing left and right Riemann sums with 4 equal subintervals to approximate displacement from $t = 0$ to $t = 4$ seconds, which statement is true?