Exercises: Average rate of change and marginal analysis

Calculate the average rate of change of $f(x) = x^{2}$ in the interval $[1, 3]$.

Calculate the average rate of change of $f(x) = 2x + 1$ in the intervals $[0, 4]$ and $[-2, 5]$. What do you observe? Why is this?

The average rate of change of the population of a town between 2010 and 2020 was $1,200$ inhabitants per year. Explain in your own words what this datum means and calculate how much the population grew in that decade.

The figure shows the distance travelled by a cyclist as a function of time.

1. a) Read from the graph the distance travelled in the first hour and calculate their average speed during that hour.

1. b) Calculate their average speed between hour $1$ and hour $3$.

1. c) In which section were they faster? Relate this to the steepness of the curve.

Calculate the average rate of change of $f(x) = x^{2} - 2x$ in the interval $[0, 2]$ and explain why the result is $0$ even though the function is not constant in that interval.

Write, using the limit definition of derivative, the expression for $f'(2)$ for the function $f(x) = x^{2} + 1$ (it is not necessary to evaluate the limit).

Calculate, applying the definition with limits, the derivative of $f(x) = 3x - 2$ at $x = 1$. Does the result depend on the point chosen? Why?

Without doing any calculations, answer with reasons:

1. a) If $\text{TVM}[a, b] < 0$, has the function increased or decreased overall between $a$ and $b$?

1. b) Can the average rate of change in an interval be negative and, at the same time, have $f(b) > f(a)$? Justify your answer.

Aoife needs to calculate the average rate of change of the profits of a bookstore between month $2$ and month $6$ after its opening. The function that models the accumulated profits (in hundreds of euros) is $f(x) = x^2 + 2x$.

On her exam, Aoife writes:

and concludes that "profit grew at an average rate of €$2000$ per month".

1. a) Identify the mistake made by Aoife and explain in your own words what the denominator conceptually represents in the formula for $\text{ARC}[a, b]$ (or with step $h$).
2. b) Correct the calculation and find the real value of the average rate of change over that period in euros/month.
3. c) Geometrically interpret the correct result in relation to the secant line to the graph of $f(x)$ passing through the points with abscissa $x = 2$ and $x = 6$.

A bicycle courier company in Cork estimates that the daily total cost (in euros) of making $q$ deliveries follows the model $C(q) = 2q^2 + kq + 50$, where $k$ is a unit variable cost due to equipment maintenance. The finance department knows that between $5$ and $10$ daily deliveries, the average rate of change of the cost is €$38$ per delivery.

1. a) Set up the equation for $\text{ARC}[5, 10]$ in terms of the parameter $k$ and deduce its exact value.
2. b) Using the obtained value of $k$, find the marginal cost function $C'(q)$ and calculate the marginal cost for $q = 10$ deliveries.
3. c) Compare the value of $C'(10)$ with $\text{ARC}[5, 10]$ and reason why the approximate cost of making delivery number $11$ is greater than the average obtained on the interval $[5, 10]$.

The figure shows the graph of a function and the secant line passing through the points $A(1, 2)$ and $B(4, 8)$.

1. a) Read the changes $\Delta x$ and $\Delta y$ from the figure and calculate $\text{ARC}[1, 4]$.

1. b) What is the relationship between this rate and the gradient of the secant line drawn?

1. c) Write the equation of the secant line passing through $A$ and $B$.

The figure illustrates how the secant lines passing through a fixed point $P$ on the graph of $f(x) = x^{2} + 1$ approach the tangent line at $P$ as the second point, $Q$, gets closer to $P$.

Let $f(x) = x^{2} + 1$ and $P = (2, 5)$. Remember that the average rate of change on an interval $[a, b]$ is

and it coincides with the slope of the secant line passing through the points with x-coordinates $a$ and $b$.

1. a) Complete the following table of $\text{TVM}[2, 2 + h]$ (calculate $f(2+h)$ and then the rate):

1. b) What number do the rates approach as $h$ gets smaller and smaller? What does that number represent with respect to the tangent line at $P$?

1. c) Write the equation of the tangent line at $P$.

The graph shows the temperature in a city over the course of a day.

1. a) Read the approximate temperature values at $6$ and $10$ hours and calculate the average rate of change in that interval, in °C/h.

1. b) Do the same for the interval from $18$ to $22$ hours.

1. c) Compare the two results: what do their signs indicate about what was happening to the temperature in each time slot?

A company that manufactures backpacks has a total cost function $C(q) = 0.02q^{2} + 10q + 800$, in euros, where $q$ is the number of backpacks produced per month.

1. a) Calculate the actual cost of producing the $61^{\text{st}}$ backpack: $C(61) - C(60)$.

1. b) Calculate the marginal cost $C'(60)$ as an approximation of the cost of that additional unit.

1. c) Compare both values and explain why economists use the derivative as "marginal cost".

A skydiver jumps from a balloon and, before opening the parachute, the distance fallen approximately follows the model $e(t) = 2t^{2} + t$, in metres, with $t$ in seconds.

1. a) Calculate their average speed between seconds $1$ and $3$.

1. b) Calculate their instantaneous speed at second $2$.

1. c) At what point does the instantaneous speed coincide with the average speed over the interval $[1, 3]$? Calculate it and comment on the result.

Let $f(x) = \frac{1}{x}$.

1. a) Calculate $\text{TVM}[1, 2]$.

1. b) Calculate $\text{TVM}[1, 1 + h]$ in terms of $h$ and simplify the expression.

1. c) What value does the above expression tend to as $h \to 0$? What does that value mean?

The closing share price for each day of a week was: Monday $\text{€}\ 10.40$; Tuesday $\text{€}\ 10.12$; Wednesday $\text{€}\ 10.35$; Thursday $\text{€}\ 10.80$; Friday $\text{€}\ 10.55$.

1. a) Calculate the daily change (absolute and percentage) for each day, rounding to two decimal places.

1. b) Calculate the total change for the week and the average daily rate of change, in euros per day.

1. c) A headline claims: "the share has risen on average by $0.4 \%$ daily". Check if this is true using the average percentage change and comment on which measure —absolute or percentage— provides better information to the investor.

Find the value of parameter $a$ such that the average rate of change of $f(x) = x^{2} + ax$ in the interval $[0, 2]$ is equal to $5$. Check the result.

The temperature $T(t)$ (in ºC) recorded at a weather station in the Sierra de Guadarrama during a winter morning is given by the function $T(t) = -0.5t^2 + 3t + 2$, where $t$ indicates the hours elapsed from $08:00$ ($t = 0$) to $14:00$ ($t = 6$).

1. a) Without calculating derivatives or numerical values yet, look at the signs of the coefficients: given that the linear term $+3t$ dominates in the first hours before the negative curvature slows down the rise, estimate whether $\text{ARC}[0, 2]$ will be positive or negative and propose a numerical range that you consider reasonable for the average rate of thermal change (in ºC/hour).
2. b) Analytically calculate $\text{ARC}[0, 2]$ and the instantaneous rate of change at $08:00$, given by $T'(0)$.
3. c) Check whether your estimate was consistent with the result and justify why $T'(0)$ is strictly greater than $\text{ARC}[0, 2]$, explaining what is happening to the warming rate over those two hours.

A handmade shoe workshop in Kilkenny produces pairs of leather shoes. The weekly total revenue function from selling $q$ pairs is $I(q) = 120q - 0.4q^2$ (in euros), while the weekly total manufacturing costs are given by $C(q) = 500 + 20q + 0.1q^2$ (in euros).

1. a) Obtain the expressions for marginal revenue $I'(q)$, marginal cost $C'(q)$, and marginal profit $B'(q) = I'(q) - C'(q)$.
2. b) If the workshop currently manufactures $60$ pairs per week, calculate the marginal profit $B'(60)$ and interpret its practical economic meaning: is it worthwhile for the company to manufacture and sell pair number $61$? Justify your answer by comparing the values of $I'(60)$ and $C'(60)$.
3. c) Reasonably determine the weekly production volume that maximizes the workshop's total profit using the marginal condition ($I'(q) = C'(q)$) and calculate the value of that maximum profit.

The figure shows, in panel A, the graph of a polynomial function with a local maximum at $x = 1$ and a local minimum at $x = 3$; panel B contains only the axes, ready for drawing.

1. a) State the intervals where $f'(x)$ is positive and where it is negative, reasoning from the growth of $f$.

1. b) At which points does $f'$ equal zero?

1. c) Sketch the graph of $f'$ in panel B, explaining the shape you have drawn and why it intersects the $X$-axis where it does.

You can do it on paper and send a photo with your answer.

A workshop manufactures $q$ bicycles per month with revenues $I(q) = 50q - 0.1q^{2}$ and costs $C(q) = 20q + 1,000$, in thousands of euros.

1. a) Calculate the marginal revenue $I'(q)$ and evaluate it at $q = 100$. What does its value mean?

1. b) Calculate the marginal profit $B'(q)$, where $B(q) = I(q) - C(q)$, and find for which value of $q$ it equals zero.

1. c) Explain, using the sign of the marginal profit before and after that value, why it corresponds to the maximum profit.

1. a) Show that the average rate of change of a linear function $f(x) = mx + n$ in any interval $[a, b]$ is constant and equal to its gradient $m$.

1. b) Show that for $f(x) = x^{2}$ it holds that $\text{TVM}[a, a + h] = 2a + h$, and deduce, by letting $h \to 0$, that $f'(a) = 2a$.

The position of a cyclist on a straight line is given by $e(t) = t^{3} - 6t^{2} + 9t$, in metres, with $t$ in seconds and $0 \leq t \leq 4$.

1. a) Calculate their average velocity in the intervals $[0, 2]$ and $[2, 4]$. What does the sign of each indicate?

1. b) Calculate the instantaneous velocity $v(t) = e'(t)$.

1. c) At which moments of the journey is the instantaneous velocity zero? Describe what the cyclist is doing at those moments.

A student calculates the derivative of $f(x) = x^{2}$ at $x = 2$ by writing:

«$f'(2) = \lim_{h \to 0} \frac{f(2) + h - f(2)}{h} = \lim_{h \to 0} \frac{h}{h} = 1$»

1. a) Explain the error they have made when applying the definition.

1. b) Write the correct calculation and find the true value of $f'(2)$.

1. c) Check the result by calculating $\text{TVM}[2, 2 + h]$ for $h = 0.1$ and for $h = 0.01$, and comment on which value they approach.

The population of a protected species in a national park follows the model $P(t) = 12,000 \times 1.02^{t}$, with $t$ in years.

1. a) Calculate the average rate of change of the population in the interval $[0, 5]$, in individuals per year, rounded to the nearest unit.

1. b) Calculate the annual relative (percentage) average rate of change in that interval, $\frac{P(5)-P(0)}{5 \times P(0)}$, and compare it with $2 \%$. What causes the difference?

1. c) Show that in any exponential model $P(t) = P_0 \times a^{t}$ the annual relative rate of change is constant, while the absolute rate increases over time. What does this imply for forecasts?

1. a) Calculate, using the definition with limits, the derivative of $f(x) = x^{2} + x$ at the point $x = 1$.

1. b) Write the equation of the tangent line to the graph of $f$ at the point $(1, f(1))$.

1. c) Check the result graphically: plot the parabola and the tangent around $x = 1$ and verify that the line "touches" the curve without crossing it at that point.

State whether these statements are true or false, giving reasons by proving them or providing a counterexample:

1. a) "If $f$ is an even function (symmetric with respect to the $Y$-axis), then $\text{TVM}[-a, a] = 0$ for all $a$ in its domain."

1. b) "If $\text{TVM}[a, b] = 0$, then the function is constant in $[a, b]$."

1. c) "If $\text{TVM}[a, b] > 0$, then $f$ is increasing in the entire interval $[a, b]$."

A rapeseed-growing cooperative in Wexford models its total milling and packaging costs using the function $C(q) = 0.05q^3 - 1.5q^2 + 50q + 200$ (in hundreds of euros), where $q$ is the quantity of rapeseed oil produced in tonnes. The technical management debates how to measure the impact of increasing production from $20$ to $21$ tonnes and proposes three methods:

• Method 1: Calculate the exact actual increment $\Delta C = C(21) - C(20)$.

• Method 2: Calculate the average rate of change $\text{ARC}[20, 21] = \frac{C(21) - C(20)}{21 - 20}$.

• Method 3: Estimate the additional cost using the instantaneous marginal cost $C'(20)$.

1. a) Explain why Methods 1 and 2 yield exactly the same number but have different conceptual meanings and units (absolute magnitude versus rate of change relative to step $h$).
2. b) Apply Methods 1 and 3. Find the real cost of that additional tonne and compare it with the marginal approximation by determining the absolute error committed ($|C'(20) - \Delta C|$).
3. c) Choose which of the methods is the standard in microeconomic analysis for modeling decisions at the margin and justify your choice. Furthermore, analyze the sign of the second derivative $C''(20)$ to explain why Method 3 underestimates the real cost and explain why the error of the marginal approximation would increase considerably if the production jump were $5$ tonnes instead of $1$.

A photovoltaic energy company in County Wexford analyzes the daily electricity generation cost. The total cost, in thousands of euros, when producing $q$ megawatt-hours (MWh) per day follows the quadratic function $C(q) = aq^2 + bq + 8$, with $a > 0$.

The analyst team only has three pieces of information:

1. The daily fixed cost (with no generation) is $8$ thousand euros.

2. The average rate of change of the cost between producing $10\text{ MWh}$ and $30\text{ MWh}$ is $18$ thousand €/MWh (that is, $\text{ARC}[10, 30] = 18$).

3. At the production level of $20\text{ MWh}$, the marginal cost of production coincides exactly with the average cost per MWh generated, that is: $C'(20) = \frac{C(20)}{20}$.

1. a) Set up the corresponding system of equations and determine the coefficients $a$ and $b$ of the cost function.
2. b) Algebraically prove that for any quadratic function $f(x) = Ax^2 + Bx + C$, the rate $\text{ARC}[x_1, x_2]$ always coincides with the instantaneous derivative at the midpoint of the interval, $f'\left(\frac{x_1 + x_2}{2}\right)$, and explain how this validates the relationship between data item 2 and data item 3 at $x = 20$.
3. c) If the wholesale electricity market price on that day is fixed and equal to $18.4$ thousand euros per MWh (revenue $I(q) = 18.4q$), is it worthwhile for the plant to generate more than $20\text{ MWh}$? Use the concept of marginal profit $B'(q)$ to justify your answer and calculate the exact amount of MWh that the plant must produce to maximize its daily profit.