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

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 $P$.

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?

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.

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 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 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 $[5, 10]$.

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 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.

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.

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.

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?