Average rate of change and marginal analysis
In economics nobody talks about "the derivative": they talk about marginal cost, marginal revenue or marginal profit. They are the same idea under a different name: how much a company's cost, revenue or profit changes if it produces one more unit, right at the level of production it is at now. This section generalises the average rate of change to any interval and turns it into the central tool of economic analysis.
Average rate of change over a general interval $[a, a+h]$
Calculate the average rate of change of $f(x) = x^{2}$ in the interval $[1, 3]$.
Calculate the average rate of change of $f(x) = x^{2}$ 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…
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 derivative as the instantaneous rate of change of any quantity
The derivative is useful for any quantity that changes with time or with another variable: position, temperature, population, number of users, price... In each case, $f'(a)$ is "how much $f$ changes per unit of $x$, right at $a$".
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?
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 $0 \leq t \leq 4$. a) Calculate their average…
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 $0 \leq t \leq 4$.
- a) Calculate their average velocity in the intervals $[0, 2]$ and $[2, 4]$. What does the sign of each indicate?
- b) Calculate the instantaneous velocity $v(t) = e'(t)$.
- c) At which moments of the journey is the instantaneous velocity zero? Describe what the cyclist is doing at those moments.
Marginal cost: the derivative of total cost
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. a)…
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.
- a) Calculate the actual cost of producing the $61^{\text{st}}$ backpack: $C(61) - C(60)$.
- b) Calculate the marginal cost $C'(60)$ as an approximation of the cost of that additional unit.
- c) Compare both values and explain why economists use the derivative as "marginal cost".
Marginal revenue and marginal profit: the optimal point
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. a) Calculate the marginal…
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.
- a) Calculate the marginal revenue $I'(q)$ and evaluate it at $q = 100$. What does its value mean?
- b) Calculate the marginal profit $B'(q)$, where $B(q) = I(q) - C(q)$, and find for which value of $q$ it equals zero.
- c) Explain, using the sign of the marginal profit before and after that value, why it corresponds to the maximum profit.
Absolute and relative change
The same unit is not always what matters: an investor does not only look at how much a share rises in euros, but at what percentage that rise represents.
The closing share price for each day of a week was: Monday $10,40\ \text{ €}$; Tuesday $10,12\ \text{ €}$; Wednesday $10,35\ \text{ €}$; Thursday $10,80\ \text{ €}$;…
The closing share price for each day of a week was: Monday $10,40\ \text{ €}$; Tuesday $10,12\ \text{ €}$; Wednesday $10,35\ \text{ €}$; Thursday $10,80\ \text{ €}$; Friday $10,55\ \text{ €}$.
- a) Calculate the daily change (absolute and percentage) for each day, rounding to two decimal places.
- b) Calculate the total change for the week and the average daily rate of change, in euros per day.
- 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 $P(t) = 12.000 \cdot 1,02^{t}$, with $t$ in years. a) Calculate the average rate of change of…
The population of a protected species in a national park follows the model $P(t) = 12.000 \cdot 1,02^{t}$, with $t$ in years.
- a) Calculate the average rate of change of the population in the interval $[0, 5]$, in individuals per year, rounded to the nearest unit.
- b) Calculate the annual relative (percentage) average rate of change in that interval, $\frac{P(5)-P(0)}{5 \cdot P(0)}$, and compare it with $2 \%$. What causes the difference?
- c) Show that in any exponential model $P(t) = P_0 \cdot a^{t}$ the annual relative rate of change is constant, while the absolute rate increases over time. What does this imply for forecasts?