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 $\text{TVM}[1, 4]$.

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 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.

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.

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?

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?

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 $P$.

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?

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.

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 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.

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.

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

The position of a cyclist on a straight line is given by $e(t) = t^{3} - 6t^{2} + 9t$, in meters, 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 company manufacturing backpacks has a total cost function $C(q) = 0.40q^{2} + 200q + 16\,000$, in pesos, where $q$ is the number of backpacks produced per month. The figure shows the cost curve and the tangent line at $q = 60$.

1. a) Calculate the actual cost of producing backpack number $61$: $C(61) - C(60)$.

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 workshop manufactures $q$ bicycles per month with revenues $I(q) = 1\,000q - 2q^{2}$ and costs $C(q) = 400q + 20\,000$, in thousands of pesos.

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 stock price for each day of a week was: Monday $\$208$; Tuesday $\$202.40$; Wednesday $\$207$; Thursday $\$216$; Friday $\$211$.

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 pesos per day.

1. c) A headline claims: "the stock 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?

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.

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 $[a, b]$."