Exercises: Concept

A content creator analyses the growth of their followers on a social network over the past week. The number of followers $S$ as a function of day $t$ is given by the expression $S(t) = t^2 + 10t + 500$. Calculate the Average Rate of Change (ARC) of their followers between day $t=1$ and day $t=4$. Explain in your own words what the result obtained means in the context of the problem.

Look at the following graph, which represents the change in the price of a cryptocurrency over a morning. Read from it the values of $f(1)$ and $f(5)$ and calculate the Average Rate of Change over the interval $[1, 5]$, interpreting it as the slope of the secant line that passes through those two points.

The speed of an electric scooter during the first few seconds of a journey follows the function $v(t) = 0,5t^2 + 2$, where $v$ is the speed in m/s and $t$ is the time in seconds.

1. a) Calculate the ARC in the intervals $[2, 3]$ and $[2, 2,1]$.
2. b) What value do you think the rate of change approaches as the interval becomes smaller and smaller around $t=2$?

Given the linear function $f(x) = 4x - 3$, use the definition of a derivative at a point using the limit:

to calculate the value of the derivative at the point $x=2$. Check whether the result coincides with the gradient of the line.

Match each of the points marked on the graph ($A$, $B$ and $C$) with the sign of the derivative of the function at that point (positive, negative or zero). Justify your answer by mentally drawing the tangent line at each point.

A small mobile phone case manufacturing company has estimated that its daily costs (in euros) depend on the number of units manufactured $x$ according to the function $C(x) = x^2 - 4x + 100$.

Calculate the Instantaneous Rate of Change of the cost when exactly 5 units are produced, applying the definition of derivative:

Find the equation of the tangent line to the graph of the function $f(x) = x^2 + 1$ at the point with x-coordinate $x=1$, knowing that the value of the derivative at that point is $f'(1) = 2$.

(Remember that the equation of the tangent line is $y - f(a) = f'(a)(x - a)$).

Calculate, using the limit definition, the derivative function of the constant function $f(x) = 7$. Based on the geometric interpretation of the derivative as the "rate of change" or "gradient", explain why it makes sense for the result to be $0$.

To analyze the sales rate of a small bookstore during its first week, the function $V(t) = t^2 + 4t + 10$ is used, where $t$ is the time in days and $V(t)$ is the cumulative number of copies sold. Marcos wants to calculate the average rate of change between day $t = 1$ and day $t = 4$, and presents the following solution:

From this result, Marcos concludes that on day $t = 2$ the bookstore sold exactly $6,75$ more books than on day $t = 1$.

1. a) Identify and explain the two conceptual errors made by Marcos: one in applying the $\text{ARC}$ formula and another in interpreting what an average represents compared to a specific daily change.
2. b) Correct the calculation, obtaining the true value of $\text{ARC}[1, 4]$.
3. c) Write a correct conclusion describing the bookstore's sales trend over that period using the corrected value.

The temperature $T$ (in degrees Celsius) recorded by an AEMET weather station in a town in the peninsular interior is given, during the early morning hours, by the function $T(t) = -0,5t^2 + 4t + 12$, where $t$ is the time in hours elapsed since 8:00 (with $0 \le t \le 5$).

1. a) Without performing prior analytical calculations, observe the quadratic expression and estimate with reasoning whether at 9:00 ($t = 1$) the temperature will be increasing or decreasing. Based on that deduction, what sign do you expect the instantaneous rate of change $T'(1)$ to have?
2. b) Calculate the average rate of change of the temperature over the time intervals $[1; 1,1]$ and $[1; 1,01]$, expressing the results with their corresponding units.
3. c) In light of the numerical results obtained in part b), estimate with reasoning the exact value of the derivative $T'(1)$. Check if it matches your prediction from part a) and interpret what this value means for the population of that municipality at 9:00.

A streaming platform has analysed the growth of its subscribers during the first months of the year. The function estimating the number of subscribers (in thousands) is given by $S(t) = t^2 + 2t + 10$, where $t$ is the time measured in months from the beginning of the year ($t=0$ is 1 January).

1. a) Calculate the Average Rate of Change (ARC) of the number of subscribers between the second month ($t=2$) and the fifth month ($t=5$).
2. b) Explain the meaning of the result obtained in the context of the problem.
3. c) Would it be correct to state that the platform gained exactly that number of subscribers each month during that period? Reason your answer.

Calculate, using strictly the definition of a derivative at a point (using the limit as $h \to 0$), the value of $f'(3)$ for the function:

The following graph shows the profile of a roller coaster on a specific section. It is known that the slope of the track at a point determines the passengers' sensation of acceleration.

Based on the geometric concept of the derivative:

1. a) Is the derivative zero at any of the three marked points ($A$, $B$, or $C$)? What does it mean for the derivative to be zero with respect to the slope of the track?
2. b) State whether the derivative at point $B$ is positive or negative. Justify whether the roller coaster car is ascending or descending at that moment.
3. c) Order the values of the derivatives at points $A, B$, and $C$ from lowest to highest.

Given the function $f(x) = \frac{4}{x}$:

1. a) Find the Instantaneous Rate of Change (IRC) at the point with x-coordinate $x = 2$ using the limit of the Average Rate of Change.
2. b) Determine the equation of the tangent line to the graph of the function at that point $x = 2$.

A small clothing shop has estimated that its daily profits (in euros) depend on the number of discounted items ($x$) according to the function $B(x) = -x^2 + 40x - 300$.

1. a) Calculate the derivative function $B'(x)$ using differentiation rules.
2. b) Calculate $B'(15)$ and $B'(25)$.
3. c) Interpret the results from the previous section: Is it advantageous for the shopkeeper to increase the number of discounted items if they currently have 15? What if they have 25? Reason your answer using the concept of instantaneous change.

The concept of a derivative allows us to approximate the value of a function near a known point. Observe the following diagram showing a function and its tangent line at the point $P(1, 2)$.

If we know that the equation of the orange tangent line is $y = 3x - 1$:

1. a) What is the value of the derivative of the function at $x = 1$? Justify your answer.
2. b) If we wanted to know the value of the function at $x = 1,1$, what value would we obtain if we used the tangent line to approximate it?
3. c) Do you think that for $x = 5$ the tangent line would still be a good approximation of the function? Explain why.

A function is said to be "differentiable" at a point if its graph does not have any jumps or sharp points at that location. Analyse the following piecewise function:

1. a) Check whether the function is continuous at $x = 1$.
2. b) Calculate the one-sided derivatives at $x = 1$ (that is, the derivative of each section at that point).
3. c) From the previous results, is the function differentiable at $x = 1$? Justify your answer by relating continuity and the existence of one-sided derivatives.

Find the points on the graph of the function $f(x) = x^3 - 3x^2 + 5$ where the tangent line is horizontal.

Hint: Remember what value the gradient (and therefore the derivative) must have for a line to be horizontal.

Based on data from the INE's Household Budget Survey, the cumulative average expenditure per household on digital services and telecommunications (in euros) over an expansion period is modeled by the function $G(t) = 2t^2 + 10t + 30$, where $t$ is the time in years elapsed since 2018 ($t = 0$).

1. a) Obtain the simplified algebraic expression of the average rate of change $\text{ARC}[2, 2+h]$ for any increment $h \ne 0$, taking the year 2020 ($t = 2$) as reference.
2. b) By calculating the limit $\lim_{h \to 0} \text{ARC}[2, 2+h]$, determine the instantaneous rate of change $G'(2)$. Explain its economic meaning in detail and include its units of measurement.
3. c) Find the equation of the tangent line to the graph of $G(t)$ at the point with abscissa $t = 2$. Use this tangent line to approximate the expenditure predicted by the model for $t = 2,25$ (the first quarter after 2020) and compare it with the actual value obtained with the function $G(2,25)$. Why is the tangent line approximation so close to the actual value in this neighborhood?

A Spanish express courier company estimates that the unit transport cost $C(x)$ (in euros) as a function of the distance $x$ (in hundreds of kilometers) follows a quadratic model of the form $C(x) = a x^2 + b x + 5$, where $a$ and $b$ are unknown parameters. The cost analysts have only two pieces of technical information:

1. At a distance of $100\text{ km}$ ($x = 1$), the instantaneous rate of change of the cost is $7\text{ €}$ for every $100\text{ km}$, that is, $C'(1) = 7$.

2. The tangent line to the cost graph at the point $x = 2$ is given by the equation $y = 11x - 3$.

1. a) From the given tangent line, determine the value of the function at that point, $C(2)$, and the value of the instantaneous derivative $C'(2)$, justifying the source of both pieces of data.
2. b) Using the formal definition of the derivative via the limit of the difference quotient $C'(x_0) = \lim_{h \to 0} \frac{C(x_0+h)-C(x_0)}{h}$, set up a system of two equations with two unknowns and calculate the value of the coefficients $a$ and $b$.
3. c) With the parameters now identified, calculate the $\text{ARC}[1, 2]$ and explain why this average value coincides exactly with the arithmetic mean of $C'(1)$ and $C'(2)$.

The number of followers of a video game streaming channel follows the function $S(t) = 2t^2 - 4t + 500$, where $t$ is the time in months.

1. a) Calculate the Average Rate of Change ($TVM$) in the generic interval $[2, 2 + h]$.
2. b) Using the previous result, deduce the value of the limit as $h \to 0$. What does this value physically represent for the streamer in month $t = 2$?
3. c) Is there any value of $h$ for which the $TVM$ in the interval $[2, 2 + h]$ is equal to $10$ followers/month?

Calculate, using strictly the definition of a derivative as a limit, the derivative function of:

From the result obtained, determine the coordinates of the point or points on the graph where the gradient of the tangent line is exactly $-3$. Explain why there cannot be points where the gradient of the tangent line is positive.

At a ballooning festival, the height of a hot-air balloon (in meters) with respect to time (in seconds) since it begins its ascent is given by the function $H(t) = \sqrt{t + 4}$. The figure shows its graph during the first 12 seconds.

1. a) Calculate the derivative $H'(t)$ using the limit definition.
2. b) Find the equation of the tangent line to the trajectory at the instant $t = 5$.
3. c) For safety, the festival regulations require that the balloon not exceed an ascent speed of $0,25$ m/s. Determine at what instant the instantaneous velocity is exactly that value.

Study the differentiability of the following function at the point $x = 2$. In order for the function to represent the production cost of a new electronic component smoothly (without jumps or sharp points), determine the values of the parameters $a$ and $b$:

Remember that for it to be differentiable at $x = 2$, it must first be continuous at that point.

Find the points of the curve $y = x^3 - 3x^2 - 9x + 2$ at which the tangent line is parallel to the x-axis. Once you have found them, use the interpretation of the derivative to reason whether at those points the function has a relative maximum, a relative minimum, or an inflection point, by analyzing the sign of the derivative near those points.

It is known that the revenue function of an agricultural cooperative is $I(x) = -x^2 + 12x$, where $x$ are the tonnes of product sold.

1. a) Calculate the derivative function $I'(x)$ and interpret it as the "Marginal Revenue" (the approximate revenue from selling one additional unit).
2. b) Show that the revenue is maximum when the Marginal Revenue is equal to zero.
3. c) What happens to the revenue function if the derivative is negative? Justify your answer in terms of increasing and decreasing.

Given the function $f(x) = |x^2 - 4|$, analyse its differentiability at the points $x = 2$ and $x = -2$. The figure shows its graph.

1. a) Express the function as a piecewise-defined function.

1. b) Calculate the one-sided derivatives at $x = 2$ using the limit definition. (At $x = -2$ the reasoning is analogous, by the symmetry of the graph.)

1. c) Explain graphically what happens at those points (the concept of a "corner point") and why the derivative does not exist even though the function is continuous.

Consider the function $f(x) = ax^2 + bx + c$. Determine the values of $a$, $b$ and $c$ knowing that:

1. The graph of $f(x)$ passes through the point $(0, 5)$.

2. The tangent line to the curve at the point with x-coordinate $x = 1$ is the line $y = 2x + 1$.

3. Use the derivative to check whether the vertex of the resulting parabola lies to the left or to the right of the $Y$-axis.

An agricultural cooperative bills the cold storage cost of fruit $C(x)$ (in hundreds of euros) according to the deposited tonnage $x$. The pricing model has two tiers and changes upon reaching $4$ tonnes:

To verify whether the cost transition at the change point ($x = 4$) is smooth or if a corner point exists, three members of the technical team suggest different methods:

• Ignacio states: “It is enough to check that the function is continuous at $x = 4$; if the two branches join at the same point, the derivative must necessarily exist.”

• Beatriz argues: “We should numerically calculate the $\text{ARC}$ over very small intervals on both sides, such as $[3,99; 4]$ and $[4; 4,01]$, to check if the slopes almost match.”

• Carla defends: “It is essential to calculate the one-sided derivatives $C'(4^-)$ and $C'(4^+)$ algebraically by applying the limit of the average rate of change as $h \to 0^-$ and $h \to 0^+$.”

1. a) Refute Ignacio's statement using a theoretical argument: why is continuity at a point a necessary but not sufficient condition for differentiability?
2. b) Carry out the calculations proposed by Beatriz (calculating both $\text{ARC}$) and by Carla (solving both limits with $h$).
3. c) Evaluate the methods of Beatriz and Carla: what conceptual limitation does Beatriz's numerical calculation have compared to the certainty provided by Carla's formal method? Conclude whether the function is differentiable at $x = 4$, whether the graph has a unique tangent line or a corner point, and explain what practical implication this has for the tariff.

A study on labor productivity $P(t)$ (in thousands of euros per year per worker) in an autonomous community models the first four years of a reform using the quadratic function $P(t) = -t^2 + 6t + 10$ for $t \in [0, 4]$. From the fourth year onwards ($t > 4$), the regional administration wishes to implement a linear digital shock plan of the form $g(t) = m \cdot (t - 4) + k$.

To avoid abrupt economic distortions in the labor market, a “smooth join” is required at $t = 4$, which calls for two mathematical conditions:

1. Productivity does not undergo sudden jumps (the overall function must be continuous at $t = 4$).

2. The instantaneous trend of productivity does not change abruptly (the overall function must be differentiable at $t = 4$, so that the one-sided derivatives coincide).

1. a) Calculate the instantaneous rate of change $P'(4^-)$ at the end of the first period using rigorously the definition of derivative as the limit of the average rate of change. In light of the sign obtained, indicate whether productivity was increasing or decreasing at that moment.
2. b) Deduce the exact values that the parameters $k$ and $m$ must have to ensure that the digital shock plan meets both conditions at $t = 4$.
3. c) Write the equation of the tangent line to the function at $t = 4$. Suppose that, for budgetary reasons, a slope $m = 1$ were applied instead of the value calculated in part b); explain with reasoning what geometric feature would appear on the graph at $t = 4$ and what misleading information calculating the $\text{ARC}[3,9; 4,1]$ would provide if one ignores that the function is no longer differentiable.