Exercises: Applications of the derivative

The number of people visiting a new contemporary art exhibition during the first 4 hours since its opening follows the function $f(t) = t^2 + 2t$, where $t$ is the time in hours. Calculate the Average Rate of Change (ARC) of the number of visitors between hour $t = 1$ and hour $t = 3$.

Given the function $f(x) = x^2 - 4x + 5$, use the definition of the derivative at a point to find the gradient of the tangent line to the graph at $x = 3$.

Find the equation of the tangent line to the function $f(x) = 2x^2 - 3x + 1$ at the point with abscissa $x = 1$.

An ice cream parlour has determined that its monthly profits (in hundreds of euros) depend on the number of employees hired $x$ according to the function $B(x) = -x^2 + 10x - 16$. Determine whether the profit is increasing or decreasing when the parlour has 3 employees.

Determine the points on the graph of the function $f(x) = x^3 - 12x + 2$ at which the tangent line is horizontal.

Study the intervals of increase and decrease of the quadratic function $f(x) = x^2 - 6x + 8$ by analyzing the sign of its first derivative.

A drone takes off vertically and its height $h$ (in metres) with respect to time $t$ (in seconds) is given by the expression $h(t) = -t^2 + 8t$. Calculate the instantaneous speed of the drone 2 seconds after takeoff.

Calculate the value of $x$ at which the function $f(x) = x^2 - 8x + 20$ reaches its relative minimum value using the derivative.

Lucía is trying to classify the stationary points of the function $f(x) = (x - 2)^4 + 3$. In her notebook, she writes the following solution:

"I calculate the first derivative: $f'(x) = 4(x - 2)^3$. I set it to zero: $4(x - 2)^3 = 0 \implies x = 2$. To see whether it is a maximum or a minimum, I calculate the second derivative: $f''(x) = 12(x - 2)^2$. Evaluating at the critical point, I get $f''(2) = 0$. Since the second derivative is zero, the function has neither a maximum nor a minimum at $x = 2$, but it must necessarily be an inflection point."

1. a) Identify and explain the two conceptual errors that Lucía makes when applying the derivative tests.
2. b) Correct the exercise by analysing the sign of $f'(x)$ to the left and to the right of $x = 2$ to determine, with justification, whether the function has a maximum, a minimum, or neither.
3. c) Check whether the point $(2, 3)$ is an inflection point or not by studying the sign of $f''(x)$ on both sides of $x = 2$.

During a summer heatwave recorded by AEMET in Córdoba, the outdoor temperature $T$ (in degrees Celsius) over the course of the 24 hours of the day ($t \in [0, 24]$) fits the model:

1. a) Estimate: Before differentiating, and based on everyday experience regarding the hottest hours in summer, propose a reasonable time interval in which you expect the maximum temperature of the day to be reached. Briefly justify your prediction.
2. b) Calculate: Using the tools of differential calculus, find the exact time $t$ at which the absolute maximum temperature occurs and the value of that temperature.
3. c) Check: Compare the exact result with your initial estimate and assess whether the mathematical model plausibly describes daily thermal evolution.

Calculate the equation of the tangent line to the graph of the rational function $f(x) = \dfrac{2x + 1}{x - 1}$ at the point $P$ with abscissa $x = 2$ (marked in the figure). Interpret the result by stating whether the function is increasing or decreasing at that point based on the value of the slope.

Given the polynomial function $f(x) = x^3 - 6x^2 + 9x + 2$, perform a monotonicity study. To do this, determine the intervals where it is increasing and decreasing, and find the coordinates of its local maximum and minimum points.

A company that sells personalized phone cases has determined that its daily profit, in euros, follows the function $B(x) = -x^2 + 40x - 300$, where $x$ is the number of cases sold.

1. a) How many cases must they sell for the profit to be maximum?

1. b) What is the amount of that maximum profit?

1. c) Between which numbers of cases sold does the company make a positive profit (that is, $B(x) > 0$)?

Find the values of the parameters $a$ and $b$ so that the function $f(x) = x^3 + ax^2 + b$ has a local minimum at the point $(2, -3)$. Once the parameters have been found, check if the function has any other local extrema.

A surveillance drone moves along a straight path. Its distance from the control point (in metres) is given by the function $s(t) = \dfrac{1}{3}t^3 - 4t^2 + 12t$, where $t$ is the time in seconds ($0 \le t \le 10$).

• Calculate the velocity function $v(t)$ and the acceleration function $a(t)$.

• Determine at which instants the drone is momentarily stationary.

• What is the speed of the drone at 5 seconds? Is it accelerating or decelerating at that moment?

Determine the points on the curve $f(x) = x^3 - 3x^2 + 2$ at which the tangent line is parallel to the line $y = 9x - 5$. Write the equations of these tangent lines.

Study the curvature (concavity and convexity) and find the points of inflection of the function $f(x) = 3x^4 - 4x^3$. Draw an approximate sketch of the shape of the function based solely on its second derivative.

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

We wish to fence a rectangular garden using the wall of a house, so that it is only necessary to fence three of its sides. If 60 metres of fencing are available, calculate the dimensions of the garden so that the enclosed area is as large as possible.

A parcel logistics platform located in Zaragoza models its daily net profit $B(x)$, expressed in thousands of euros, as a function of the number $x$ of active delivery vans (with $x \in [1, 7]$), using the polynomial function:

1. a) Determine the intervals of increase and decrease of the daily profit within the fleet's operating range $[1, 7]$.
2. b) Find the number of vans that yields a relative maximum profit and confirm this extremum using the second derivative test.
3. c) The operations director claims that putting the entire fleet of 7 vans into circulation will always yield more total net profit than leaving vans idle. Evaluate the profit at the endpoints of the interval $[1, 7]$, compare them with the maximum found, and refute or confirm the director's statement using numerical data.

A financial analyst needs to reconstruct a cubic function $f(x) = ax^3 + bx^2 + cx + d$ that describes the accumulated profits of a renewable energy company during its first years of activity. Only the following information is available:

1. The company starts with neither profit nor loss at the beginning of its activity, so its graph passes through the origin $O(0, 0)$.

2. The curve has an inflection point exactly at the abscissa $x = 1$.

3. The function has a relative minimum at the point $P(2, -4)$.

1. a) Set up the system of linear equations that allows you to calculate the coefficients $a, b, c$, and $d$, justifying in detail which mathematical property each equation comes from.
2. b) Solve the system and write the explicit analytic expression of $f(x)$.
3. c) Find out whether another relative extremum exists in the company's trajectory, calculate its abscissa, and classify it using the sign of $f''(x)$ or $f'(x)$.

Given the polynomial function $f(x) = ax^3 + bx^2 + cx + d$, determine the values of the coefficients $a, b, c$, and $d$ knowing that the following conditions are simultaneously satisfied:

• The graph of the function passes through the point $P(0, 2)$.

• It has a relative extremum at the point with abscissa $x = 1$.

• The tangent line to the curve at the point with abscissa $x = -1$ is parallel to the line with equation $y = 12x - 5$.

• The function has an inflection point at $x = -1$.

Study the continuity and differentiability of the following piecewise function throughout its domain. If parameters exist, determine the values of $a$ and $b$ so that the function is differentiable at $x = 2$:

Once the parameters have been found, calculate the equation of the normal line to the curve at the point $x = 0$.

A intercity bus company has determined that the total cost (in euros) of operating its fleet for one day depends on the average speed $v$ (in km/h) at which the buses travel, according to the function:

The figure shows the graph of $C(v)$ and, shaded, the interval of speeds $[20, 60]$ used in part c).

Find:

1. a) Find the average speed at which the buses must travel so that the operating cost is minimal.

1. b) Prove, using the second derivative, that the value found is indeed a minimum.

1. c) If, due to traffic regulations, the average speed must lie in the interval $[20, 60]$ km/h, what would be the maximum cost and the minimum cost in that interval?

We wish to fence a rectangular enclosure intended for a playground in a housing development. The enclosure must have an area of $600 \text{ m}^2$. One of the sides of the rectangle adjoins a building and does not need fencing. The price of the fence for the two lateral sides is $20 \text{ €/m}$, while the side parallel to the building uses a reinforced fence costing $60 \text{ €/m}$.

Determine the dimensions of the enclosure that minimise the total cost of the fence and calculate this minimum cost.

The following graph represents the derivative function $f'(x)$ of a continuous function $f(x)$. From observing the graph of $f'(x)$, deduce with reasoning for the original function $f(x)$:

1. The intervals where the function is increasing and decreasing.

2. The x-coordinates of its local maximum and minimum points.

3. The intervals of concavity and convexity (by studying the growth of $f'(x)$).

4. Knowing that $f(0) = 0$, draw an approximate sketch of the graph of $f(x)$.

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

An influencer has analysed the impact of their videos on a social network. The cumulative number of views $V(t)$ (in thousands) of their latest video follows the function:

where $t$ is the time in days since publication ($0 \le t \le 15$).

1. On which day does the video reach the maximum rate of growth of views? (Note: You are asked to maximise the growth rate, i.e., the derivative).

2. On which day is the maximum cumulative number of views reached before the video loses interest?

3. What is the average rate of change of the number of views between day 1 and day 5?

Consider the rational function $f(x) = \frac{x^2 + kx}{x - 1}$.

1. Calculate the value of $k$ so that the function has a local extremum at the point with x-coordinate $x = 2$.

2. For that value of $k$, study whether the extremum is a maximum or a minimum.

3. Find the equations of the asymptotes of the function for the obtained value of $k$.

4. Determine the points on the curve where the gradient of the tangent line is equal to $0$.

The total profit function of an agricultural cooperative (in thousands of euros) is given by $B(x) = -0,5x^2 + 20x - 18$, where $x$ represents the tonnes of product sold. The figure shows the graph of $B(x)$.

1. a) Calculate the marginal profit function $B'(x)$. What does this concept represent in economic terms?

1. b) If current production is $10$ tonnes, approximately how much would profit increase if one additional tonne were sold?

1. c) Determine the production level that maximizes total profit.

1. d) From how many tonnes onward does the marginal profit start to be negative? Interpret the result.

A textile company calculates that the average cost of manufacturing each batch of $x$ hundreds of garments (for $x \in [1, 8]$) is given by the rational function:

(expressed in euros per garment). To find the production level that achieves the absolute minimum unit average cost on that closed interval, two members of the finance department debate which method to use:

• Method 1: Differentiate $C(x)$, set it to zero to find the critical value $x_0 \in (1, 8)$, and calculate $C''(x_0)$. Since $C''(x_0) > 0$, conclude directly that the absolute minimum cost over the entire interval $[1, 8]$ is reached at $x_0$.

• Method 2: Find the interior critical points where $C'(x) = 0$ within $(1, 8)$ and directly evaluate the function $C(x)$ at those interior points and at the boundary endpoints, $C(1)$ and $C(8)$, choosing the lowest of all the values obtained.

1. a) Analyse the theoretical validity of both procedures: why is Method 1 incomplete and risky for ensuring the existence of an absolute extremum on a closed interval? What geometric condition does it ignore?
2. b) Apply Method 2 to rigorously calculate the production volume $x$ that minimizes the absolute unit average cost and the corresponding cost.
3. c) Reason how the conclusions of both methods would be affected if the company changed its production capacity and the working interval became $[5, 8]$.

An olive oil cooperative in Jaén has estimated its daily total production function $P(x)$, measured in hundreds of litres, as a function of the number of workers $x$ hired at the oil mill (with $x \in [0, 10]$):

In economics, marginal productivity is defined as the first derivative of production, $P'(x)$, and represents the increase in production obtained by adding one additional worker.

1. a) Obtain the analytic expression of marginal productivity and calculate the second derivative $P''(x)$. Find the number of workers for which marginal productivity is maximized and explain what singular geometric element of the graph of $P(x)$ this point represents.
2. b) Study the intervals of concavity and convexity of $P(x)$ on $[0, 10]$. Economically interpret why the transition from a convex function to a concave function reflects the "law of diminishing marginal returns".
3. c) Management proposes hiring 10 workers to "obtain the maximum possible volume of oil". By conducting a full study of the absolute extrema of $P(x)$ on the closed interval $[0, 10]$, determine whether this decision really maximizes total daily production or if there is excess staff saturating the oil mill.