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 $x = 1$.

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.

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?

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.

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.

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

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

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.

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.

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 $[5, 8]$.

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 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, $B(x) > 0$)?

During a summer heatwave recorded by Met Éireann in Kilkenny, 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.