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 $x = 1$.

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.

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.

An ice cream parlor has determined that its monthly profits (in hundreds of pesos) 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 parlor 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.

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.

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.

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 $x = 0$.

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 $0$.

A drone takes off vertically and its height $h$ (in meters) 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.

A surveillance drone moves along a straight path. Its distance from the control point (in meters) 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?

A company selling personalized phone cases has determined that its daily profits, in pesos, follow the function $B(x) = -x^2 + 40x - 300$, where $x$ represents the number of cases sold.

• How many cases must they sell to maximize profit?

• What is this maximum profit?

• From what quantity of cases sold does the profit start to decrease?

The total profit function of an agricultural cooperative (in thousands of pesos) is given by $B(x) = -10x^2 + 400x - 360$, where $x$ represents the metric tons of product sold.

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

2. If current production is $10$ metric tons, approximately how much would profit increase if an additional metric ton were sold?

3. Determine the production level that maximizes total profit.

4. From how many metric tons onward does the marginal profit start to be negative? Interpret the result.

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?

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 meters of fencing are available, calculate the dimensions of the garden so that the enclosed area is as large as possible.

A intercity bus company has determined that the total cost (in pesos) 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.