Exercises: Derivative of a polynomial

Calculate the derivative function of the following polynomial function: $f(x) = 5x^4 - 3x^3 + 7x^2 - 2x + 9$.

Given the function describing the trajectory of a particle $s(t) = \frac{1}{4}t^4 - \frac{2}{3}t^3 + \frac{1}{2}t^2$, where $s$ is measured in metres and $t$ in seconds, determine the expression for its instantaneous velocity $v(t)$.

Find the value of the derivative of the function $g(x) = -2x^3 + 4x^2 - x + 5$ at the point with x-coordinate $x = -1$.

Determine the stationary points (those where the derivative is zero) of the quadratic function $h(x) = 3x^2 - 12x + 7$.

Calculate the second derivative, $f''(x)$, of the polynomial function $f(x) = x^5 - 4x^4 + 2x^2 - 8$.

The production cost, in euros, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 150 + 12x - 0,05x^2$. Find the marginal cost function, $C'(x)$.

Determine the gradient of the tangent line to the curve $y = x^3 - 5x + 2$ at the point where it intersects the y-axis.

Calculate the derivative of the function $p(x) = (2x^2 - 3)(x + 4)$ by first expanding the product of the polynomials.

In a class homework assignment, Carlos had to differentiate the polynomial function $f(x) = 4x^3 - 7x + 5^2$. In his notebook, he wrote the following procedure:

1. a) Identify and explain in detail the conceptual error that Carlos made when differentiating the term $5^2$. Why is it not correct to apply the power rule $(x^n)' = n \cdot x^{n-1}$ to a constant base?
2. b) Write the correct expression for the derivative function $f'(x)$.
3. c) Calculate the value of $f'(2)$ and interpret its geometric meaning with respect to the tangent line to the curve at $x = 2$.

A weather report from AEMET models the temperature $T$ (in degrees Celsius) recorded in a town during a clear morning using the polynomial $T(t) = -0,5t^2 + 5t + 10$, where $t$ is the elapsed time in hours since 7:00 ($0 \le t \le 8$).

1. a) Without calculating the derivative yet, reasonably estimate whether at 9:00 ($t = 2$) the temperature will be increasing noticeably or decreasing, basing your reasoning on the natural progression of the day and on the signs of the coefficients of the polynomial.
2. b) Find the derivative function $T'(t)$ and calculate the exact instantaneous rate of change at 9:00 ($t = 2$), clearly indicating its units of measurement ($^{\circ}\text{C}/\text{h}$).
3. c) Check whether the numerical value and the sign obtained confirm your previous estimate and explain what physical meaning it would have if the derivative were exactly $0$.

The profile of a suspension bridge cable is approximated by the parabola $y = 0,01x^2 - 0,6x + 15$, where $x$ and $y$ are measured in metres. Calculate the gradient of the cable at the point located $40$ metres from the origin.

Find the coordinates of the points on the graph of the function $f(x) = x^3 - 3x^2 + x - 1$ at which the tangent line is parallel to the angle bisector of the first quadrant.

The number of registered users of a productivity application, in thousands, evolves according to the function $U(t) = -t^3 + 9t^2 + 48t + 200$, where $t$ is the time in months since its launch. Determine the growth rate of the number of users at time $t = 5$ months and state whether at that moment the rate of growth is increasing or decreasing.

Determine the values of the parameters $a$ and $b$ such that the parabola $y = ax^2 + bx + 5$ passes through the point $(1, 2)$ and has a tangent line at that point with a slope equal to $-1$.

A projectile is launched vertically upwards. Its height in metres after $t$ seconds is given by $h(t) = -4,9t^2 + 29,4t + 1,5$. Calculate the time it takes to reach its maximum height, knowing that at this point its instantaneous velocity is zero.

Find the equation of the tangent line to the function $f(x) = -x^2 + 4x - 1$ at its vertex.

The level of noise pollution at an urban junction from 8:00 to 18:00 is modelled by the function $N(t) = 0,1t^3 - 1,5t^2 + 6t + 60$, where $t$ is the time in hours elapsed since 8:00. Determine the time $t$ at which the noise level is decreasing most rapidly.

Find the x-coordinate of the points on the curve $y = \frac{1}{3}x^3 - \frac{5}{2}x^2 + 6x$ where the tangent line forms an angle of $45^\circ$ with the positive x-axis.

To find the points with a horizontal tangent of the polynomial curve $f(x) = 2x^3 - 3x^2 - 12x + 8$, two students propose different approaches:

• Hugo proposes solving the equation $f(x) = 0$ to find the intercepts with the horizontal axis and assuming that the stationary points will be located exactly at the midpoint of each pair of consecutive intercepts, based on the symmetry he remembered from parabolas.

• Irene proposes calculating the derivative function $f'(x)$ and directly solving the equation $f'(x) = 0$.

1. a) Critically compare both approaches: justify why Hugo's strategy is not valid for this cubic polynomial and clarify for which specific type of polynomials his geometric intuition would actually work.
2. b) Apply the correct method to find the x-coordinates of all the stationary points of $f(x)$.
3. c) Calculate the complete coordinates $(x, y)$ of these stationary points and classify them with justification (relative maximum or minimum) by studying the sign of $f'(x)$ to the left and to the right of each one.

Based on records from Red Eléctrica de España (REE), the solar photovoltaic power generated on the peninsular grid during the central hours of a spring day can be approximated by the cubic function:

where $P(t)$ is the generated power in gigawatts ($\text{GW}$) and $t$ is the solar hour of the day, valid in the interval $10 \le t \le 16$.

1. a) Model the instantaneous rate of change of solar production by obtaining the derivative function $P'(t)$. Specify in which physical units $P'(t)$ is measured.
2. b) Evaluate $P'(12)$ and $P'(15)$. From the sign and the value obtained in each case, interpret what happens to solar energy generation at 12:00 and at 15:00.
3. c) Determine at what exact time of day the rate of change of production vanishes ($P'(t) = 0$), reaching the peak solar maximum of the day, and calculate the maximum power generated at that moment.

Determine the coefficients $a, b, c, d$ of the cubic polynomial function $f(x) = ax^3 + bx^2 + cx + d$ knowing that it passes through the origin, has a stationary point at $x = 2$, and has a point of inflection at $(1, -2)$.

A company manufactures and sells $x$ units of a tech item. The revenue function is given by $I(x) = 1200x - 2x^2$ and the total cost function is $C(x) = x^3 - 15x^2 + 400x + 1500$. Determine the number of units that must be produced to maximise profit and calculate this maximum profit.

Find the equations of the tangent lines to the parabola $y = x^2 - 2x + 3$ passing through the exterior point $P(1, -2)$.

Two test vehicles travel in a straight line. Their positions relative to a reference point are given by $s_1(t) = t^3 - 6t^2 + 15t$ and $s_2(t) = 2t^2 - 4t + 10$, with $t \ge 0$ in seconds and $s$ in metres. Determine at what times both vehicles travel with the same acceleration and calculate the distance separating them at that moment.

Find the equation of a parabola of the form $y = ax^2 + bx + c$, given that it is tangent to the line $y = -2x + 5$ at the point with x-coordinate $x = 1$ and passes through the point $(-1, 15)$.

An environmental study models the concentration of a pollutant in a lake using the function $C(t) = -0,05t^4 + 0,8t^3 - 3,6t^2 + 10$, where $t$ represents the months elapsed since the beginning of the year. Find out in which months the concentration of the pollutant is increasing and in which it is decreasing.

Prove mathematically that the polynomial function $f(x) = x^3 + 2x + 5$ is strictly increasing over its entire domain $\mathbb{R}$ and, therefore, has no tangent line parallel to the x-axis.

Given the parabolas $f(x) = x^2$ and $g(x) = -x^2 + 8x - 16$, determine the equations of all lines that are tangent to both curves simultaneously.

The goal is to reconstruct the analytical expression of a third-degree polynomial $P(x)$ whose coefficients are unknown, but for which the following information about its behavior and derivatives is available:

1. Its second derivative is $P''(x) = 6x - 12$.

2. It has a stationary point at the x-coordinate $x = 1$.

3. At this stationary point, the function takes the value $P(1) = 8$.

1. a) Knowing that differentiating a polynomial successively decreases its degree by one unit at each step, deduce by reverse path the general expression for $P'(x)$ from $P''(x)$ and find the unknown constant using the stationary point condition.
2. b) Following the same reverse reasoning, recover the complete formula for $P(x)$ by determining its constant term with the data $P(1) = 8$.
3. c) Prove that this polynomial has a second stationary point, find its coordinates $(x, y)$, and classify both stationary points (as local maximum or minimum), justifying it using the sign of the second derivative $P''(x)$ at each x-coordinate.

In a debate about the properties of cubic functions, Sergio makes the following statement: "Since the derivative function of any third-degree polynomial $f(x) = ax^3 + bx^2 + cx + d$ (with $a \neq 0$) is a second-degree polynomial, and according to the quadratic formula quadratic equations have two solutions, every third-degree polynomial must necessarily have two stationary points."

1. a) Rigorously refute Sergio's argument by explaining what error he makes regarding the solutions of a quadratic equation in the set of real numbers. Construct an explicit counterexample of a third-degree polynomial (with simple numerical coefficients) that has no stationary points, and prove it using its derivative.
2. b) Consider the family of polynomials depending on the real parameter $k$:

Calculate the derivative $g'(x)$ and determine with justification for which values of the parameter $k$ the function $g(x)$:

• Has exactly two distinct stationary points.

• Has a single stationary point with a horizontal tangent.

• Is strictly increasing on all of $\mathbb{R}$ because it has no stationary points.