Derivative of a polynomial

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

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

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

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.

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

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.

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:

• Cian 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.

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

1. a) Critically compare both approaches: justify why Cian'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.

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.

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

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.

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 $(1, -2)$.

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.

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.

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, $C'(x)$.

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.

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

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 $P(1, -2)$.