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 meters 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 pesos, of manufacturing $x$ units of a handcrafted product is given by $C(x) = 3\,000 + 240x - 1x^2$. 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.

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 meters. Calculate the slope of the cable at the point located $40$ meters 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 upward. Its height in meters 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.

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 meters. 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.