Exercises: Graphical interpretation of the derivative

Calculate the equation of the tangent line to the graph of the function $f(x) = x^2 - 4x + 5$ at the point with x-coordinate $x = 3$.

Determine the equation of the normal line to the curve $f(x) = x^3 - 2x$ at the point with x-coordinate $x = -1$.

Find the coordinates of the points on the curve $f(x) = \frac{x^3}{3} - 2x^2 + 3x$ where the tangent line is horizontal.

Look at the graph of the function $f(x)$ and the tangent line drawn at the point with x-coordinate $x = 1$. From the coordinates of the points $A$ and $B$ shown on the line, determine the value of the derivative $f'(1)$.

Look at the graph of the polynomial function $f(x)$. Without calculating its analytical expression, determine the sign of the derivative $f'(x)$ at the points with x-coordinates $x = -2$, $x = 0$ and $x = 2$, stating whether the function is increasing or decreasing at those points.

Find the equation of the tangent line to the graph of the rational function $f(x) = \frac{2x+1}{x-3}$ at the exact point where the graph intersects the y-axis.

The figure shows the graph of a quadratic function $f(x)$. From its visible geometrical elements (vertex and axes intercepts), deduce the analytical expression of its derivative function $f'(x)$.

Determine at which exact point on the graph of the radical function $f(x) = \sqrt{x+4}$ the tangent line has a gradient equal to $m = \frac{1}{4}$.

The trajectory of an inspection drone in a vertical plane follows the curve defined by $y = x^3 - 6x^2 + 9x + 2$, where $x$ and $y$ are measured in meters. Determine at which points along its path the direction of its movement is parallel to the line with equation $y = -3x + 5$.

An architect is designing the main arch of a bridge that takes the shape of a parabola with equation $f(x) = ax^2 + bx + c$. It is known that the arch passes through the point $(0, 5)$ and that at the point $(2, 9)$ the tangent line has the equation $y = 4x + 1$. Calculate the exact values of the parameters $a$, $b$ and $c$.

A light ray strikes a parabolic mirror whose cross-section is given by the function $f(x) = \frac{1}{4}x^2$. If the ray hits at the point with x-coordinate $x = 2$, find the equation of the normal line to the surface at that point (which is fundamental in geometrical optics to calculate the angle of reflection).

The figure shows the graph of the derivative function $f'(x)$ of a polynomial function $f(x)$. From it, determine analytically the intervals where the original function $f(x)$ is increasing and decreasing, and state the x-coordinates at which $f(x)$ has its relative extrema.

The number of subscribers to a science channel on a video platform grows according to the exponential function $S(t) = 500 e^{0.2t}$, where $t$ is measured in weeks. Find the equation of the tangent line to the curve at time $t = 5$ and use it to estimate linearly the number of subscribers at $t = 6$.

Determine the coordinates of the points on the curve defined by $f(x) = \frac{x^3}{3} - \frac{x^2}{2} - x + 1$ where its tangent line forms an exact angle of $45^\circ$ with the positive x-axis.

Given the polynomial functions $f(x) = x^2 - 3x + 2$ and $g(x) = -x^2 + 7x - 6$, first find the intersection points of their graphs. Next, calculate the equation of the tangent line to the function $f(x)$ at the intersection point with the largest x-coordinate.

In the design of a mountain road, a straight section given by the equation $y = 2x$ (for $x \le 1$) must join smoothly with another straight section with equation $y = -x + 6$ (for $x \ge 3$) by means of a parabolic transition curve $f(x) = ax^2 + bx + c$ strictly defined on the interval $[1, 3]$. Calculate the values of $a$, $b$ and $c$ so that the road is continuous and differentiable at both junction points ($x=1$ and $x=3$).

Find the equations of all the tangent lines to the curve $f(x) = x^2 - 2x + 3$ that pass through the origin $(0,0)$. Note: keep in mind that the origin does not lie on the curve.

Determine the exact values of the parameters $a$, $b$, $c$ and $d$ so that the cubic function $f(x) = ax^3 + bx^2 + cx + d$ has a relative extremum at the point $(0, 2)$ and its tangent line at the point $(1, 0)$ is strictly parallel to the line $y = -3x$.

Prove algebraically that the graph of the rational function $f(x) = \frac{x^2 + x + 1}{x-1}$ has no tangent line anywhere in its domain that is parallel to the line with equation $y = 2x$.

Calculate the exact area of the triangle formed by the x-axis, the tangent line and the normal line to the logarithmic curve $f(x) = \ln(x)$ at the point with x-coordinate $x = e$.

Determine for which set of values of the parameter $k$ the rational function $f(x) = \frac{x^2 + kx + 1}{x-2}$ has no horizontal tangent lines at any point in its domain.

The graphs of the quadratic functions $f(x) = x^2 + ax + b$ and $g(x) = -x^2 + 2x - 1$ are tangent to each other at the point with x-coordinate $x = 1$ (that is, they share the same point and the same tangent line at that point). Calculate the values of $a$ and $b$, and state the equation of this common tangent line.

The figure represents the graph of the derivative function $f'(x)$ (composed of two line segments) of a continuous function $f(x)$. Based exclusively on this graph, identify the x-coordinates of the relative extrema of $f(x)$, classifying them as maximums or minimums, and state the x-coordinate of the point of inflection of the original function.

You can do it on paper and send a photo with your answer.

Let $f(x) = x^3 - 3x^2 + kx$ be the function. Determine the value of the parameter $k$, given that the tangent line to the graph of $f(x)$ at its point of inflection is parallel to the angle bisector of the second and fourth quadrants.