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 $x = 3$.

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

Cian has to find the equation of the normal line to the graph of the function $f(x) = x^2 - 4x + 6$ at the point with abscissa $x = 3$. On his exam, he writes the following procedure:

«1. I calculate the ordinate of the point: $f(3) = 3^2 - 4\times 3 + 6 = 3$. The point of tangency is $(3, 3)$.

1. I find the derivative function: $f'(x) = 2x - 4$, so the slope of the tangent line is $f'(3) = 2\times 3 - 4 = 2$.

2. Since the normal line is perpendicular to the tangent line, its slope is the opposite, that is, $m_{\text{normal}} = -2$.

3. I apply the point-slope equation: $y - 3 = -2\times(x - 3) \implies y = -2x + 9$.»

1. a) Identify the conceptual mistake Cian made in step 3 and explain the correct geometric relationship between the slopes of two perpendicular lines when neither of them is horizontal nor vertical.
2. b) Correct the procedure and calculate the correct equation of the normal line.

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).

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.

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 $f'(x)$.

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 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.

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 $m = \frac{1}{4}$.

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.

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 $c$.

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 $x=3$).

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.