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

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.

Given the quadratic function $f(x) = -x^2 + 4x - 1$:

1. a) Without using derivatives, determine the coordinates of the vertex of the parabola and its orientation (opening upwards or downwards). From the relative position of the abscissa $x = 3$ with respect to the vertex, estimate in a reasoned manner whether the slope of the tangent line at $x = 3$ must be positive, negative, or zero, and what sign you would expect for the slope of the normal line.
2. b) Calculate analytically using the derivative the equations of the tangent line and the normal line to the graph of $f$ at the point with abscissa $x = 3$.
3. c) Check whether the analytical results confirm your estimates from part a) and deduce, by observing the $y$-intercept of the obtained tangent line, whether this line intersects the $Y$-axis above or below the vertex of the parabola.

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 metres. 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 $x=3$).

During a heatwave day in Cork, Met Éireann (the Irish Meteorological Service) records the evolution of the ambient temperature $T$ (in degrees Celsius, $^{\circ}\text{C}$) between 10:00 and 18:00 solar hours ($t \in [10, 18]$), which is modeled by the function:

1. a) Obtain the expression of the derivative function $T'(t)$ and interpret its physical meaning in terms of the instantaneous rate of change of temperature.
2. b) Find the equation of the tangent line to the graph of $T$ at $t = 12$ and at $t = 16$.
3. c) Based on the sign of the slope of each of these two tangents, reason whether the temperature was rising or falling at those instants and calculate at what exact solar hour the maximum temperature of the day occurred, justifying why the tangent line at that point is necessarily horizontal.

Consider the cubic function $f(x) = x^3 + a\times x + b$, where $a$ and $b$ are unknown real numbers.

1. a) Determine the values of $a$ and $b$ knowing that the tangent line to the graph of $f$ at the point with abscissa $x = 1$ is $y = 5x - 7$. Justify in detail the two independent conditions (one of contact or membership and another of differentiability) that you must set up.
2. b) Once the function is determined with the obtained coefficients, find out if there is any other point on the graph of $f$ whose tangent line is parallel to the given line $y = 5x - 7$. If it exists, calculate its coordinates and the equation of that tangent line.

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.

You want to find the equations of the tangent lines to the curve $f(x) = x^2 + 1$ that pass through the exterior point $P(1, -2)$. Two students propose different strategies:

• Aoife's method (differential approach): She sets up a generic point of tangency $T(a, f(a))$ on the curve, writes the equation of the tangent line in terms of $a$ using the derivative, and requires that the point $P(1, -2)$ satisfy this equation to solve for the possible values of $a$.

• Cian's method (algebraic approach): He writes the family of lines passing through $P(1, -2)$, given by $y - (-2) = m\times(x - 1)$, equates its expression with that of the parabola to set up the system of intersections ($x^2 + 1 = m(x - 1) - 2$), and requires that the discriminant of the resulting quadratic equation be zero ($\Delta = 0$) to force a single double point of contact.

1. a) Solve the problem applying both methods and verify that they lead to exactly the same tangent lines.
2. b) Critically compare both methods and answer with justification: if the given function were $g(x) = x^3 - 2x$ instead of a quadratic function, which of the two methods would no longer be directly applicable and for what conceptual reason?

Determine the real coefficients $a$, $b$, $c$, and $d$ of the polynomial function $f(x) = a\times x^3 + b\times x^2 + c\times x + d$ knowing that it simultaneously satisfies the following three conditions:

1. The tangent line to the graph of $f$ at the point with abscissa $x = 0$ has the equation $y = 2x - 1$.

2. The normal line to the graph of $f$ at the point with abscissa $x = 1$ is parallel to the line with equation $x + 8y = 0$.

3. The graph of $f$ passes through the point $(2, 3)$.

After finding the definitive analytical expression of $f(x)$:

1. a) Justify what direct geometric information the $y$-intercept and the slope of the tangent line given in condition 1 provide regarding the coefficients $c$ and $d$.
2. b) Deduce analytically whether there is any other point on the graph of $f$ where the tangent line has exactly the same slope as at $x = 0$ and, if so, calculate the abscissa of that point.