Exercises: Tangent line to a curve

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

Find the equation of the tangent line to the curve $f(x) = x^2 + 1$ at the point with x-coordinate $x = 1$, as shown in the figure, writing it in the form $y - f(1) = f'(1) \cdot (x - 1)$.

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

At which point on the graph of the function $f(x) = x^2 - 4x + 3$ is the tangent line horizontal? State the coordinates of the point.

Find the point on the parabola $f(x) = x^2$ at which the tangent line is parallel to the line $y = 4x - 1$.

State, giving reasons, whether the following statement is true or false: "The tangent line to a curve at a point cannot intersect the curve again at another point." If it is false, give a counterexample using the function $f(x) = x^3$ at the point with x-coordinate $x = 1$.

Marta tries to calculate the equation of the tangent line to the curve $f(x) = x^2 - 3x + 2$ at the point with abscissa $x = 2$. On her exam, she writes the following:

"$f'(x) = 2x - 3$

Equation: $y - f(2) = f'(x)\cdot(x - 2)$

Since $f(2) = 0$, I substitute: $y - 0 = (2x - 3)(x - 2)$

Therefore, the tangent line is $y = 2x^2 - 7x + 6$".

1. a) Identify the conceptual error Marta makes and explain why her final result contradicts the geometric definition of a tangent line itself.
2. b) Correct the exercise by properly calculating the explicit equation of the tangent line and that of the normal line at $x = 2$.

Consider the function $f(x) = 4 - x^2$.

1. a) Before computing any derivative, make the following qualitative estimates by reasoning about the shape of the parabola and its vertex:

1. Will the tangent line at $x = 1$ have a positive or negative slope? Justify it based on the increase/decrease of the function.

2. Will the normal line at that point have a positive or negative slope?

3. Will the y-intercept ($n$) of the tangent line be greater or less than $4$?

1. b) Algebraically calculate the equation of the tangent line and that of the normal line to the graph at $x = 1$.
2. c) Check whether the numerical values obtained for the slope and the y-intercept confirm your estimates from part a).

A cyclist training in a velodrome travels $s(t) = 2t^2 + 3t$ metres during the first $t$ seconds.

1. a) Find the equation of the tangent line to the graph of $s(t)$ at time $t = 2$.

1. b) What physical quantity does the gradient of this tangent line represent? State its value and its units.

The figure shows the branch of the hyperbola $f(x) = \frac{1}{x}$ corresponding to $x > 0$ and the point with x-coordinate $x = 2$. Find the equation of the tangent line at this point and verify that it intersects the y-axis at $y = 1$.

Find the equation of the tangent line to the curve $f(x) = \ln x$ at the point with x-coordinate $x = 1$. How is this line related to the values of $\ln x$ when $x$ is very close to $1$?

Find the points on the graph of $f(x) = x^3 - 3x + 2$ at which the tangent line is parallel to the line $y = 9x + 5$, and state the coordinates of each one.

Consider the function $f(x) = x^2 - 2x + 3$.

1. a) Find the equation of the tangent line at the point with x-coordinate $x = 3$.

1. b) Determine the points of intersection of this tangent line with the coordinate axes.

1. c) Calculate the area of the triangle formed by the tangent line and the two axes.

The normal line to a curve at a point is the line perpendicular to the tangent line at that point. Find the equations of the tangent line and the normal line to the parabola $f(x) = x^2$ at the point with x-coordinate $x = 1$.

An AEMET station in the Sierra de Guadarrama records the ambient temperature $T$ (in $^{\circ}\text{C}$) during a spring day. Between $08:00$ and $18:00$, the data fit the function $T(t) = -0,1t^2 + 2,6t - 4,9$, where $t$ is the time in hours elapsed since midnight ($8 \le t \le 18$).

1. a) Interpret the physical meaning of the derivative $T'(a)$ and of the tangent line to the curve at any instant $t = a$. State the units in which this slope is measured.
2. b) Find the equation of the tangent line to the temperature graph at $10:00$ in the morning ($t = 10$).
3. c) Use the tangent line found as a linear approximation to estimate the forecast temperature at $10:15$ ($t = 10,25$). Compare the result with the exact value provided by the model $T(10,25)$ and explain why the tangent line is a useful short-term prediction tool.

Find the coefficients $a$, $b$, and $c$ of the polynomial function $f(x) = ax^2 + bx + c$ knowing that it simultaneously satisfies three geometric conditions:

1. Its graph intersects the y-axis at the point $(0, 3)$.

2. The tangent line to the curve at the point with abscissa $x = 2$ is completely horizontal.

3. At the point with abscissa $x = -1$, the normal line to the curve is parallel to the line with equation $x - 6y + 12 = 0$.

First, write the mathematical translation of each condition in terms of $f$ and $f'$, and solve the resulting system in detail.

From the point $P(0, -4)$, outside the parabola $f(x) = x^2$ and shown in the figure, tangent lines to the curve are to be drawn.

1. a) Write down the equation of the tangent line to the parabola at a general point with x-coordinate $x = a$.

1. b) Apply the condition that it passes through $P$ and find the equations of the two possible tangent lines, stating the points of tangency.

Determine the point on the curve $f(x) = \ln x$ at which the tangent line passes through the origin, and write down the equation of this tangent. (Hint: formulate the tangent line at a general point with x-coordinate $x = a$ and impose the condition that it passes through $(0, 0)$.)

The parabolas $f(x) = x^2$ and $g(x) = 2 - x^2$ intersect forming an angle, which is the angle formed by their tangent lines at the point of intersection, as shown in the figure.

1. a) Verify that the two curves intersect at the points with x-coordinates $x = -1$ and $x = 1$.

1. b) Calculate the gradients of the tangents to both curves at the point with x-coordinate $x = 1$.

1. c) Using the formula for the angle between two lines, $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 \cdot m_2} \right|$, find the angle formed by the two curves at that point.

It is required to estimate the value of $\sqrt{102}$ without a calculator.

1. a) Find the equation of the tangent line to $f(x) = \sqrt{x}$ at the point with x-coordinate $x = 100$.

1. b) Use this line to give an estimate of $\sqrt{102}$ to two decimal places.

1. c) Given that the actual value rounded to four decimal places is $10,0995$, comment on whether the tangent approximation is an overestimate or an underestimate and explain why.

The goal is to determine the equations of the tangent lines to the parabola $f(x) = x^2 - 4x + 5$ passing through the exterior point $P(2, -3)$ (first check that $P$ does not belong to the curve). Two students propose different approaches:

• Method 1 (Analytic geometry): Write the pencil of lines passing through $P$, $y + 3 = m(x - 2)$, substitute into the parabola to form a quadratic equation in $x$, and set the discriminant to zero ($\Delta = 0$) to enforce tangency.

• Method 2 (Differential calculus): State the generic tangent line at an unknown point $(a, f(a))$, impose that it passes through $P(2, -3)$, and solve for the values of $a$.

1. a) Apply Method 1 and obtain the equations of the tangent lines.
2. b) Apply Method 2 and verify that exactly the same solutions are reached.
3. c) Compare both methods and justify which of the two is preferable if the curve were $g(x) = x^3 - 2x$ or $h(x) = e^x$, explaining precisely why the other method ceases to be applicable or becomes impracticable.

Consider the family of parabolas $f_k(x) = x^2 + k$, where $k \in \mathbb{R}$ is a constant. From the exterior point $P(0, -1)$, tangent lines are drawn to the curve.

1. a) Derive the equation of the tangent line at a generic point $(a, f_k(a))$ and impose that it passes through $P(0, -1)$ to obtain an equation in the unknown $a$.
2. b) Reasonably discuss the number of tangent lines that can be drawn from $P(0, -1)$ depending on the values of the parameter $k$ (distinguishing when there are two, one, or none).
3. c) For the values of $k$ where two distinct tangent lines exist, prove that these lines are perpendicular to each other for a single value of $k$. Calculate this value and the explicit equations of both perpendicular tangents.