Information About a Function

Given the rational function $f(x) = \frac{x^2 + 3}{x^2 - 9}$, carry out a preliminary study of its basic elements by determining:

1. a) The domain of the function.

1. b) The points where it crosses the axes of coordinates.

1. c) The existence of symmetries (state whether it is even, odd or has no symmetry).

Determine the period of the trigonometric function $f(x) = \sin(2x)$. Next, state with reasons whether the function exhibits any symmetry with respect to the origin or the y-axis.

Given the function $f(x) = \frac{2x}{x-1}$, determine:

1. The equations of its vertical and horizontal asymptotes.

2. The position of the curve with respect to these asymptotes by studying the one-sided limits and limits at infinity.

Given the function $f(x) = \frac{x^2 - 1}{x}$, carry out a study of its infinite branches:

1. a) Check whether it has vertical asymptotes.

1. b) Check whether it has horizontal asymptotes.

1. c) If it has no horizontal asymptote, find out whether it has an oblique asymptote and find its equation.

Determine all the asymptotes (vertical, horizontal and oblique) of the function $f(x) = \frac{2x^2 + 1}{x - 3}$. If an oblique asymptote exists, calculate its equation in the form $y = mx + n$ and study the position of the curve with respect to this asymptote as $x \to \infty$.

A team of engineers is monitoring the temperature of an electronic component. The evolution of the temperature $T$ (in degrees Celsius) with respect to time $t$ (in minutes) is given by the function:

$T(t) = \dfrac{40t + 100}{t + 2}$ with $t \geq 0$

You are asked to:

1. a) Find the initial temperature of the component ($t=0$).

1. b) Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem (long-term temperature).

A team of engineers monitors the charge level of a new lithium battery using the function $B(t) = \frac{100t}{t^2 + 9} + 10$, where $B$ is the percentage of charge and $t$ is the time in hours ($t \ge 0$).

1. a) At what time does the battery reach its maximum charge level? Justify that it is a maximum.

1. b) What value does the battery charge approach if it is left plugged in indefinitely? Justify your answer by calculating limits.

Study the monotonicity (increasing and decreasing intervals) and calculate the relative extrema (maximums and minimums) of the following polynomial function:

$f(x) = x^3 - 3x + 2$

Analyse the curvature of the function $f(x) = x^4 - 6x^2 + 5$. To do this:

1. Find the second derivative $f''(x)$.

2. Determine the intervals of concavity ($\cup$) and convexity ($\cap$).

3. Find the x-coordinates of the points of inflection.

Carry out a complete study of the monotonicity and relative extrema of the rational function $f(x) = \frac{x^2 - 4x + 4}{x - 1}$. Determine precisely the intervals where the function is increasing and decreasing, as well as the coordinates of its maximums and minimums, if any.

Consider the polynomial function $f(x) = \frac{1}{4}x^4 - 2x^2 + 3$. Find the intervals of concavity and convexity, as well as the exact coordinates of its inflection points. Use the symbol $\cup$ to denote the intervals where the function is convex (curving upward) and $\cap$ for the intervals where it is concave (curving downward).

Consider the function $f(x) = \ln(x^2 + k)$, where $k$ is a real parameter.

1. a) Determine the value of $k$ so that the function has a point of inflection at $x = 1$.
2. b) For that value of $k$, study the domain of the function and check whether it exhibits any symmetry (even or odd).

Given the piecewise function:

$f(x) = \begin{cases} x^2 + 2x & \text{if } x \le 0 \\ \frac{ax}{x+1} & \text{if } x > 0 \end{cases}$

1. a) Calculate the value of $a$ so that the function is differentiable at $x = 0$.
2. b) For the obtained value of $a$, investigate whether the function has a horizontal asymptote as $x \to +\infty$.

Given the function $f(x) = (x^2 - 1) \times e^{-x}$, carry out a comprehensive study for its graphical representation by analysing:

1. Domain and axis intercepts.

2. Behavior at infinity (limits and asymptotes).

3. Monotonicity intervals and relative extrema (x- and y-coordinates).

4. Points of inflection and concavity/convexity.

5. Use all the above information to sketch the graph of the function, highlighting in particular the horizontal asymptotic behavior.

The following information is known about a function $f(x)$ that is continuous on all of $\mathbb{R}$:

• It crosses the $X$ axis at $x = -2$ and $x = 2$.

• It has a relative minimum at $(0, -4)$.

• It is decreasing on $(-\infty, 0)$ and increasing on $(0, +\infty)$.

• $\lim_{x \to +\infty} f(x) = +\infty$ and $\lim_{x \to -\infty} f(x) = +\infty$.

Draw a sketch of the graph that satisfies all these conditions. To help you, here are the axes with the known points; copy the axes onto your paper and draw the curve.

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

Sketch the graph of a continuous function $f(x)$ that simultaneously satisfies the following conditions obtained from its analytic study:

• Domain: $\mathbb{R} - \{0\}$

• Axes intercepts: $(2, 0)$ and $(-2, 0)$.

• Vertical asymptote at $x = 0$, with $\lim_{x \to 0^-} f(x) = -\infty$ and $\lim_{x \to 0^+} f(x) = +\infty$.

• Horizontal asymptotes: $y = -1$ as $x \to +\infty$ and $y = 1$ as $x \to -\infty$.

• $f'(x) < 0$ on its entire domain.

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

Given the graph of the derivative function $f'(x)$ shown below (a straight line passing through the points $(-2, 0)$ and $(0, 4)$), state with reasons:

1. a) The intervals where the original function $f(x)$ is increasing and decreasing.
2. b) The x-coordinate of the relative extrema of $f(x)$ and their type (maximum or minimum).
3. c) The interval of concavity or convexity of $f(x)$.

The following graph corresponds to the derivative function $f'(x)$ of a continuous function $f(x)$ defined on all of $\mathbb{R}$. Using the visual information provided:

1. State the intervals where the original function $f(x)$ is increasing and decreasing.

2. Determine the x-coordinates of the relative extrema of $f(x)$, classifying them as maximums or minimums.

3. Study the curvature of $f(x)$ and locate its points of inflection.

4. Given that $f(0) = 1$, sketch the graph of the original function $f(x)$, justifying the shape obtained.

A team of engineers is designing the profile of a jump ramp for a skateboarding competition. The side profile follows the curve of the function $f(x) = \ln(x^2 - 2x + 2)$ in the interval $[0, 3]$.

1. Find the point on the ramp that is at the lowest height above the ground.

2. Determine the points on the ramp where the slope is maximum and minimum. What physical interpretation do these points have in the context of skateboarding?

3. Investigate whether the function has any point of inflection in this interval and explain how the skater's sensation changes when passing through it.

We wish to inscribe a rectangle of maximum area in the region of the first quadrant bounded by the $OX$-axis, the $OY$-axis and the graph of the function $f(x) = 4e^{-0.5x}$.

1. Express the area of the rectangle as a function of the x-coordinate $x$ of one of its vertices located on the $OX$-axis.

2. Calculate the dimensions of this rectangle so that its area is maximum.

3. Draw a sketch of the situation using the following visual diagram.

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

Study the continuity and differentiability of the function $f(x) = x \times |x - 2|$. From this study:

1. Express the function as a piecewise defined function.

2. Calculate the intervals where the function is increasing and decreasing.

3. Determine the relative and absolute maximums and minimums in the interval $[0, 4]$.

4. Study the curvature and points of inflection. Pay special attention to the point where the function changes its definition.

Consider the function $f(x) = \frac{ax^2 + bx}{x + c}$. Determine the values of the real parameters $a$, $b$, and $c$ knowing that the following conditions are simultaneously met:

1. The line $x = 2$ is a vertical asymptote of the graph of the function.

2. The function has a relative extremum at the point with abscissa $x = 3$.

3. The tangent line to the graph at the point of intersection with the $OY$ axis is parallel to the line $y = -2x + 5$.

Once the parameters are found, study the existence of an oblique asymptote and determine the relative position of the curve with respect to it.

Let the piecewise function be defined by:

$f(x) = \begin{cases} \frac{x^2 - kx}{x-1} & \text{if } x < 0 \\ \ln(x^2 + 1) + ax + b & \text{if } x \ge 0 \end{cases}$

1. Calculate the values of $a$ and $b$ so that the function is differentiable at $x = 0$, knowing that in the first piece the parameter $k$ takes the value $1$.

2. For the values found, investigate whether the function has an oblique asymptote as $x \to -\infty$. If so, find it.

3. Does the function have any relative extremum in the interval $(0, +\infty)$? Justify your answer.

Consider the function $f(x) = \frac{x^2 - 4}{x^2 - 1}$.

1. Analyse its symmetries and domain.

2. Determine all its asymptotes and the position of the graph with respect to them.

3. Find the points on the graph where the tangent line is horizontal.

4. Prove that the function is concave in its entire domain except in the interval $(-1, 1)$, where it is convex.

5. Sketch the graph of the function, combining all the previous elements.

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