Curve Sketching: Analyzing 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. The domain of definition of the function.

2. The points of intersection with the coordinate axes.

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

Determine the period of the trigonometric function $f(x) = \sin(2x)$. Then, state with reasons whether the function has any kind of 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 the limits at infinity.

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

1. Check if it has vertical asymptotes.

2. Check if it has horizontal asymptotes.

3. If it does not have a horizontal asymptote, find out if 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 when $x \to \infty$.

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

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

You are asked to:

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

2. 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 instant does the battery reach its maximum charge level?
2. b) What value does the battery charge tend to if it is left connected indefinitely? Justify the answer by calculating limits.

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

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

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 of increase and decrease, as well as the coordinates of its maxima and minima if it has any.

Analyze 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.

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 points of inflection. Use the symbol $\cup$ to denote the intervals where the function is convex (upward curvature) and $\cap$ for the intervals where it is concave (downward curvature).

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

• It intersects 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.

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

• Domain: $\mathbb{R} - \{0\}$

• Intersections with the axes: $(2, 0)$ 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 asymptote $y = 1$ when $x \to \infty$ and when $x \to -\infty$.

• $f'(x) < 0$ over its entire domain.

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

1. a) The intervals of increase and decrease of the original function $f(x)$.
2. b) The abscissa 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 over all $\mathbb{R}$. From the visual information provided:

1. State the intervals of increase and decrease of the original function $f(x)$.

2. Determine the abscissae of the relative extrema of $f(x)$, classifying them as maxima or minima.

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

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

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 value of $a$ obtained, study whether the function has any horizontal asymptote when $x \to +\infty$.

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

1. Express the function as a piecewise function.

2. Calculate the intervals of increase and decrease.

3. Determine the relative and absolute maxima and minima in the interval $[0, 4]$.

4. Study the curvature and the points of inflection. Take special care with the point where the function changes its definition.

Let the piecewise function be:

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

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, study whether the function has any oblique asymptote when $x \to -\infty$. If so, find it.

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

Consider the function $f(x) = \ln(x^2 + k)$, 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 this value of $k$, study the domain of the function and check if it has any type of symmetry (even or odd).

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 $x = 4$.

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 have been found, study the existence of an oblique asymptote and determine the relative position of the curve with respect to it.

Given the function $f(x) = (x^2 - 1) \cdot e^{-x}$, carry out an exhaustive study for its graphical representation by analyzing:

1. Domain and points of intersection with the coordinate axes.

2. Behavior at infinity (limits and asymptotes).

3. Intervals of monotonicity and relative extrema (abscissae and ordinates).

4. Points of inflection and convexity.

5. Use all the previous information to draw the graph of the function, especially highlighting the horizontal asymptotic behavior.

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

1. Analyze 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 over its entire domain except in the interval $(-1, 1)$, where it is convex.

5. Sketch the graph of the function integrating all the previous elements.

A team of engineers is designing the profile of a jump ramp for a skateboarding competition. The lateral 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 with respect to the ground.

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

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

A rectangle of maximum area is to be inscribed 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 abscissa $x$ of one of its vertices located on the $OX$-axis.

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

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