Types of functions

Given the polynomial function $f(x) = x^3 - 3x^2 + 4$, carry out a basic study of its behaviour following these steps:

1. a) Calculate the axis intercepts.
2. b) Find the intervals of increase and decrease using the first derivative $f'(x)$.
3. c) Determine the coordinates $(x, y)$ of its relative maxima and minima.

Determine the values of the parameters $a$ and $b$ so that the function $f(x) = ax^3 + bx^2 - x + 4$ has an inflection point at $x = 1$ and the slope of the tangent line at that point is $1$. Once you have found the parameters:

1. a) Study the monotonicity of the resulting function.

1. b) Determine whether it has any relative maximum or minimum.

1. c) Justify whether the function has asymptotes.

Consider the rational function $f(x) = \frac{2x + 4}{x - 3}$. To understand its structure before graphing it, analyze its main elements. The figure shows part of its graph; the tick marks on the axes are 1 unit apart.

1. a) Determine its domain.

1. b) Calculate the equations of its vertical and horizontal asymptotes.

1. c) Study the symmetry of the function. Is it even, odd, or does it show no symmetry?

Given the rational function $f(x) = \frac{2x^2}{x^2 - 9}$, carry out a complete study for its graphical representation following these steps:

1. a) Find the domain of definition and the points where the graph crosses the coordinate axes.

1. b) Study whether the function has any symmetries.

1. c) Determine the vertical and horizontal asymptotes and the position of the curve with respect to them.

1. d) Calculate the intervals of increase and decrease, and determine the relative maxima and minima.

1. e) Sketch the graph of the function based on the previous results.

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

Consider the function $f(x) = \frac{x^2 + 3}{x - 1}$. Find:

1. Find the equations of its asymptotes. If an oblique asymptote exists, determine the position of the curve with respect to it as $x \to +\infty$ and $x \to -\infty$.

2. Find the intervals of concavity and convexity, and check whether there are points of inflection.

3. Use the information on increasing/decreasing intervals and extrema to draw an accurate sketch of the graph.

Determine the parity of the following functions, justifying mathematically whether they are even ($f(x) = f(-x)$), odd ($f(-x) = -f(x)$), or neither:

1. a) $f(x) = x^2 \times \cos(x)$
2. b) $g(x) = \frac{x^3}{x^2 + 1}$
3. c) $h(x) = e^x - x$

To study the asymptotes and the relative position of the curve of the rational function $f(x) = \dfrac{2x^2 + 3x - 1}{x + 2}$, two students propose different strategies:

• Method A (Alba): Perform polynomial division between the numerator and denominator to decompose the function into the form $f(x) = c(x) + \dfrac{r(x)}{d(x)}$.

• Method B (Bruno): Apply the classical limits $m = \lim_{x \to \pm\infty} \dfrac{f(x)}{x}$ and $n = \lim_{x \to \pm\infty} [f(x) - m \times x]$.

1. a) Compare both methods and justify why Alba's method is more efficient in this case to simultaneously determine the equation of the oblique asymptote and the position of the graph relative to it at $+\infty$ and $-\infty$ without needing to calculate additional limits.
2. b) Apply Alba's method to find the asymptote line and analytically determine whether the curve lies above or below this asymptote at each of the two infinities.

Functions with absolute values often have points where the graph changes direction abruptly. Let $f(x) = |x - 4|$ be a function.

1. a) Express the function as a piecewise-defined function.

1. b) State the point on the graph where the function is continuous but not differentiable (corner point).

1. c) Sketch its graph, indicating the vertex.

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

Given the absolute value function $f(x) = |x^2 - 4x + 3|$:

1. a) Express the function as a piecewise-defined function.

1. b) Study the differentiability of the function at the points $x = 1$ and $x = 3$. What happens geometrically at those points?

1. c) Find the relative extrema of the function and graph it.

The domain of a function is fundamental for sketching its graph. Find the domain of the following transcendental functions:

1. a) $f(x) = \ln(x^2 - 1)$
2. b) $g(x) = e^{\frac{1}{x}}$
3. c) $h(x) = \sqrt{x + 5}$

Briefly explain why restrictions exist in each case.

A biologist studies the evolution of a bacteria population in a controlled environment. The number of bacteria (in thousands) is given by the function $P(t) = \frac{40t}{t + 2}$, where $t$ is the time in hours since the start of the experiment ($t \geq 0$). The figure shows the graph of $P(t)$.

1. a) What is the population at the start of the experiment ($t = 0$)?

1. b) What is the limit of the population as time grows indefinitely? Interpret the result in terms of asymptotes.

1. c) Does the population increase or decrease over time? Justify your answer using the derivative $P'(t)$.

Let $f(x) = (x - 2)e^x$ be the function. Carry out a detailed study of it in order to sketch its graph, including:

1. Domain and axis intercepts.

2. Horizontal asymptotes and behaviour at infinity.

3. Intervals of monotonicity and relative extrema.

4. Points of inflection and curvature.

Analyse and sketch the logarithmic function $f(x) = \ln(x^2 + 1)$. In your analysis, you must include:

1. Proof that the function is even. How does this affect its graphical representation?

2. Determination of the intervals of increase and decrease.

3. Study of the curvature and finding the points of inflection.

4. Does the function have any type of asymptotes? Justify your answer.

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

Given the function $f(x) = \dfrac{\ln(x)}{x}$, answer the following questions by combining logical deduction and formal calculation:

1. Estimation phase (without differentiating or calculating formal limits yet):

a) From the domain of the logarithmic function and knowing that $\ln(x) \to -\infty$ as $x \to 0^+$, estimate the behavior of the curve near the $Y$-axis. Will there be a vertical asymptote? Where will the branch head towards?

b) The notes emphasize that the logarithm "grows much slower than any polynomial". Based on this, estimate what value $f(x)$ will approach as $x \to +\infty$ and deduce whether a horizontal asymptote will exist.

c) Knowing that $f(1) = 0$, that $f(x) < 0$ for $x \in (0, 1)$, and that for $x > 1$ the function is positive but must approach the horizontal asymptote as $x \to +\infty$, deduce whether at least one relative maximum must necessarily exist and estimate in which interval of $x$ it should be located.

1. Verification phase:

d) Analytically confirm your estimates by calculating $\lim_{x \to 0^+} f(x)$ and $\lim_{x \to +\infty} f(x)$.

e) Calculate the derivative $f'(x)$, determine the exact coordinates $(x, y)$ of its relative extremum, and check whether your previous deduction regarding the existence and position of the maximum was correct.

Consider the piecewise function:

$$f(x) = \begin{cases} \frac{1}{x-2} & \text{if } x < 0 \\ x^2 - 2x - \frac{1}{2} & \text{if } x \geq 0 \end{cases}$$

1. Study the continuity and differentiability of the function at $x = 0$.

2. Find the asymptotes of the function in its domain of definition.

3. Calculate the local minima and maxima of each section in its corresponding interval.

4. Sketch the graph of the function.

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

During a winter thermal inversion episode in Dublin, the concentration of nitrogen dioxide ($NO_2$) in the air, measured in micrograms per cubic meter ($\mu\text{g/m}^3$), is monitored over a full day ($t \in [0, 24]$ hours). The air quality department models its evolution using a piecewise function that links the morning accumulation phase due to traffic with the evening dispersion:

$$C(t) = \begin{cases} 40 + 10t - 0.5t^2 & \text{if } 0 \le t \le 12 \\ 124 \times e^{-0.1(t-12)} - 36 & \text{if } 12 < t \le 24 \end{cases}$$

1. a) Check whether the model guarantees continuity at the transition point ($t = 12$). Next, study differentiability at that point and determine whether the graph exhibits a corner point or a smooth transition. Interpret the physical meaning of this fact.
2. b) Find at what time of day the maximum pollutant concentration is reached and calculate its exact value.
3. c) If the exponential dispersion model were to continue indefinitely ($t \to +\infty$), would it have a horizontal asymptote? Analytically justify why the exponential term collapses to zero and what the theoretical residual pollution level would be according to that asymptote.

Consider the piecewise-defined function:

$$f(x) = \begin{cases} \dfrac{x^2 - 4}{x+2} & \text{if } x < -2 \\ a \times e^{x+2} + b x & \text{if } -2 \le x < 0 \\ \dfrac{x+1}{x^2+1} & \text{if } x \ge 0 \end{cases}$$

Carry out a complete study of the function following these steps:

1. Find the values of $a$ and $b$ so that the function is continuous at $x = -2$ and has a relative extremum at $x = -1$.

2. For the values found, study differentiability at $x = -2$ and $x = 0$.

3. Calculate the asymptotes of the function on its entire domain.

4. Determine the intervals of increase and decrease, as well as the relative maxima and minima on the interval $(-\infty, 0)$.

Given the function $f(x) = x^4 - 2x^2$, study its curvature by following these steps:

1. a) Calculate the second derivative $f''(x)$.
2. b) Find the intervals of concavity ($\cup$) and convexity ($\cap$).
3. c) Find the points of inflection of the function.

Consider the function $f(x) = (x-2)^2 \times e^x$.

1. Calculate the axis intercepts and the asymptotes of the function.

2. Determine the intervals of monotonicity and local extrema (maximums and minimums).

3. Find the intervals of curvature and the points of inflection.

4. Knowing that $e^{-2} \approx 0.135$ and $e^0 = 1$, sketch the graph of the function.

5. Calculate the area of the triangle formed by the origin of coordinates and the points on the graph where the tangent is horizontal.

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 eight conditions:

• Its domain is $\mathbb{R} - \{1\}$.

• It has a vertical asymptote at $x = 1$ such that $\lim_{x \to 1^-} f(x) = +\infty$ and $\lim_{x \to 1^+} f(x) = -\infty$.

• It has an oblique asymptote $y = x$ as $x \to +\infty$.

• It has a horizontal asymptote $y = -2$ as $x \to -\infty$.

• $f'(x) < 0$ on the interval $(-\infty, -2)$ and $f'(x) > 0$ on $(-2, 1) \cup (1, +\infty)$.

• It has a relative minimum at the point $(-2, -4)$.

• $f''(x) > 0$ on $(-3, 1)$ and $f''(x) < 0$ on $(-\infty, -3) \cup (1, +\infty)$.

• It intersects the $x$-axis only at the points $x = 0$ and $x = 2$.

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