Real numbers

Classify the following real numbers, indicating which number sets they belong to: natural numbers ($\mathbb{N}$), integers ($\mathbb{Z}$), rational numbers ($\mathbb{Q}$), or irrational numbers ($\mathbb{I}$). Briefly justify your answer for decimal numbers and roots:

• $a) \ - \sqrt{16}$

• $b) \ \frac{15}{3}$

• $c) \ 2.131313...$

• $d) \ \sqrt{7}$

• $e) \ \frac{\pi}{2}$

• $f) \ -4.25$

Simplify the following expressions using the properties of powers and roots, giving the result in the form of a power with a fractional exponent:

• $a) \ \sqrt[3]{x^2} \times \sqrt{x}$

• $b) \ \frac{x^3}{\sqrt[4]{x}}$

• $c) \ (\sqrt[5]{x^2})^3$

Simplify the following expression involving radicals and powers, expressing the final result as a single power of base $a$ with a rational exponent:

Note: Assume that $a > 0$.

The design of a triangular metal part requires its sides to measure $\sqrt{18}$ cm, $\sqrt{50}$ cm, and $\sqrt{98}$ cm.

a) Simplify each of the radicals by extracting factors.

b) Calculate the exact perimeter of the part, expressing it as a single radical.

c) If the material used costs $150$ pesos per centimeter of perimeter, calculate the total cost rounded to two decimal places.

Using the properties of logarithms, expand the following expression in terms of $\log x$, $\log y$, and $\log z$:

Then, given that $\log x = 0.5$, $\log y = 1.2$, and $\log z = 0.8$, calculate the numerical value of $L$.

Represent the following intervals and rays on the real number line, and express them using set notation (using inequalities). For $I_3$, which already appears with inequalities, also write it in interval notation.

• $I_1 = (-3, 2]$

• $I_2 = [0, +\infty)$

• $I_3 = \{x \in \mathbb{R} \mid -1 < x < 4\}$

Given the following subsets of the real line:

$A = \{ x \in \mathbb{R} : |x - 2| < 3 \}$

$B = [ -1, 4 ]$

a) Express the set $A$ using interval notation.

b) Calculate and express using interval notation the set resulting from the intersection $A \cap B$.

c) Calculate and express using interval notation the set resulting from the union $A \cup B$.

Calculate the distance between the following pairs of points on the number line using the concept of modulus:

• $a)$ $x = 5$ and $y = -2$

• $b)$ $x = -8$ and $y = -3$

• $c)$ $x = \frac{1}{2}$ and $y = \frac{3}{4}$

Find the values of $x$ that satisfy the following equations and inequalities involving absolute values. Express the result as an interval:

• $a) \ |x| = 7$

• $b) \ |x - 2| = 5$

• $c) \ |x| < 3$

• $d) \ |x + 1| \leq 2$

A biotechnology laboratory maintains an enzyme sample in a climate chamber. To prevent the enzymes from denaturing, the temperature $T$ (in degrees Celsius) must satisfy the following condition:

Solve the inequality and determine the temperature range (interval) in which researchers can work without damaging the sample.

In the manufacturing process of microchips for satellites, the length of a copper track must be $12.5$ microns. For thermal reasons, a margin of error (tolerance) is allowed such that the difference between the actual length $x$ and the theoretical one is at most $0.05$ microns.

a) Express this condition by means of an inequality with absolute value.

b) Determine the interval of acceptable values for the length of the track using interval notation.

c) Graphically represent this interval on the real number line.

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

In an electronic component store, the price of a racing drone model fluctuates according to market supply. It is known that the price $p$ in pesos satisfies the condition $|p - 9\,000| \leq 1\,200$.

• $a)$ Determine the interval of possible prices for this drone.

• $b)$ What are the minimum and maximum prices that a user could pay?

Find the set of real values satisfying the following inequality with double modulus:

Represent the solution on the number line and express it as a single open interval.

Determine, giving reasons, the set of values of $x \in \mathbb{R}$ satisfying the following compound inequality, expressing the final result as a union of intervals:

Take into account domain restrictions and analyse the different cases depending on the sign of the expression inside the modulus.

To determine the domain of a future function, we need to solve the following quadratic inequality:

a) Find the roots of the associated polynomial.

b) Determine the solution set in interval form.

c) Justify the result by analysing the sign of the polynomial in each section of the number line.

The function $f(x) = \sqrt{x^2 - kx + 9}$ is defined. Determine the set of values of the real parameter $k$ for which the domain of the function is the entire real line $\mathbb{R}$.

For the largest value in this set, determine whether there is any horizontal or oblique asymptote as $x \to +\infty$.

Given the function $f(x) = \ln(|x^2 - 4| - 5)$, determine its domain of definition in exact form. Express the result as a union of intervals, paying special attention to the values where the argument of the logarithm becomes zero or negative. Analyse the symmetry of this domain.

Express the following neighborhoods as open intervals and represent them graphically on the real number line:

1. a) $E(2; 3)$ (neighborhood with center 2 and radius 3)

1. b) $E(-1; 0.5)$ (neighborhood with center $-1$ and radius $0.5$)

1. c) $E^*(0; 4)$ (punctured neighborhood with center 0 and radius 4)

In designing a high-precision part for an aerospace engine, the radius of a cylinder, denoted by $x$, must be sufficiently close to the nominal radius of $5\text{ cm}$ so that the error in its cross-sectional area, $A(x) = \pi x^2$, does not exceed $0.01\text{ cm}^2$.

Using the formal definition of a deleted neighbourhood $E^*(a, \delta)$, find the maximum value of the tolerance $\delta$ that the radius $x$ can have to ensure that the area remains within the neighbourhood $E(25\pi, 0.01)$. Express $\delta$ in terms of $\pi$ in exact form.

Prove, using exclusively the metric definition of a limit ($\varepsilon - \delta$), that:

To do this, given $\varepsilon > 0$, find an expression for $\delta$ in terms of $\varepsilon$ (you may assume an initial bound for $\delta$, for example $\delta \leq 1$).

Consider the sequence defined by the general term $a_n = \frac{3n^2 - 1}{n^2 + 2n}$.

1. a) Calculate the limit $L$ of the sequence as $n \to \infty$.
2. b) Determine from which index $n_0 \in \mathbb{N}$ all terms of the sequence are at a distance from $L$ of less than one thousandth ($\varepsilon = 10^{-3}$).

An investor analyzes the growth of a financial asset whose value in thousands of pesos is given by the function $V(t) = \frac{240t^2 + k}{t^2 + 1}$, where $t \geq 0$ is the time in years.

1. a) Given that the initial value of the asset was $60\,000$ pesos, determine the value of the constant $k$.
2. b) What will be the value of the asset in the "very long term"? Justify your answer using the concept of a limit at infinity.
3. c) Will the value of the asset exceed $240\,000$ pesos at any point? Prove it analytically using an inequality.

A temperature sensor on a data server indicates that the CPU is at $42.6^{\circ}C$. If the actual value measured with a precision thermometer is $42.45^{\circ}C$, calculate:

• $a)$ The absolute error of the sensor measurement.

• $b)$ The relative error made, expressed as a percentage.

In astrophysics, the distance between two celestial bodies is often measured in astronomical units (AU). Suppose two measurements of the distance to an exoplanet have been made: $d_1 = 4.56 \times 10^8$ km and $d_2 = 4.562 \times 10^8$ km.

a) Calculate the absolute error and the relative error made if we take the first measurement as an approximation of the second.

b) Express the result of the relative error as a percentage and comment on whether the approximation is acceptable for a scientific context.