Real Numbers, Inequalities, and Neighborhoods

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

• $a) \ - \sqrt{16}$

• $b) \ \frac{15}{3}$

• $c) \ 2,131313...$

• $d) \ \sqrt{7}$

• $e) \ \frac{\pi}{2}$

• $f) \ -4,25$

Represent the following intervals and half-lines on the real line, and express them using set notation (using inequalities):

• $I_1 = (-3, 2]$

• $I_2 = [0, +\infty)$

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

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 analyzing the sign of the polynomial in each section of the real line.

Given the following subsets of the real line:

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

$B = [ -1, 4 ]$

a) Express the set $A$ as an interval.

b) Calculate and express as intervals the set resulting from the intersection $A \cap B$.

c) Calculate and express as intervals the set resulting from the union $A \cup B$.

Simplify the following expressions using the properties of powers and roots, leaving the result as a power with a fractional exponent:

• $a) \ \sqrt[3]{x^2} \cdot \sqrt{x}$

• $b) \ \frac{x^3}{\sqrt[4]{x}}$

• $c) \ (\sqrt[5]{x^2})^3$

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

Given: Assume that $a > 0$.

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

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 $\$2,5$ for each inch of perimeter, calculate the total cost, rounding to two decimal places.

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

Next, if we know that $\log x = 0,5$, $\log y = 1,2$ and $\log z = 0,8$, calculate the numerical value of $L$.

Calculate the distance between the following pairs of points on the real 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 equalities and inequalities with a modulus. Express the result as an interval:

• $a) \ |x| = 7$

• $b) \ |x - 2| = 5$

• $c) \ |x| < 3$

• $d) \ |x + 1| \leq 2$

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 so that the difference between the actual length $x$ and the theoretical length is, at most, $0,05$ microns.

a) Express this condition using an inequality with a modulus.

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

c) Represent this interval graphically on the real line.

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

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

In an electronic components store, the price of a racing drone model fluctuates according to market supply. It is known that the price $p$ in dollars satisfies the condition $|p - 150| \leq 20$.

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

• $b)$ What is the minimum price and the maximum price a user could pay?

Find the set of real values that satisfy the following inequality with a double modulus:

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

Determine with reasoning the set of values of $x \in \mathbb{R}$ that satisfy the following simultaneous inequality, expressing the final result as a union of intervals:

Take into account the domain restrictions and analyze the different cases according to the sign of the contents of the modulus.

Express the following neighborhoods as open intervals and represent them graphically:

• $a) \ E(2, 3)$ (Neighborhood with center 2 and radius 3)

• $b) \ E(-1, 0.5)$

• $c) \ E^*(0, 4)$ (Deleted neighborhood with center 0 and radius 4)

In the design of a high-precision part for an aerospace engine, the radius of a cylinder, denoted by $x$, must be such that its difference from the nominal radius of $5\text{ cm}$ is so small 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 neighborhood $E^*(a, \delta)$, find the maximum value of the tolerance $\delta$ that the radius $x$ can have to ensure that the area remains in the neighborhood $E(25\pi, 0,01)$. Express the result of $\delta$ in terms of $\pi$ exactly.

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

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

A temperature sensor in 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's 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: $d_1 = 4,56 \cdot 10^8$ miles and $d_2 = 4,562 \cdot 10^8$ miles.

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 environment.

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

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 value of the index $n_0 \in \mathbb{N}$ all terms of the sequence are at a distance from $L$ of less than one thousandth ($\varepsilon = 10^{-3}$).

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 if there is any horizontal or oblique asymptote as $x \to +\infty$.

An investor analyzes the growth of a financial asset whose value in thousands of dollars is given by the function $V(t) = \frac{4t^2 + k}{t^2 + 1}$, where $t \geq 0$ is the time in years.

1. a) If it is known that the initial value of the asset was $1000$ dollars, 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 ever exceed $4000$ dollars? Prove this analytically using an inequality.