Real numbers (revision)

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:

• $a) \ \sqrt[3]{x^2} \cdot \sqrt{x}$

• $b) \ \frac{x^3}{\sqrt[4]{x}}$

• $c) \ (\sqrt[5]{x^2})^3$

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

$$L = \log \left( \frac{100 \cdot x^3 \cdot \sqrt{y}}{z^2} \right)$$

Then, given that $\log x = 0,5$, $\log y = 1,2$, and $\log z = 0,8$, calculate the numerical value of $L$.

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:

$$|2T - 15| \le 25$$

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

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

$$|x - 2| + |x + 3| < 7$$

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

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:

$$\lim_{x \to 3} (x^2 - 2x) = 3$$

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 $\delta \leq 1$).

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: $d_1 = 4,56 \cdot 10^8$ km and $d_2 = 4,562 \cdot 10^8$ 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.