Approximation and Errors

Round and truncate the number $\frac{17}{6}$ to thousandths. Calculate the absolute error in both approximations and state with justification which of the two is more precise.

The irrational number $\pi$ has been approximated as $3.14$. Is it an underestimate or an overestimate? Justify your answer and find a bound for the absolute error made.

A financial institution calculates daily interest on an account using rounding to two decimal places in each daily transaction, whereas a second institution stores all decimals internally and only rounds the total balance to two decimal places at the end of the month (consider a 30-day month). If the exact daily interest generated in the account is $1.234$ pesos, calculate the total accumulated absolute error differentiating both methods at the end of the month.

When measuring the length of a table of $1.25$ m, a value of $1.28$ m was obtained. On the other hand, when measuring the distance between two cities of $245$ km, a value of $246$ km was obtained. Calculate the relative error in both measurements and determine which of the two is more precise.

The golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$ is frequently approximated in graphic design by the value $1.618$. Calculate the absolute error and the relative error of this approximation, using all digits from your calculator for the exact value. Express the relative error as a percentage.

In a microbiology laboratory, the diameter of a cell is measured as $0.000015$ m with an absolute error of less than $10^{-7}$ m. At the same time, an astronomical observatory measures the distance to a nearby star as $4.2 \times 10^{16}$ m with an absolute error of less than $10^{14}$ m. Which measurement has greater actual precision? Justify your answer by calculating and comparing the relative error of both.

Write the number $0.000045602$ in standard form leaving exactly three significant figures. Next, state a bound on the absolute error made when making this approximation.

Determine the real interval containing the exact value of a measurement $x$ if we know that its approximation is $5.4$ and the absolute error bound is $E_a < 0.05$.

A digital thermometer shows a body temperature of $37.5^\circ\text{C}$, guaranteeing a relative error of less than $0.2\%$. What is the maximum absolute error the device could be making?

Perform the mathematical operation $3.2 \times 10^4 + 5.1 \times 10^3$, expressing the final result in standard form with the appropriate number of significant figures.

The mass of a planet is $5.97 \times 10^{24}$ kg and that of its natural satellite is $7.35 \times 10^{22}$ kg. Calculate the total mass of the planetary system in standard form and determine a bound for the absolute error of the sum, assuming that the given data are already approximations rounded to the last decimal place shown.

The speed of light in a vacuum is a physical constant whose exact value is $299792458$ m/s. Express this quantity in standard form rounded to four significant figures. What is the bound on the absolute error and the relative error of this new abbreviated representation?

An old geometric calculation software uses the approximation $\frac{22}{7}$ for the number $\pi$. How many exact decimal places does this approximation provide compared to the real value of $\pi$? Calculate the percentage relative error made by the software.

We approximate the irrational value of $\sqrt{10}$ using the rational fraction $\frac{19}{6}$. Calculate the absolute error made with this approximation. Then, find a fraction with a denominator less than $20$ that provides a better approximation (that is, with a strictly smaller absolute error) and prove analytically that it is better.

A computer algorithm approximates the exponential function for values near zero using the second-degree Taylor polynomial: $e^x \approx 1 + x + \frac{x^2}{2}$. If we evaluate this algorithm at $x = 0.1$, calculate the absolute error and the percentage relative error of this approximation compared with the value given by the scientific calculator for $e^{0.1}$. How many correct significant figures does this polynomial approximation guarantee?

When approximating a real number $x$ by the value $3.14$, an absolute error of less than $0.005$ is made. When approximating another real number $y$ by the value $2.71$, an absolute error of less than $0.005$ is also made. Determine the guaranteed error interval for the sum $x + y$ and for the difference $x - y$.

Prove algebraically that if two positive real numbers $x$ and $y$ are approximated by $x_0$ and $y_0$ with a relative error less than $\varepsilon$ in both cases, then the relative error of the product $x \times y$ is bounded approximately by $2\varepsilon$ (you may neglect higher-order terms above $\varepsilon$, that is, terms containing $\varepsilon^2$).

Newton's law of universal gravitation is given by the expression $F = G \frac{m_1 m_2}{d^2}$. If in an experiment we measure the masses $m_1, m_2$ and the distance $d$ with a maximum relative error of $2\%$ in each of the three quantities, and we treat the constant $G$ as an exact mathematical value, determine with justification the bound on the maximum relative error that will be made when calculating the force $F$.

A rectangular room is to be tiled. The approximate measurements of the sides, taken with a tape measure, are $a = 4.2$ m and $b = 3.5$ m. Knowing that the instrument introduces a maximum error of $0.05$ m in each measurement, calculate the nominal area of the room and determine the strict interval of real values in which this area lies.

In a land surveying project, the two legs of a right-angled triangular plot of land are measured, giving $c_1 = 30.0$ m and $c_2 = 40.0$ m. The precision of the surveying instrument guarantees a relative error of less than $1\%$ in each measured linear length. Determine the nominal length of the hypotenuse $h$ and bound the absolute error and the maximum relative error that can be made when calculating it indirectly.

We want to prepare a chemical solution in which the mass of the solute must be $m = 15.00$ g and the volume of the solvent $V = 250$ ml. The laboratory balance has a guaranteed precision of $\pm 0.01$ g and the measuring cylinder has a margin of error of $\pm 1$ ml. Calculate the nominal concentration $C = \frac{m}{V}$ expressed in g/ml and state the strict interval of real values within which this concentration lies.