Problem 59 - Calculus Single Variable

Question

In Exercises \(59-62,\) determine the interval that \(y\) must lie in to agree with \(x\) to \(q\) decimal places. \(x=0.449, q=3\)

Step-by-Step Solution

Answer

The interval for \( y \) is \([0.4485, 0.4495)\).

Step 1: Understand the Decimal Precision

We need the variable \( y \) to agree with \( x = 0.449 \) to \( q = 3 \) decimal places. This means \( y \) should match every digit and decimal place of \( x \) up to and including the third decimal place.

Step 2: Define the Interval Based on Precision

Since \( y \) must agree with \( x \) to three decimal places, \( y \) should be in the range where its first three decimal places are 0.449. This implies that \( y \) must be at least 0.4485 and less than 0.4495.

Step 3: Construct the Interval

The interval for \( y \), using the endpoints determined in the previous step, is \( [0.4485, 0.4495) \). This means \( y \) can be anything starting from 0.4485 up to, but not including, 0.4495.

Key Concepts

Interval Notation

Interval notation is a useful way for expressing a range of numbers. It's essential to have a clear understanding of this concept.
When you see notation like \([0.4485, 0.4495)\), it describes the range of values that a number can take. In this specific interval:

• The square bracket "[" indicates that 0.4485 is included in the interval. This is known as a closed boundary.
• The round parenthesis ")" shows that 0.4495 is not included in the interval. This is an open boundary.

Thus, the interval [0.4485, 0.4495) includes all numbers starting from 0.4485 and going up to—but not reaching—0.4495. Understanding how intervals are constructed and interpreted helps in solving problems related to ranges and limits.

Decimal Places

Decimal places indicate the number of digits to the right of the decimal point. In context, maintaining precision in decimal places means ensuring that digits match exactly up to a specified position.
For example, if you want variable \( y \) to agree with \( x = 0.449 \) to 3 decimal places, then the first three digits right after the decimal in \( y \) should be 4, 4, and 9. This precision ensures uniformity in calculations and comparisons.

• First decimal place: Tenths position.
• Second decimal place: Hundredths position.
• Third decimal place: Thousandths position.

Paying attention to these positions ensures that numbers are precise, especially important in fields requiring high accuracy like finance and engineering.

Inequalities

Inequalities are statements about the relative size or order of two numbers or expressions. They help describe ranges where certain values lie, often seen with interval notation.
The inequality \(0.4485 \leq y < 0.4495\) tells us that \( y \) must be greater than or equal to 0.4485, but less than 0.4495. This helps set boundaries on the acceptable values \( y \) can take.

Recognizing symbols:

• \( \leq \) means "less than or equal to."
• \( < \) means "less than."

Understanding and solving inequalities is crucial in defining valid ranges for variables and ensuring calculations stay accurate.

Precision in Calculations

Precision in calculations refers to the exactness of a number or measurement. It’s a crucial aspect in applying mathematical concepts accurately, particularly in decimal calculations.
Precision is essential when you want consistent results. In our example, we ensure that \( y \) matches \( x \) to three decimal places, thereby defining precise boundaries for \( y \).

A few aspects to consider about precision include:

• Reporting results to the specified number of decimal places.
• Understanding rounding rules to maintain precision.
• Being aware that more decimal places typically mean a higher level of precision.

Accuracy is key, especially in scientific and engineering contexts where even small errors can lead to significant discrepancies in outcomes.