Problem 41 - Calculus Single Variable

Question

For any real number \(x\), the greatest integer in \(x\) is denoted by \(\lfloor x\rfloor\) and defined to be the unique integer satisfying \(\lfloor x\rfloor \leq x<\lfloor x\rfloor+1 .\) For example, \(\lfloor 3.2\rfloor=3\) and \(\lfloor-3.2\rfloor=-4 .\) Notice that \(\lfloor x\rfloor=x\) if and only if \(x\) is an integer. The function \(x \mapsto\lfloor x\rfloor\) is called the greatest integer function. (The expression \(\lfloor x\rfloor\) is sometimes read as "the floor of \(x . ")\) The integer part of \(x,\) denoted by \(\operatorname{Int}(x),\) is that part of the decimal expansion of \(x\) to the left of the decimal point. Express \(\lfloor x\rfloor\) as a multicase function of \(\operatorname{Int}(x) .\) Graph \(x \mapsto\lfloor x\rfloor\) and \(x \mapsto \operatorname{Int}(x)\) for \(-3 \leq x \leq 3\)

Step-by-Step Solution

Answer

\(\lfloor x \rfloor = \operatorname{Int}(x)\) for \(x \geq 0\) or integer \(x\); otherwise, \(\lfloor x \rfloor = \operatorname{Int}(x) - 1\) for negative \(x\).

Step 1: Understand the Greatest Integer Function

The greatest integer function, denoted as \(\lfloor x \rfloor\), maps a real number \(x\) to the largest integer that is less than or equal to \(x\). For example, for \(x = 3.2\), the greatest integer function returns \(3\), since \(3\) is the largest integer less than \(3.2\). When \(x\) is negative, like \(-3.2\), the function returns \(-4\), which is the largest integer less than \(-3.2\). This is because \(-3\) is greater than \(-3.2\).

Step 2: Defining Integer Part of a Number

The integer part of \(x\), denoted \(\operatorname{Int}(x)\), is the integer part to the left of the decimal point in the decimal representation of \(x\). It is the same as applying the floor function for non-negative numbers but differs for negative \(x\). For example, \(\operatorname{Int}(-3.2) = -3\).

Step 3: Expressing \(\lfloor x \rfloor\) using \(\operatorname{Int}(x)\)

We express \(\lfloor x \rfloor\) as a multi-case function using \(\operatorname{Int}(x)\): - If \(x\) is non-negative (\(x \geq 0\)), then \(\lfloor x \rfloor = \operatorname{Int}(x)\).- If \(x\) is negative, then when there is no fractional part (when \(x\) is exactly an integer, like \(-3.0\)), \(\lfloor x \rfloor = \operatorname{Int}(x)\). - Otherwise, \(\lfloor x \rfloor = \operatorname{Int}(x) - 1\). Thus, this can be written as:\[\lfloor x \rfloor = \begin{cases}\operatorname{Int}(x), & x \geq 0 \, \text{or} \, x \in \mathbb{Z} \\operatorname{Int}(x) - 1, & x < 0 \, \text{and} \, x otin \mathbb{Z}\end{cases}\]

Step 4: Graphing \(\lfloor x \rfloor\) and \(\operatorname{Int}(x)\)

The graph of \(x \mapsto \lfloor x \rfloor\) consists of horizontal line segments that jump down at each integer value of \(x\). For each integer \(n\), the segment extends from \(n\) to \(n+1\) and has a value of \(n\).For \(\operatorname{Int}(x)\), the graph is similar but for negative \(x\), \(\operatorname{Int}(x)\) equals \(x\) (since no subtraction by 1 is needed). Thus, there are no downward jumps for negative numbers.Both functions are compared from \(-3\) to \(3\), visibly differing only for the negative non-integer values of \(x\).

Key Concepts

Integer Part of a Number

The integer part of a number is essentially the whole number portion of a decimal, located before the decimal point. To better understand, imagine a number line. If you take any real number \(x\), the integer part of \(x\), denoted as \(\operatorname{Int}(x)\), is straightforward for positive numbers. For instance, in \(4.7\), the integer part is 4. For the number \(-2.5\), \(\operatorname{Int}(x)\) would be \(-2\).

When dealing with positive numbers, the integer part mirrors the greatest integer function or floor function, as both would round down to the same integer value that precedes the decimal point. However, differences arise when we consider negative numbers: the integer part gives you the number preceding the decimal point, avoiding any rounding down like in the floor function process for negatives.

For easier recollection, always remember:

• Positive numbers: \(\operatorname{Int}(x) = \lfloor x \rfloor\).
• Negative numbers: \(\operatorname{Int}(x)\) simply strips the decimal part without further rounding down.

Multicase Function

A multicase function presents different expression results based on varying conditions of input, similar to choosing a route based on road conditions. It's useful in mathematics for defining functions like the floor function with multiple scenarios. For the floor function, \(\lfloor x \rfloor\), conditions determine the results:

• For \(x\geq 0\) or when \(x\) is an integer, \(\lfloor x \rfloor = \operatorname{Int}(x)\).
• If \(x\) is negative and not an integer (e.g., \(-3.7\)), the function becomes \(\operatorname{Int}(x) - 1\).

Expressing functions like this with cases allows them to handle differing conditions seamlessly. It's pivotal to understand these distinctions to appreciate how inputs affect outputs, particularly for values straddling integer boundaries. Remember that multicase functions can perfectly capture these logical scenarios and are vital to grappling with piecewise functions.

Graphing Piecewise Functions

Graphing piecewise functions can initially seem tricky, but breaking them into manageable segments simplifies matters. Consider the greatest integer function \(x \mapsto \lfloor x \rfloor\). Imagine it as a series of steps on a staircase.

For each integer \(n\), a horizontal segment stretches from \(n\) to \(n+1\), hugging the line \(y=n\). As soon as you cross \(n+1\), the function 'steps down,' dropping to \(n+1\) for its next segment, hence forming a step-like graph.

Conversely, the graph of \(x \mapsto \operatorname{Int}(x)\) mirrors this, with a key variance visible in negative ranges. In these negative sections:

• The greatest integer function swiftly 'falls' below each integer,
• while the integer part function maintains levelness, as it doesn't subtract 1 for negative fractions.

Drawing these line segments for each integer highlights where and how these two functions diverge, especially for negative non-integers. Always ensure clarity by plotting key points, and then connect these dots with perfectly horizontal lines for integer ranges. This clarity visually reinforces understanding of function nuances.