Problem 34 - Precalculus: Mathematical for Calculus

Question

Find and prove an inequality relating \(100 n\) and \(n^{3}\).

Step-by-Step Solution

Answer

The inequality \( n^3 > 100n \) holds for \( n > 10 \).

Step 1: Identify the inequality structure

We want to find a basic inequality comparing the two expressions, 100n and n^3. We'll start by considering the inequality:\[ n^3 > 100n \]

Step 2: Simplify the inequality

To simplify the inequality \( n^3 > 100n \), we divide both sides by \( n \) (assuming \( n eq 0 \)):\[ n^2 > 100 \]

Step 3: Solve the inequality

Next, we solve the inequality \( n^2 > 100 \). By taking the square root of both sides, we find:\[ n > \sqrt{100} \]Since \( \sqrt{100} = 10 \), this gives us:\[ n > 10 \]

Step 4: Test integer values

Now we test integer values of \( n \) greater than 10 to ensure our inequality holds true. If \( n = 11 \), then:\[ n^3 = 11^3 = 1331 \] and \[ 100n = 1100 \]Since \( 1331 > 1100 \), the inequality \( n^3 > 100n \) holds. Similarly, for \( n = 12 \), \[ n^3 = 12^3 = 1728 \] which is clearly greater than \( 1200 = 100 imes 12 \). We can verify for greater integers, and the pattern persists, confirming the inequality holds.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions composed of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents. For example, in the inequality problem, we deal with polynomial expressions such as \(100n\) and \(n^3\). Here, \(n^3\) is the cubic polynomial expression with the term raised to the power of three, while \(100n\) is a linear polynomial expression.Understanding polynomials allows you to identify the degree of an expression, which is based on the highest power of the variable. In our exercise, \(100n\) is of degree 1 and \(n^3\) is of degree 3.

• A polynomial of degree 1 behaves linearly, which means its growth is constant.
• A polynomial of degree 3 grows much faster as the variable increases, because it involves the variable raised to the third power.

Recognizing this difference helps us comprehend how these expressions interact in inequalities, like ensuring when \(n^3\) surpasses \(100n\).

Solving Inequalities

Solving inequalities involves finding the range of values for which an inequality holds true. The process may require simplifying expressions, factoring, or using properties of inequalities.In our problem, the primary task was to solve \(n^3 > 100n\). Here’s how it was approached:

• First, we rearranged the inequality into a simpler form by dividing both sides by \(n\), assuming \(n eq 0\). This gives us \(n^2 > 100\).
• Next, solving \(n^2 > 100\) involves taking the square root of both sides, resulting in \(n > 10\).

When solving inequalities, especially polynomial ones, remember:- Solve by isolating the variable whenever possible.- Always consider the possibility of multiple roots or values, especially in higher-degree polynomials.- Confirm that step manipulations do not violate any mathematical rules, such as division by zero.

Integer Solutions

Integer solutions refer to solutions of an inequality or equation that are whole numbers. In many mathematical problems, we are particularly interested in integer solutions for practical or theoretical reasons.During our solution verification for \(n^3 > 100n\), we specifically look at integer values greater than 10:

• If \(n = 11\), then \(n^3 = 1331\) and \(100n = 1100\). Clearly, \(1331 > 1100\).
• For \(n = 12\), \(n^3 = 1728\) is greater than \(1200 = 100 \times 12\).

Finding integer solutions helps in verifying if an inequality holds for a sequence of values. It also tests the practical application of mathematical concepts.As you work on similar exercises, use integer checks to assure your understanding of the inequality's behavior across the number line. This introduces a tangible check into abstraction, showcasing the conditions under which the inequality holds true.