Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that $$ |x| \leq 1 \Rightarrow\left|x^{4}+\frac{1}{2} x^{3}+\frac{1}{4} x^{2}+\frac{1}{8} x+\frac{1}{16}\right|<2 $$

Step-by-Step Solution

Answer

For all \(x\) where \(|x| \leq 1\), the expression is less than 2.

Step 1: Understand the Problem

We need to show that for any value of \(x\) that satisfies \(|x| \leq 1\), the expression \( \left| x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2 + \frac{1}{8}x + \frac{1}{16} \right| \) is less than 2.

Step 2: Evaluate the Expression Bounds

Since \(|x| \leq 1\), each term in the expression can be bounded by using the maximum absolute value when \(x = 1\) or \(x = -1\). Calculate the expressions for both bounds.

Step 3: Substitute and Simplify

Substitute \(x = 1\) first and compute the value: \[ 1^4 + \frac{1}{2}(1)^3 + \frac{1}{4}(1)^2 + \frac{1}{8}(1)^1 + \frac{1}{16} \]. Simplify to get \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = 1.9375 \].

Step 4: Check Negative Bound

Now substitute \(x = -1\) and compute the value: \[ (-1)^4 + \frac{1}{2}(-1)^3 + \frac{1}{4}(-1)^2 + \frac{1}{8}(-1)^1 + \frac{1}{16} \]. Simplify to get \[ 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} = 0.8125 \].

Step 5: Conclusion

Since both expressions \(1.9375\) and \(0.8125\) from the evaluated bounds are less than 2, it follows that for \(|x| \leq 1\), the expression \( \left| x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2 + \frac{1}{8}x + \frac{1}{16} \right| \) is indeed less than 2.

Key Concepts

Polynomial Inequality

When dealing with polynomial inequalities, like the one here, our main aim is to establish that a polynomial expression remains within certain bounds under specified conditions. A polynomial of degree 4, like our expression \((x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2 + \frac{1}{8}x + \frac{1}{16})\), can take numerous values depending on the input.

• This means the way it grows or shrinks as \(x\) changes is more complex compared to linear equations.
• Identifying inequalities involves understanding how the degree of each term influences the overall behavior of the expression.

The coefficients in front of \(x\) significantly impact the polynomial's value. Larger coefficients mean that term will grow faster. When solving these inequalities, we often substitute extreme values within the allowed range - here \(|x| \leq 1\) - to assess the expression's bounds.

Absolute Values

Absolute values measure the 'distance' of a number from zero, ignoring any negative sign. So, when we say \(|x| \leq 1\), we're bounding the values of \(x\) between -1 and 1. This is crucial for understanding the problem because it limits our exploration.

• Absolute values can significantly simplify inequality problems by focusing solely on the magnitude, not the direction, of the value.
• It applies to each term separately, therefore bounding each term within manageable limits.

In our solution, knowing that \(|x| \leq 1\) enables us to consider only the extreme cases \(x = 1\) and \(x = -1\). This simplification is valuable because it means we do not need to consider all possible values of \(x\) individually.

Bounds and Limits

Finding bounds is about understanding the maximum and minimum values an expression can reach based on given conditions. Limits help us evaluate behavior as we approach these extremes. In this exercise, calculating the bounds involves substituting boundary values from \(|x| \leq 1\):

• Calculate the expression using \(x = 1\).
• Calculate again using \(x = -1\).

These calculations give us the highest and lowest possible values for our expression within the given limits. Understanding limits involves analyzing how the expression behaves as \(x\) approaches its bounds. Calculating these helps verify whether the polynomial remains within the desired range - explicit calculations showed that our expression was bounded by 0.8125 and 1.9375, both less than 2.

Evaluation of Expressions

Evaluating expressions with substituted numerical values is a practical step in verifying polynomial inequalities. Each term of the polynomial contributes to the entire expression's value. By evaluating, you transform any algebraic expression into a concrete number for comparison:

• Substitution allows us to conveniently and accurately compute values.
• It helps us confirm that the expression satisfies the inequality under the assumed bounds.

In our example, substituting \(x = 1\) and \(x = -1\) allowed us to calculate fixed results (1.9375 and 0.8125). Comparing these results to the boundary value of 2 was essential in proving the inequality holds for all \(|x| \leq 1\). Hence, expression evaluation bridges the step from abstract inequality to concrete proof.