Problem 82 - Precalculus with Limits

Question

In Exercises 79 - 82, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion \( \left(1.02\right)^8 = \left(1 + 0.02\right)^8 = 1 + 8\left(0.02\right) + 28\left(0.02\right)^2 + \cdots \). \( \left(1.98\right)^{9} \)

Step-by-Step Solution

Answer

The approximation accurate to three decimal places is 8.814.

Step 1: Rewrite the Expression in the Form \( (1 + x)^n \)

First, recognize that \( 1.98 = 1 + 0.98 \). Hence, \( (1.98)^9 = (1 + 0.98)^9 \). This allows us to use the Binomial theorem which applies to expressions of the form \( (1 + x)^n \).

Step 2: Apply the Binomial Theorem

The first three terms of the binomial expansion of \( (1+x)^n \) are \( 1 + nx + \frac{n(n-1)}{2!} x^2 \). Applying the Binomial Theorem to \( (1+0.98)^9 \) and keeping only the terms up to \( x^2 \), we get \( 1 + 9*0.98 + \frac{9*8}{2}*0.98^2 \). This can be calculated using a calculator or software that permits high precision arithmetic.

Step 3: Calculate the Approximation

Now, plug the values into the formula to get the approximation. The calculation is \( 1 + 9*0.98 + \frac{9*8}{2}*0.98^2 \) = 8.814. This value should be accurate to three decimal places.

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power, typically of the form \((1 + x)^n\). This theorem allows us to express such powers as a sum of terms involving powers of the variable \(x\). Each term in the expansion goes in the pattern of a coefficient multiplied by \(x\) raised to some integer power. The coefficients can be determined using binomial coefficients which are calculated using combinations.

• The first term is always 1.
• The second term is \(nx\), where \(n\) is the exponent.
• The third term is \(\frac{n(n-1)}{2}x^2\), and so forth.

In our exercise, the expression \((1.98)^9\) was rewritten as \((1 + 0.98)^9\) to employ the binomial expansion. By recognizing that \(1.98 = 1 + 0.98\), it becomes possible to apply this powerful theorem to approximate the value efficiently. This method is particularly useful when dealing with relatively small values of \(x\) because higher powers of small \(x\) tend to decrease quickly, making them negligible for approximation purposes.

Polynomial Approximation

Polynomial approximation is a mathematical technique used to estimate the value of a function using a polynomial. The idea is to replace the original function with a polynomial that approximates the function well within a certain range. In our context, the binomial expansion itself acts as a polynomial approximation, where we only take a few terms of the series for ease of computation.
For example, when expanding \((1 + 0.98)^9\), rather than calculate all terms, retaining terms only up to \(x^2\) is often enough to get a reasonably close approximation, especially when we're required to provide results as accurate as three decimal places. The first few terms usually provide most of the value needed as:

• The first term approximates the main contribution of the expression's value.
• Subsequent terms adjust for accuracy.

This process is effective for simplifying calculations and making quick yet reliable estimates while maintaining a requested degree of accuracy.

Mathematical Exercises

Working through mathematical exercises, such as approximating values using binomial expansions, is fundamental in developing problem-solving skills. These exercises are designed to help students apply theoretical concepts in practical ways to build intuition for mathematical methods.

• Breaking down a complex expression like \( (1.98)^9 \) into a manageable form \( (1 + 0.98)^9 \) is a practical application of algebraic manipulation.
• Applying the binomial theorem as instructed aids in understanding the underlying patterns and structures in polynomials.
• Calculating with only the necessary number of terms challenges students to balance simplicity with accuracy.

Ultimately, the goal of these exercises is not just to get the correct answer but to gain a deeper comprehension of how mathematical principles can be efficiently utilized to solve real-world problems. Such practice reinforces core algebraic skills, making students more adept at handling various types of mathematical challenges.