Problem 55 - A Graphical Approach to Precalculus with Limits

Question

It can be shown that $$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\cdots$$ for any real number \(n\) (not just positive integer values) and any real number \(x\), where \(|x|<1\). Use this result to approximate each quantity to the nearest thousandth. $$(1.01)^{3 / 2}$$

Step-by-Step Solution

Answer

The approximate value is 1.015.

Step 1: Identify given values

We recognize that in the exercise, we are asked to approximate \((1.01)^{3/2}\).Using the provided formula \((1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3} + \cdots\), we will approximate \((1.01)^{3/2}\) by considering \(n = 3/2\) and \(x = 0.01\).

Step 2: Calculate first few terms

Substitute \(x = 0.01\) and \(n = \frac{3}{2}\) into the formula:\[(1+x)^{n} = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots\]Calculate each term:- The first term is 1.- The second term is \(\frac{3}{2} \times 0.01 = 0.015\).- The third term is \(\frac{3}{2} \times \frac{1}{2} \times \frac{1}{2} \times (0.01)^2 = \frac{3}{2} \times \frac{1}{2} \times 0.0001 = 0.000075\).

Step 3: Approximate value

Add up the terms: - Sum = 1 + 0.015 + 0.000075 Compute the total: - Sum = 1.015075. Round this to the nearest thousandth: - The result is 1.015.

Key Concepts

Binomial Expansion

The concept of binomial expansion is a powerful algebraic tool used to express powers of sums in a series of terms. When we talk about the binomial expansion of \((1 + x)^n\), we refer to expressing it as a series:

• \(1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots\).

This expression is derived from the Binomial Theorem, which originally dealt with positive integer values of \(n\). However, it can also be applied to any real numbers, including fractions and negatives, as long as the absolute value of \(x\) is less than 1. This allows for a wider range of calculations and approximations, offering a detailed insight into how a small change in \(x\), like 0.01, impacts the overall value.Understanding binomial expansion enables us to solve complex real-life problems involving growth, decay, and more, by breaking down calculations into manageable terms.
Using binomial expansion, one can readily approximate values without the use of calculators, making it especially useful in theoretical contexts.

Approximations

Approximations are techniques used to find a close, yet not exact, solution to mathematical problems. In the context of the binomial series, they are incredibly useful for obtaining results quickly with an accepted level of accuracy.They involve using a truncated series, or a finite number of terms, to estimate a calculation that would otherwise be challenging to work out exactly.In the problem at hand, approximating \((1.01)^{3/2}\) can be easily done by calculating only the first few terms of the expansion:

• The first three terms were deemed enough to achieve an accuracy to the nearest thousandth.
• The expansion permitted simplification that was both swift and precise for our needs.

By applying this technique, we efficiently reduced the computation and rounded our final result to 1.015.Approximations are crucial in contexts where small errors are acceptable, allowing complex calculations to be simplified and loaned to mental math or quick estimations.

Real Numbers

Real numbers comprise all the numbers on the number line, including both rational numbers (like integers and fractions) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).They provide a comprehensive way to measure continuous quantities, offering a broader set of values over whole numbers or fractions alone.Real numbers are a fundamental part of mathematical calculations and theories, as they allow us to handle operations involving roots, logarithms, and polynomial expressions seamlessly. In our particular exercise, both \(1.01\) and \(\frac{3}{2}\) are considered real numbers, which illustrates their flexibility and applicability to practical and theoretical problems.Using real numbers in binomial expansions signifies more flexibility and adjustable computations, proving advantageous in many applications such as physics, engineering, and finance.The rule that \(|x|<1\) when using the binomial expansion with non-integer powers ensures convergence and accuracy of approximations.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations that together define a mathematical statement. They form the backbone of algebra and are used to represent real-world scenarios in mathematical language. An algebraic expression like \((1+x)^n\) in our exercise succinctly captures a complex process behind expansion and approximation.

• The power \(n\), which can be any real number, and the term \(x\), which should adhere to \(|x|<1\), both play important roles.
• The series of product-term expressions guide us in precise step-by-step calculations.

Understanding algebraic expressions in the context of the binomial series allows us to simplify complex processes into understandable terms.It's beneficial in translating physical phenomena into models that scientists and mathematicians can manipulate and solve.In dealing with algebraic expressions, clear operations lead to better analytical skills and deeper insights into the relationships between numbers, allowing us to skillfully manage and solve equations.