Problem 34 - Calculus - AP Edition

Question

The approximation \((1+x)^{k} \approx 1+k x\) is commonly used by engineers for quick calculations. (a) Derive this result, and use it to make a rough estimate of \((1.001)^{37}\) (b) Compare your estimate to that produced directly by your calculating device. (c) If \(k\) is a positive integer, how is the approximation \((1+x)^{k} \approx 1+k x\) related to the expansion of \((1+x)^{k}\) using the binomial theorem?

Step-by-Step Solution

Answer

The approximation \((1+x)^{37} \approx 1.037\) is close to the calculated value \(1.03747\). Generally, it simplifies the binomial expansion by ignoring higher order terms.

Step 1: Understand the Approximation

The approximation \((1+x)^{k} \approx 1+kx\) is used when \(x\) is very small in absolute value. This means that higher powers of \(x\) or combinations of \(k\) and \(x\) are ignored, simplifying the expression.

Step 2: Apply the Approximation to the Given Problem

For \((1.001)^{37}\), identify \(x = 0.001\) and \(k = 37\). Using the approximation, it becomes \(1 + 37 \times 0.001 = 1.037\).

Step 3: Calculate the Exact Value Using a Calculator

Use a calculator to find \((1.001)^{37}\) directly. This will give you a precise value; for example, \((1.001)^{37} \approx 1.03747\).

Step 4: Analyze the Congruence Between the Approximation and Binomial Expansion

The binomial theorem states that \((1+x)^k = 1 + kx + \frac{k(k-1)}{2}x^2 + \cdots\). For very small \(x\), the higher order terms \((x^2, x^3, \ldots)\) contribute insignificantly, making \((1+x)^{k} \approx 1+kx\), which aligns with our approximation.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows for the expansion of expressions raised to a power. Specifically, it states that \[(1+x)^{k} = 1 + kx + \frac{k(k-1)}{2}x^2 + \frac{k(k-1)(k-2)}{6}x^3 + \ldots \] This formula includes terms involving powers of \(x\), where each term's coefficient is calculated using combinations.
In simpler terms, it reveals how to expand powers of binomials into sums of powers of individual terms. When \(x\) is very small, as in the original exercise, the first two terms can dominate the expression, making calculations manageable.
The higher Power terms like \(x^2, x^3, \ldots\) become much less significant. This insight into the theorem helps in understanding why \(1+kx\) serves as a reasonable approximation for small values of \(x\).

Approximation Techniques

Approximation Techniques are methods used to find an approximate solution to complex mathematical problems. Using approximations make calculations faster and simpler, especially when high precision is not strictly necessary.
The approximation \((1+x)^{k} \approx 1+kx\) is particularly useful when \(x\) is very small. This means that the change in \((1+x)^k\) is mainly determined by \(kx\) alone.
Such techniques are critical in engineering contexts where speed over precise computation can be more practical. The simplifications in approximation also aid in mental calculations.

• Ensures quicker results.
• Reduces complexity.
• Continues to provide adequate accuracy for small \(x\).

When deciding whether to use an approximation over the exact formula, consider how small the contributing terms in the expansion are.

Exponential Growth

Exponential Growth is a fundamental concept that describes the increase in quantity according to a constant percentage over time. In essence, you multiply the original amount by a constant factor for every set time period. This is a key idea captured in expressions of the form \((1+x)^k\).
In many disciplines, exponential functions describe fast increase behaviors such as population growth, interest calculations in finances, and certain biological processes. For small \(x\), \((1+x)^{k} \approx 1+kx\) gives a good estimate of this growth without needing an exact calculation.

• Provides an understanding of rapid increases.
• Examples include compound interest and population models.
• Understanding such growth patterns is crucial in resource planning and prediction scenarios.

This approximate approach effectively models situations where the growth is moderate, but the approximation holds strong with small rates and brief time frames.