Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.


a. (1.0002)⁵⁰

Verified step by step guidance

1. Identify the expression to approximate: (1.0002)⁵⁰. Here, we can see that x = 0.0002 and k = 50.
2. Recognize that the approximation (1 + x)ᵏ ≈ 1 + kx is useful when x is small, which is the case here.
3. Substitute the values of x and k into the approximation formula: 1 + kx = 1 + 50 * 0.0002.
4. Calculate the product kx: 50 * 0.0002 = 0.01.
5. Add the result to 1 to complete the approximation: 1 + 0.01.

Key Concepts

Binomial Approximation

The binomial approximation (1 + x)ᵏ ≈ 1 + kx is a simplification used when x is small and k is a constant. It allows for quick estimates of expressions raised to a power without complex calculations. This approximation is derived from the binomial theorem and is particularly useful for small perturbations around 1.

Small x Assumption

The small x assumption is crucial for the validity of the binomial approximation. It implies that x is close to zero, making higher-order terms in the binomial expansion negligible. This assumption simplifies calculations and is often used in physics and engineering to approximate values efficiently.

Exponentiation

Exponentiation is the mathematical operation involving numbers raised to a power, denoted as (1 + x)ᵏ. Understanding how to manipulate and approximate powers is essential in calculus, especially when dealing with series expansions and approximations. It forms the basis for many calculus concepts, including derivatives and integrals.