Approximation (a) Find limx →0 (1-cosx)/(x^2) (b) Use your answer to part (a) to derive the appr

step 1 In problem 126 for part a we want to find the limit for this function when x approaches 0. Find

Approximation (a) Find limx →0 (1-cosx)/(x^2) (b) Use your...

Approximation (a) Find $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$ (b) Use your answer to part (a) to derive the approximation $\cos x \approx 1-\frac{1}{2} x^{2}$ for $x$ near $0 .$ (c) Use your answer to part (b) to approximate $\cos (0.1) .$ (d) Use a calculator to approximate $\cos (0.1)$ to four decimal places. Compare the result with part (c).

Approximation
(a) Find $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$
(b) Use your answer to part (a) to derive the approximation $\cos x \approx 1-\frac{1}{2} x^{2}$ for $x$ near $0 .$
(c) Use your answer to part (b) to approximate $\cos (0.1) .$
(d) Use a calculator to approximate $\cos (0.1)$ to four decimal places. Compare the result with part (c).

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus that describe the behavior of functions as inputs approach a particular value. In the context of trigonometric functions, evaluating the limit as x approaches 0 helps us understand the instantaneous or local behavior of the function. This concept underpins many methods in analysis, such as derivative definitions and approximations, and is essential for establishing standard limits involving trigonometric expressions.

Maclaurin Series (Taylor Series Expansion at Zero)

The Maclaurin series is a type of Taylor series expansion around the point x = 0, which represents functions as infinite sums of powers of x. This expansion is particularly useful for approximating functions like cosine near 0, where higher order terms can be ignored for small values of x. The series provides a systematic method to derive approximations like cos x ? 1 - (1/2)x², showing how the function behaves locally and allowing for error estimation in practical computations.

Small-Angle Approximations

Small-angle approximations capitalize on the fact that for angles close to zero the higher order terms in a function's expansion become insignificant. For the cosine function, this means that cos x can be closely approximated by the first two terms of its Maclaurin series for small x. This simplification is commonly used in physics, engineering, and applied mathematics when an exact value is unnecessary or when computational efficiency is required.

Numerical Approximation and Error Comparison

Numerical approximation involves using methods and tools, such as calculators or computers, to evaluate functions to a desired degree of accuracy. Comparing numerical approximations with analytical approximations, like those derived from Taylor series expansions, helps in assessing the magnitude of the error and validating the appropriateness of the approximation for a given application. This comparison is critical to ensure that the simplifications made in analytical methods yield sufficiently accurate results in practical scenarios.

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