(a) Find limx →0 (1-cosx)/(x^2). (b) Use the result in part...

(a) Find $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$. (b) Use the result in part (a) to derive the approximation $\cos x \approx 1-\frac{1}{2} x^{2}$ for $x$ near 0 (c) Use the result in 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).

(a) Find $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}$.
(b) Use the result in part (a) to derive the approximation $\cos x \approx 1-\frac{1}{2} x^{2}$ for $x$ near 0
(c) Use the result in 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

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. They are used to understand the continuity, differentiability, and the general behavior of functions near specific points.

Trigonometric Limits

Trigonometric limits focus on the behavior of trigonometric functions as their input approaches a value, typically zero, where small-angle approximations are effective. A classic example includes evaluating limits like (1 - cos(x))/(x^2), which plays an essential role in deriving approximations for trigonometric functions.

Taylor Series

The Taylor series is an expansion of a function into an infinite sum of terms calculated based on the function's derivatives at a given point. This series, and its special case the Maclaurin series when centered at zero, is a powerful tool for approximating functions and understanding local behavior around a point.

Small-Angle Approximation

Small-angle approximations are simplifications used when the variable is close to zero, allowing trigonometric functions to be approximated by the first few terms of their Taylor series. For example, the cosine function can be approximated as 1 - (x^2)/2 for small values of x, simplifying calculations in engineering and physics.

Error Analysis

Error analysis involves evaluating the difference between an approximate value and the exact value. When using approximations like those derived from Taylor series, error analysis helps determine the reliability of these approximations by estimating how close the computed values are to the true function values.

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