91Ó°ÊÓ

When using a tangent line to approximate a function near a certain point, there is usually a discrepancy between this approximation and the actual function. This discrepancy is called the tangent line error. In this context, it is defined as:

• \[ E(x) = f(x) - [f(a) + f'(a)(x-a)] \]

This formula represents the difference between the actual function value at \(x\) and the value predicted by the tangent line equation at that point.
Fundamentally, this error arises due to the fact that a line can only closely mimic a curve around a small proximity to a point. As you move further away from the point of tangency, the error usually increases, especially if the function is not linear.

The Taylor series is a powerful mathematical tool used to approximate functions around a specific point \(a\). By considering not just the value and the first derivative, but higher-order derivatives as well, the Taylor series provides a better approximation.
The general term for the Taylor series is:

• \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \]

In the context of the given exercise, the Taylor series is employed to express the error as a series expansion, showing that the primary source of deviation from the tangent line is tied to the second derivative term. By truncating this series after the second derivative term, we obtain a practical error approximation for functions that can sustainably be approximated over a small interval around \(a\).

The second derivative of a function, denoted as \(f''(x)\), provides further insight into the function’s curvature. It can help determine how the function is bending at a given point.
If the second derivative is zero, the tangent line may provide a better approximation over a larger interval. However, if it is non-zero, it indicates curvature, causing the tangent line to diverge faster from the actual function.
For the cosine function \(f(x) = \cos x\) at point \(a = 0\), the second derivative is:

• \[ f''(x) = -\cos x \]
• At \(x = 0\): \[ f''(0) = -1 \]

This value implies that the tangent line rapidly diverges from the cosine function as we move away from \(x = 0\). Consequently, the second derivative plays a crucial role in determining the magnitude of the error in the linear approximation and links directly to the coefficient \(k\) found in the error approximation.

Approximating the cosine function, especially around \(x = 0\), requires understanding its behavior. At this point, the function has simple derivatives which lend themselves well to the tools of the Taylor series.
When \(x = 0\), we know:

• \(\cos 0 = 1\)
• \(\sin 0 = 0\), therefore \(f'(0) = 0\)
• \(f''(0) = -1\)

Thus, for small deviations from zero, the cosine function can be effectively approximated by its tangent line \(y = 1\) and a second-order term from the Taylor series:

• \[ E(x) \approx \left(\frac{f''(0)}{2}\right)(x-a)^2 = \left(\frac{-1}{2}\right)x^2 \]

This approximation reveals the drop-off rate of cosine from its value at zero. For points extremely close to zero, the linear term suffices, but further away, this quadratic term becomes significant, encapsulating the cosine’s downward dip.