The problem consists of three main tasks related to the third-order Taylor polynomial $P_3(x)$ generated by $f(x) = \cos x$ at $x = \frac{\pi}{4}$:

1. (a) Show that the values of $\cos x$ for angles between $40^\circ$ and $50^\circ$ can be approximated using $P_3(x)$ to within $10^{-5}$, with $1^\circ = \frac{\pi}{180}$ radians.

2. (b) For what angles can we bound the error term $R_3(x)$ of the polynomial $P_3(x)$ by $10^{-5}$? This involves finding the range of $x$ (in degrees) where the approximation holds within the specified error.

3. (c) Find the order of the Taylor polynomial $P_n(x)$ generated by $\cos x$ at $x = \frac{\pi}{4}$ that provides five decimal place accuracy for angles between $30^\circ$ and $60^\circ$.

Let's approach each part step by step:

Solution Outline

(a) Showing approximation accuracy of $P_3(x)$
The third-order Taylor polynomial for $\cos x$ centered at $x = \frac{\pi}{4}$ is: $P_3(x) = \cos \frac{\pi}{4} - \sin \frac{\pi}{4} (x - \frac{\pi}{4}) - \frac{1}{2} \cos \frac{\pi}{4} (x - \frac{\pi}{4})^2 + \frac{1}{6} \sin \frac{\pi}{4} (x - \frac{\pi}{4})^3$ Calculate $P_3(x)$ and then use the remainder term $R_3(x)$ for error estimation.

(b) Finding the range of $x$ where $R_3(x) \leq 10^{-5}$
The remainder term for a third-order Taylor expansion is: $R_3(x) = \frac{f^{(4)}(c)}{4!} (x - \frac{\pi}{4})^4$ Determine the range of $x$ where $R_3(x) \leq 10^{-5}$.

(c) Determining order of $P_n(x)$ for five decimal accuracy between $30^\circ$ and $60^\circ$
Estimate the required order $n$ so that the Taylor polynomial provides five decimal places of accuracy over this range.

Would you like a detailed calculation for each part, or do you have specific questions about any of these tasks?

Here are five related questions to expand understanding:

1. How is the Taylor series for $\cos x$ derived?
2. What does the remainder term $R_n(x)$ represent in Taylor series approximations?
3. How do we determine the range for a specific error tolerance in Taylor series?
4. Why is the Taylor polynomial centered at $x = \frac{\pi}{4}$ particularly useful in this context?
5. What are the implications of increasing the order of the Taylor polynomial for accuracy?

Tip: Converting angles to radians is essential when working with Taylor series approximations of trigonometric functions, as the series is derived with respect to radians.