Question

In general, how many terms do the Taylor polynomials \(p_{2}\) and \(p_{3}\) have in common?

Short answer

Answer: The Taylor polynomials \(p_2\) and \(p_3\) have 3 terms in common.

Step-by-step solution

Understanding Taylor polynomials

Taylor polynomials are used to approximate a function, and they are defined as: $$p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^n(a)}{n!}(x-a)^n$$ Where \(n\) is the degree of the polynomial, \(a\) is the point at which the function is approximated, and \(f^k(a)\) is the \(k\)th derivative of the function evaluated at \(a\). In this problem, we are comparing the Taylor polynomials \(p_2\) and \(p_3\).

Identify the terms in \(p_2\) and \(p_3\)

According to the definition, the Taylor polynomials \(p_2\) and \(p_3\) are expressed as: $$p_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$$ $$p_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^3(a)}{3!}(x-a)^3$$

Compare the terms of \(p_2\) and \(p_3\)

Comparing the terms in \(p_2(x)\) and \(p_3(x)\), we can see that they have the following terms in common: 1. \(f(a)\) 2. \(f'(a)(x-a)\) 3. \(\frac{f''(a)}{2!}(x-a)^2\) These three terms are shared by both polynomials. The only additional term in \(p_3(x)\) is \(\frac{f^3(a)}{3!}(x-a)^3\), which is not present in \(p_2(x)\).

Conclusion

Since the Taylor polynomials \(p_2\) and \(p_3\) have three terms in common, this is our final answer.