Question

\mathrm{Conjecture } Consider the function \(f(x)=x^{2} e^{x}\) (a) Find the Maclaurin polynomials \(P_{2}, P_{3},\) and \(P_{4}\) for \(f\). (b) Use a graphing utility to graph \(f, P_{2}, P_{3},\) and \(P_{4}\). (c) Evaluate and compare the values of \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\) for \(n=2,3,\) and 4 (d) Use the results in part (c) to make a conjecture about \(f^{(n)}(0)\) and \(P_{n}^{(n)}(0)\)

Short answer

To form the conjecture, focus on the differences observed between \(f^{(n)}(0)\) and \(P_n^{(n)}(0)\) for different values of n.

Step-by-step solution

Finding Maclaurin Polynomials

Remember that the Maclaurin series for a function is given by: \[f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + \ldots\] To find the Maclaurin polynomials \(P_2, P_3, and P_4\), compute the derivatives of \(f(x)\) up to the 4th order at x=0 using the product rule. Then plug these values into the Maclaurin series formula.

Graphing f and its Polynomials

Use any graphing utility to plot the function \(f(x)=x^2 e^x\) and its Maclaurin polynomials \(P_2, P_3, P_4\). Observe how well these polynomial approximations fit the original function.

Evaluating and Comparing Function and Polynomial

Evaluate \(f^{(n)}(0)\) and \(P_n^{(n)}(0)\) for \(n=2,3,\) and 4. For \(f^{(n)}(0)\), use the values computed in the first step. For \(P_n^{(n)}(0)\), realize that the nth derivative of a polynomial of degree n evaluated at 0 will just be the coefficient of the nth degree term.

Make a Conjecture

Analyze the results from Step 3, where you compared the values of \(f^{(n)}(0)\) and \(P_n^{(n)}(0)\) for \(n=2,3,\) and 4. If you observed a pattern, use this to make a conjecture about the general relationship between \(f^{(n)}(0)\) and \(P_n^{(n)}(0)\)