Find the Maclaurin polynomial of degree \(3\) for \(e^x\).

Find the Maclaurin polynomial of degree \(5\) for \(\sin x\).

Find the Maclaurin polynomial of degree \(4\) for \(\cos x\).

Find the Maclaurin polynomial of degree \(3\) for \(\ln(1+x)\).

Find the Taylor polynomial of degree \(2\) for \(f(x)=\sqrt{x}\) about \(a=4\).

Use the degree \(3\) Maclaurin polynomial for \(e^x\) to approximate \(e^{0.1}\).

Use Taylor's remainder bound to bound the error when approximating \(e^{0.2}\) by \(1+x+x^2/2\) at \(x=0.2\).

Find the Taylor series about \(0\) for \(\frac{1}{1-x}\), and state its interval of convergence.

Find the Taylor polynomial for \(f(x)=x^2\) about \(a=1\).

A learner writes the degree \(2\) Maclaurin polynomial for \(\cos x\) as \(1-x^2\). Explain the error and give the correct polynomial.