State the degree \(N\) Taylor polynomial for \(f\) about \(c\).

What centre defines a Maclaurin polynomial?

Find the linear Maclaurin polynomial for \(e^x\).

Find the quadratic Maclaurin polynomial for \(\cos x\).

Find \(P_2(x)\) for \(f(x)=x^2+2x+1\) about \(0\).

Find \(P_2(x)\) for \(e^x\) about \(0\).

Find the cubic Maclaurin polynomial for \(\sin x\).

Find the linear Taylor polynomial for \(\sqrt{x}\) about \(4\).

Find the quadratic Taylor polynomial for \(\ln x\) about \(1\).

Find \(P_3(x)\) for \(e^{2x}\) about \(0\).

Explain why \(P_N(c)=f(c)\) for a Taylor polynomial about \(c\).

Why is a quadratic Taylor term important near a stable equilibrium?

Find the quadratic Taylor polynomial for \(1/x\) about \(1\).

Find the cubic Taylor polynomial for \(x^4\) about \(1\).

Construct \(P_3(x)\) about \(2\) if \(f(2)=5\), \(f'(2)=1\), \(f''(2)=-4\), and \(f'''(2)=12\).

Choose a better centre, \(0\) or \(5\), to approximate \(\sqrt{x}\) near \(x=4.8\), and justify.

Why is the degree \(3\) Maclaurin polynomial for \(\cos x\) the same as degree \(2\)?

Diagnose the error: \(P_2=f(c)+f'(c)(x-c)+f''(c)(x-c)^2\).

Why is \(P_N\) exact for every \(x\) when \(f\) is a polynomial of degree at most \(N\)?

Explain one reason a linear approximation may fail even near the centre.