State Taylor's theorem using \(P_N\) and \(R_N\).

In the Lagrange remainder, what is \(\xi\)?

Write the Lagrange remainder for degree \(N\) in words or symbols.

For a linear Taylor polynomial, which derivative appears in the remainder?

Write \(e^x\) as its linear Maclaurin polynomial plus Lagrange remainder.

If \(|f'''(t)|\le 6\), bound the quadratic remainder at \(x=0.2\) about \(0\).

For \(f(x)=\sin x\), write the remainder after \(P_1(x)=x\).

If \(|f^{(4)}(t)|\le 5\) and \(|x-c|\le 0.1\), bound \(|R_3|\).

Bound the error in \(\cos x\approx 1-\frac{x^2}{2}\) for \(|x|\le 0.1\) using \(|f^{(3)}|\le 1\).

Explain how Taylor's theorem justifies linear response near \(c\).

Why does a Taylor remainder bound need an interval, not only the centre?

For \(e^x\approx 1+x\), bound the error on \(0\le x\le 0.2\) if \(e^t<2\).

Bound the error in \(\sin x\approx x-\frac{x^3}{6}\) for \(|x|\le 0.1\) using \(|f^{(4)}|\le 1\).

Show that if \(|f''(t)|\le 8\), then linear error is at most \(4|x-c|^2\).

If \(|R_1|\le x^2\), how small must \(|x|\) be to guarantee error below \(0.01\)?

Why does the degree \(N\) Taylor remainder start with order \(N+1\)?

Using \(|R_2|\le \frac{|x|^3}{6}\), is \(|x|\le 0.1\) guaranteed below tolerance \(10^{-4}\)?

Diagnose the error: \(R_N=\frac{f^{(N)}(\xi)(x-c)^N}{N!}\).

Why does Taylor's theorem not automatically prove an infinite Taylor series equals \(f\)?

Describe a rigorous strategy for proving a degree \(2\) truncation is accurate on \(|x-c|\le h\).