In \(a_0+a_1(x-c)+a_2(x-c)^2+\cdots\), what is \(a_1\)?

State the Taylor coefficient formula for \(a_n\).

Find \(a_0\), \(a_1\), and \(a_2\) in \(7+3(x-2)-5(x-2)^2\).

If \(f(4)=9\) and \(f'(4)=-1\), find \(a_0\) and \(a_1\) for the Taylor series about \(4\).

If \(f(0)=2\), \(f'(0)=6\), and \(f''(0)=10\), find \(a_0\), \(a_1\), and \(a_2\).

Find the coefficient of \((x+3)^4\) in \(1-2(x+3)^2+8(x+3)^4\).

Simplify \(4+2x+6\) and identify \(a_0\) and \(a_1\) about \(0\).

Write \(x^2\) in powers of \((x-2)\), and give \(a_0\), \(a_1\), and \(a_2\).

For \(f(x)=x^3\), find \(a_0\), \(a_1\), \(a_2\), and \(a_3\) about \(0\).

For \(a_n=\frac{(-1)^n}{n+2}\), list \(a_0\), \(a_1\), \(a_2\), and \(a_3\).

A Taylor series about \(1\) begins \(4+6(x-1)-3(x-1)^2\). Find \(f(1)\), \(f'(1)\), and \(f''(1)\).

Explain why the quadratic coefficient is not usually equal to the second derivative.

Find the first four coefficients of \(e^{3x}\) about \(0\).

Write \(2x^2-x+5\) in powers of \((x-1)\), and find \(a_0\), \(a_1\), and \(a_2\).

Interpret \(U(x)=U_0+0(x-c)+\frac{k}{2}(x-c)^2+\cdots\) near equilibrium.

Find \(a_0\), \(a_1\), \(a_2\), and \(a_3\) for the Maclaurin series of \(\sin x\).

For coefficients \(a_n=n!\), explain why \(\sum a_nx^n\) has radius \(0\).

Diagnose the error: \(a_3=f'''(c)\) in every Taylor series.

Show that coefficients depend on the centre by writing \(x^2\) about \(0\) and about \(1\).

Explain how \(a_0\), \(a_1\), and \(a_2\) encode value, slope, and curvature in a Taylor model.