White Boards From Introduction to Taylor Polynomials.

We’re build Taylor Polynomials for the function \(\displaystyle f(x)=e^x\) where \(a=0\).
(\((0,f(0))\) is the point where we know the values of \(f\) and all of the derivatives.)

The first degree Taylor polynomial is \(p_1=1+x\)

Developing the Second Degree Taylor Polynomial, \(p_2(x)\)

Recall that the derivatives of \(f(x)\) at \(x=0\) must match the derivatives of \(p_2(x)\) at \(x=0\). We can start with \(p_1(x)\) and just add an \(x^2\) term. We let the coefficient of \(x^2\) be \(c\)

Developing \(p_3(x)\)

We start with \(p_2(x)\) and just add an \(x^3\) term. We let the coefficient of \(x^3\) be \(d\). Here we solve of \(d\) using the fact that Taylor polynomials share derivative vales with \(f(x)\) at \(x=a\).

Developing \(p_4(x)\)

We start with \(p_3(x)\) and just add an \(x^4\) term. We let the coefficient of \(x^4\) be \(g\).

Notice that I avoiding doing the arithmetic. Sometimes a pattern is easier to see …