## 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 …