Find the unit vectors that are tangent and normal to the curve at the given point (four vectors in all). Then sketch vectors and curve together. (I will put the equation and point later)

Question

anonymous · Answer

$$y=\sum_{n=0}^{\infty}\frac{ x^n }{ n! }$$

The point is (0,1)

anonymous · Answer

Do you recognize that sum as the power series representation of some nice function?

anonymous · Answer

I guess I don't. I'm pretty clueless :(

anonymous · Answer

Thats the same as e^x. Try writing out the expansion of e^x about x=0 and see

anonymous · Answer

I'm afraid I don't know how to make out the expansion either..

anonymous · Answer

The expansion of a function f about x=0 is 
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^{n}$$ and in this case the expansion converges to the function for all x.

anonymous · Answer

so the equation equals to e^x?

anonymous · Answer

Yes.

anonymous · Answer

is it right when I answer the unit vector tangent of the curve is <1,1> and the unit vector normal is <1,-1>?