# Problem Set 9

1. Easy
Find the 3rd order Taylor polynomial of the following functions:
1. $$f(x,y) = \sin(x)\cos(y)$$ based at the point $$(0,0)$$.
2. $$f(x,y) = \frac{1}{1+x-y}$$ based at the point $$(0,0)$$.
3. $$f(x,y) = \log(1+x-y)$$ based at the point $$(0,0)$$.
4. $$f(x,y) = x+\cos(\pi y)+x\log y$$ based at the point $$(3,1)$$
5. $$f(x,y,z) = x^2y +z$$ based at $$(1,2,1)$$. Why should the remainder be zero?
6. $$f(x,y)=\cos(x^2)+xy^2-\frac{1}{1-xy}$$ based at the point $$(0,0).$$

2. Easy
Let $$f,g: \mathbb R^n \to \mathbb R$$ be $$C^\infty$$ functions, and let $$T(f), T(g)$$ be the Taylor series of $$f$$ and $$g$$ respectively. Show that $$T(f+g) = T(f) + T(g)$$.

3. Medium
Let $$p: \mathbb R^n \to \mathbb R$$ be a multi-variate polynomial. If $$T(p,\mathbf a)$$ is the Taylor series of $$p$$ about the point $$\mathbf a \in \mathbb R$$, show that $$T(p,\mathbf a) = p$$.

4. Hard
Derive the following version of the product rule'' for partial derivatives; if $$\alpha$$ is any multi-index, then: $\partial^{\alpha}(fg) = \sum_{\beta+\gamma = \alpha} \frac{\alpha!}{\beta!\gamma!} \partial^{\beta}f \partial^{\gamma}g$

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