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