Question: Polynomial approximation works in higher dimensions just as it does in one dimension. Linearization is the first-degree Taylor polynomial; the second derivative test comes from the quadratic approximation, as mentioned in the book. This problem deals with higher-order generalizations. We will use the function \(g(x, y)=e^{2 x} \sin 3 y\) as our demonstration function.

1. Write down the one-variable 3rd-order Taylor expansion of \(e^{2 x}\) and \(\sin (3 x)\) at the origin.
2. Use those polynomials to write down a polynomial approximation of \(g\) in which the total degree powers of the terms is no more than 3 .
3. Compute the values of the first three levels of partial derivatives evaluated at the origin (just the values is fine).
4. If you think of \(h \frac{\partial f}{\partial x}+k \frac{\partial f}{\partial y}\) as an operator acting on \(f\) and that it could be written instead as \(\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right) f\), then we could also think of squaring it, cubing it, etc., meaning that we apply it two times, three times, etc. Note that \(h\) and \(k\) are thought of as vector differences, i.e., we are creating an approximation to \(f(a+h t, b+k t)\) at the point \((a, b)\) and so \(h\) and \(k\) are the vector directional changes. They appear with the partial derivatives thanks to the chain rule. Indeed,
   \[[f(a+h t, b+k t)]^{\prime \prime}=h^{2} f_{x x}+2 h k f_{x y}+k^{2} f_{y y}=\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right)^{2} f\]
   where \([f]^{\prime \prime}\) denotes the derivative with respect to \(t\). What is \(\left(h \frac{\partial}{\partial x}+k \frac{\partial}{\partial y}\right)^{3} f ?\)
5. The multivariable formula for Taylor's theorem can be written as

where \(c\) is a number between 0 and \(1 ;\left.\right|_{(a, b)}\) means evaluate at \((a, b) ;\) and the last term is how we estimate the error, just as with one variable functions. Write down the third order Taylor polynomial for \(g\) expanded at the origin. How does it compare to the one computed above?

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