# How do you find the quadratic taylor polynomial q(x,y) approximating #f(x,y)= e^(x) cos (5y)# about (0,0)?

Refer to explanation

The quadratic Taylor approximation is

Hence we have that

Hence we have that

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To find the quadratic Taylor polynomial ( q(x,y) ) approximating ( f(x,y) = e^x \cos(5y) ) about the point ( (0,0) ), follow these steps:

- Calculate the partial derivatives of ( f(x,y) ) with respect to ( x ) and ( y ) at ( (0,0) ).
- Use these partial derivatives to compute the coefficients of the quadratic terms in the Taylor polynomial.
- The quadratic Taylor polynomial ( q(x,y) ) is given by the following expression:

[ q(x,y) = f(0,0) + f_x(0,0) \cdot x + f_y(0,0) \cdot y + \frac{1}{2} f_{xx}(0,0) \cdot x^2 + f_{xy}(0,0) \cdot xy + \frac{1}{2} f_{yy}(0,0) \cdot y^2 ]

- Substitute the calculated values of ( f(0,0) ), ( f_x(0,0) ), ( f_y(0,0) ), ( f_{xx}(0,0) ), ( f_{xy}(0,0) ), and ( f_{yy}(0,0) ) into the expression to obtain the quadratic Taylor polynomial ( q(x,y) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the interval of convergence of the MacClaurin series of #f(x)=1 / (3-2x)#?
- How do you find the maclaurin series expansion of #cos (x)^2#?
- How do you find a power series representation for # (x/(2-x))^3# and what is the radius of convergence?
- Given 1- cosx, how do you find the Taylor polynomial?
- How do you find the maclaurin series expansion of #x^3/(1+x^2)#?

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