# How do you use the taylor series to find a quadratic approximation of #2x^2 - 3xy + x# at (1,1)?

We therefore need the following partial derivatives:

This means that, working to only the quadratic terms:

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To find a quadratic approximation of (2x^2 - 3xy + x) at the point ((1,1)) using Taylor series, follow these steps:

- Find the first and second-order partial derivatives of the function with respect to (x) and (y).
- Evaluate these derivatives at the point ((1,1)).
- Use the Taylor series expansion formula for a function of two variables up to the quadratic term.

The Taylor series expansion of a function (f(x,y)) up to the quadratic term around the point ((x_0,y_0)) is given by:

[f(x,y) \approx f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) + \frac{1}{2}[f_{xx}(x_0,y_0)(x-x_0)^2 + 2f_{xy}(x_0,y_0)(x-x_0)(y-y_0) + f_{yy}(x_0,y_0)(y-y_0)^2]]

Now, let's proceed with finding the required derivatives:

[f(x,y) = 2x^2 - 3xy + x]

First-order partial derivatives:

[f_x = 4x - 3y + 1] [f_y = -3x]

Second-order partial derivatives:

[f_{xx} = 4] [f_{xy} = -3] [f_{yy} = 0]

Now, evaluate these derivatives at the point ((1,1)):

[f(1,1) = 2(1)^2 - 3(1)(1) + 1 = 0] [f_x(1,1) = 4(1) - 3(1) + 1 = 2] [f_y(1,1) = -3(1) = -3] [f_{xx}(1,1) = 4] [f_{xy}(1,1) = -3] [f_{yy}(1,1) = 0]

Substitute these values into the Taylor series expansion formula:

[f(x,y) \approx 0 + 2(x-1) - 3(y-1) + \frac{1}{2}[4(x-1)^2 - 6(x-1)(y-1)]]

Simplify the expression to get the quadratic approximation:

[f(x,y) \approx 2x - 3y + 2(x-1)^2 - 3(x-1)(y-1)]

This is the quadratic approximation of (2x^2 - 3xy + x) at the point ((1,1)).

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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.

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