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Problem 110

# Find the equation of the tangent plane to the graph of $$z=f(x, y)=x^{2}+2 y^{2}-1$$ at the points (a) $$\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(0,0)$$ (b) $$\left(\mathrm{x}_{0}, \mathrm{y}_{0}\right)=(1,1)$$.

Expert verified
The tangent planes to the graph of $$z = x^2 + 2y^2 - 1$$ at the given points are: (a) at (0, 0): $$z = -1$$ (b) at (1, 1): $$z = 2x + 4y$$
See the step by step solution

## Step 1: Finding the Partial Derivatives

We first need to determine the partial derivatives of the function. Let's compute the partial derivative with respect to x and y: $$\frac{\partial z}{\partial x} = \frac{\partial}{\partial x}(x^2 + 2y^2 - 1)$$ $$\frac{\partial z}{\partial x} = 2x$$ $$\frac{\partial z}{\partial y} = \frac{\partial}{\partial y}(x^2 + 2y^2 - 1)$$ $$\frac{\partial z}{\partial y} = 4y$$

## Step 2: Finding the Tangent Plane Equation

The general equation for the tangent plane to the graph of a function is given by: $$z - z_0 = \frac{\partial z}{\partial x}(x - x_0) + \frac{\partial z}{\partial y}(y - y_0)$$ where (x_0, y_0, z_0) is the point where we want to find the tangent plane. We now have all the components needed, and can plug in the partial derivatives and the given points:

## Step 3: Tangent Plane at (0, 0)

Plugging (x_0, y_0) = (0, 0) into the function to find z_0: $$z_0 = f(x_0, y_0) = f(0, 0) = 0^2 + 2*0^2 - 1 = -1$$ Now, plugging the point and the partial derivatives into the tangent plane equation: $$z - (-1) = 2x(0 - x) + 4y(0 - y)$$ $$z + 1 = 0$$ So, the equation for the tangent plane at (0, 0) is: $$z = -1$$

## Step 4: Tangent Plane at (1, 1)

Plugging (x_0, y_0) = (1, 1) into the function to find z_0: $$z_0 = f(x_0, y_0) = f(1, 1) = 1^2 + 2*1^2 - 1 = 2$$ Now, plugging the point and the partial derivatives into the tangent plane equation: $$z - 2 = 2(1 - x) + 4(1 - y)$$ $$z - 2 = 2 - 2x + 4 - 4y$$ So, the equation for the tangent plane at (1, 1) is: $$z = 2x + 4y$$ The equations of the tangent planes to the graph of $$z = x^2 + 2y^2 - 1$$ at the given points are: (a) at (0, 0): $$z = -1$$ (b) at (1, 1): $$z = 2x + 4y$$

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