Question

For f(x,y)=√x - y² student submitted image, transcription available below, find the equation of the tangent plane to the surface z=f(x,y) at the point (5,1)

Answer 1

To find the equation of the tangent plane at the point (5,1), calculate the partial derivatives of the function at that point and apply them into the general formula for the tangent plane.

Step-by-step explanation:

To find the equation of the tangent plane to the surface represented by z = f(x,y) = √x - y2 at the point (5,1), we need to calculate the partial derivatives of the function f with respect to x and y.

First, find fx(5,1), the partial derivative of f with respect to x:
fx(x,y) = (1/2)x-1/2
fx(5,1) = (1/2)5-1/2 = 1/(2√5)

Next, find fy(5,1), the partial derivative of f with respect to y:
fy(x,y) = -2y
fy(5,1) = -2(1) = -2

Now that we have these derivatives, we can use them in the formula for the tangent plane at the given point (5,1,f(5,1)):

z - z0 = fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0)
z - (√5 - 1) = (1/(2√5))(x - 5) - 2(y - 1)

After substituting the values, we obtain the equation of the tangent plane:
z = (1/(2√5))(x - 5) - 2(y - 1) + (√5 - 1)