Find the tangent plane to the surface determined by z = x√y for (x, y) = (1, 4). (Hint: You mustfirst calculate the value of z at the given point.

Step 1: Calculate the value of z at the given point (x, y) = (1, 4)

We have the equation z = x√y. Substituting x = 1 and y = 4, we get:

z = 1√4 = 1*2 = 2

So, the point in 3D space is (1, 4, 2).

Step 2: Find the partial derivatives of the function

The partial derivative of z with respect to x is √y and with respect to y is 1/2√y.

Substituting x = 1 and y = 4 into these derivatives, we get:

∂z/∂x = √4 = 2 ∂z/∂y = 1/2√4 = 1/2*2 = 1

Step 3: Write the equation of the tangent plane

The equation of the tangent plane at a point (a, b, c) to a surface z = f(x, y) is given by:

z = f(a, b) + fx(a, b)(x - a) + fy(a, b)(y - b)

Substituting the values we have:

z = 2 + 2*(x - 1) + 1*(y - 4)

Simplifying this, we get:

z = 2x + y - 2

So, the equation of the tangent plane to the surface at the point (1, 4, 2) is z = 2x + y - 2.