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Throughout this file, T(a,b)(x, y) denotes the function whose graph is the tangent plane to the graph of a given function f (x, y) at the point x = a, y = b, z = f (a, b). The general formula for the tangent plane is the following:
T(a,b)(x, y) = f (a, b) + โf โx
(a, b)(x โ a) + โf โy
(a, b)(y โ b). (โ)
Example 1. Find T(2,โ1)(x, y) for the function
f (x, y) =
9 โ x^2 โ y^2.
Note that f (2, โ1) = 2. Compute the partial derivatives:
โf โx
x โ 9 โ x^2 โ y^2
โf โy
y โ 9 โ x^2 โ y^2
Evaluate these partial derivatives at (2, โ1):
โf โx
โf โy
Use the general equation (โ) to get
T(2,โ1)(x, y) = 2 โ (x โ 2) +
(y + 1) = โx + 0. 5 y + 4. 5.
The graph of this function is the desired tangent plane
z = T(2,โ1)(x, y) or x โ 0. 5 y + z = 4. 5.
Note that the normal vector to this plane is n = ใ 1 , โ 0. 5 , 1 ใ.
Understanding of the result: The graph of the function
z =
โ 9 โ x^2 โ y^2
is the upper hemisphere of radius 3 centered at the origin. It is known from geometry that a tangent plane to a sphere is orthogonal to the radius at the point of tangency. In our case, the point is (a, b, f (a, b)) = (2, โ 1 , 2). The position vector of this point is actually the radius of the sphere, and it is indeed parallel to n.
x
y
z
Example 2. Let f (x, y) = โ 3 xy+y^2. Use T(โ 1 ,2)(x, y) to approximate the value f (โ 0. 8 , 2 .1). Compare the approximation with the actual value.
Note that f (โ 1 , 2) = 10. Compute the partial derivatives:
โf โx = โ 3 y, โf โy = โ 3 x + 2y.
Evaluate the derivatives at (โ 1 , 2):
โf โx
โf โy
Use (โ) to write the equation of the tangent plane:
T(โ 1 ,2)(x, y) = 10 โ 6(x + 1) + 7(y โ 2) = โ 6 x + 7y โ 10.
This function provides the best linear approximation of f (x, y) near (โ 1 , 2). Therefore,
f (โ 0. 8 , 2 .1) โ T(โ 1 ,2)(โ 0. 8 , 2 .1) = 10 โ 6(0.2) + 7(0.1) = 9. 5.
This is indeed close to the actual value f (โ 0. 8 , 2 .1) = 9.45.
Example 3. Find the points in the graph of f (x, y) = y^2 eโx^ where the tangent plane is parallel to the plane 3 x โ 2 y + 5z = 1.
Suppose a desired point has coordinates (a, b) in the xy-plane. Then the third coordinate is z = f (a, b) = b^2 eโa. Find the tangent plane at the point (a, b). First, compute the partial derivatives
โf โx = โy^2 eโx, โf โy = 2yeโx.
Evaluate the derivatives at (a, b):
โf โx (a, b) = โb^2 eโa, โf โy (a, b) = 2beโa.
The equation of the tangent plane is
z = f (a, b) โ b^2 eโa(x โ a) + 2beโa(y โ b),
or โb^2 eโax + 2beโay โ z = d,
where d is a constant (it depends on a and b; it is easy to find the explicit expression, but we do not need it in this problem). The normal vector to this tangent plane is
n = ใโb^2 eโa, 2 beโa, โ 1 ใ.
Planes are parallel if and only if their normal vectors are parallel. Hence, n has to be parallel to ใ 3 , โ 2 , 5 ใ: ใโb^2 eโa, 2 beโa, โ 1 ใ โ ใ 3 , โ 2 , 5 ใ.
