Enter the z-coordinate, function value, partial derivatives, and differences in x and y coordinates into the calculator to determine the equation of the tangent plane.

Equation Of The Tangent Plane Formula

The following formula is used to calculate the equation of the tangent plane to a surface at a given point. 

z = f(x0, y0) + fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0)

Variables:

  • z is the z-coordinate of the point on the tangent plane
  • f(x0, y0) is the value of the function at the point (x0, y0)
  • fx(x0, y0) is the partial derivative of the function with respect to x at the point (x0, y0)
  • fy(x0, y0) is the partial derivative of the function with respect to y at the point(x0, y0)
  • (x – x0) and (y – y0) are the differences between the x and y coordinates of a point on the tangent plane and the point of tangency (x0, y0) respectively

To calculate the equation of the tangent plane, evaluate the function and its partial derivatives at the point of tangency. Multiply the partial derivatives by the differences between the x and y coordinates of a point on the tangent plane and the point of tangency. Add these products and the value of the function at the point of tangency to get the z-coordinate of the point on the tangent plane.

What is a Equation Of The Tangent Plane?

The equation of the tangent plane is a mathematical formula that represents a plane which just touches a surface at a given point, without cutting into it. This plane is parallel to the surface at the point of tangency. In calculus, it is used to approximate the surface near that point, and its equation can be derived from the gradient of the function that defines the surface.

How to Calculate Equation Of The Tangent Plane?

The following steps outline how to calculate the Equation of the Tangent Plane.

  1. First, determine the value of the function at the point (x0, y0), denoted as f(x0, y0).
  2. Next, determine the partial derivative of the function with respect to x at the point (x0, y0), denoted as fx(x0, y0).
  3. Next, determine the partial derivative of the function with respect to y at the point (x0, y0), denoted as fy(x0, y0).
  4. Finally, use the formula z = f(x0, y0) + fx(x0, y0)(x – x0) + fy(x0, y0)(y – y0) to calculate the equation of the tangent plane.
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

f(x0, y0) = 5

fx(x0, y0) = 2

fy(x0, y0) = 3

x – x0 = 1

y – y0 = 2