Tangent Plane Calculator

Step-by-Step Guide to Using the Tangent Plane Calculator

The Tangent Plane Calculator helps you find the equation of the tangent plane to a surface defined by a function at a specific point P(x, y, z). Follow these steps to use the calculator effectively.

Step 1: Input the Coordinates of Point P

• Enter the x-coordinate of point P: Use the input field labeled “x-coordinate of point P”. This is a required field, so ensure you provide a numerical value.
• Enter the y-coordinate of point P: Use the input field labeled “y-coordinate of point P”. This field is also required, and it must be a numerical value.
• Enter the z-coordinate of point P: Use the input field labeled “z-coordinate of point P”. Like the other coordinates, this field is required and expects a number.

Step 2: Select the Function Type

In the provided field labeled “Select Function Type”, choose the expression that represents the surface on which you want to find the tangent plane. The options include:

• z = x² + y²
• z = xy
• z = x²y
• z = sin(x)cos(y)

This is a required selection. Make sure you choose the function that accurately describes the surface.

Step 3: Calculate Partial Derivatives

After entering the coordinates and selecting the function, the calculator will compute the partial derivatives:

• ∂z/∂x at point P: This value is calculated using the specified function. The partial derivative with respect to x will vary depending on the chosen function type.
• ∂z/∂y at point P: Similarly, this value is calculated based on the selected function, representing the partial derivative with respect to y.

Step 4: Obtain the Tangent Plane Equation

The calculator uses the computed partial derivatives and the input coordinates to formulate the tangent plane equation. This equation is derived as follows:

• Tangent Plane Equation: The equation is expressed as:

  z = pointZ + partialX * (x – pointX) + partialY * (y – pointY)

Step 5: Determine the Normal Vector

The normal vector to the tangent plane is automatically calculated and provided by the calculator in the form:

• Normal Vector: Represented as n = ⟨-partialX, -partialY, 1⟩.

By following these steps, you will effectively use the Tangent Plane Calculator to find both the equation of the tangent plane and the normal vector at point P on the given surface.