11. Partial Derivatives and Tangent Planes

d. Tangent Plane to the Graph of a Function

1. Geometric Derivation

On the previous page, we introduced the tangent lines to the \(x\)-trace and \(y\)-trace of a function of two variables, \(f(x,y)\), at \((x,y)=(a,b)\). By definition, the tangent plane to the graph at \((x,y)=(a,b)\) is the plane containing these two lines.

On this page, we display the tangent plane graphcally. On the next page we will find the formula for the tangent plane.

The plot on the left below is again the graph of the function \(f(x,y)=-x^2-y^2\) discussed in the exercises on the previous page. The \(x\) and \(y\) sliders are now on the right along with a 2D-slider to move both coordinates at once.

• Set the \(x\) and \(y\) sliders to \((x,y)=(6,5)\).
• Display the \(x\)-trace and \(y\)-trace through \((x,y)=(6,5)\).
• Add the tangent lines to the \(x\)-trace and \(y\)-trace through \((x,y)=(6,5)\).
• Finally, add the tangent plane at \((x,y)=(6,5)\).
• You can change the point of tangency by typing values for \(x\) or \(y\), dragging either 1D slider or dragging the 2D slider.
• You can animate either coordinate by selecting the Play buttons.
• Video Game:   Use your mouse to try and catch the yellow dot in the 2D slider while one or both of the Play buttons are checked.
• You can rotate the 3D plot with your mouse.
• We will find the equation of the tangent plane at \((x,y)=(6,5)\) in an exercise on the next page.

The plot on the left below is the graph of the function \(f(x,y)=-x^2y^3\).

• Set the \(x\) and \(y\) sliders to \((x,y)=(3,2)\).
• Display the \(x\)-trace and \(y\)-trace through \((x,y)=(3,2)\).
• Add the tangent lines to the \(x\)-trace and \(y\)-trace through \((x,y)=(3,2)\).
• Finally, add the tangent plane at \((x,y)=(3,2)\).
• We will find the equation of the tangent plane at \((x,y)=(3,2)\) in an example on the next page.