Partial Derivatives and Slopes
The plane x = 1 intersects the paraboloid in a parabola. Find the slope of the tangent to the parabola at (1,2,5). Graph this tangent line.
Above we see , let's now add the x = 1 plane.
Let us now add the point (1,2,5)
We now desire the line which goes through the point (1,2,5) and is tangent to the parabola which is the intersection of x = 1 and .
Notice that the partial derivative is: since we want the slope at x = 1, y = 2, and z = 5. This is 4. Now we want the line where m is the slope and b is the z intercept. We simply plug in our point y=2 and z=5 and solve for b which is equal to -3. Therefore the equation for the line on the x=1 plane which is tangent to the parabola of intersection between the plane x=1 and the surface is . We use this to find the point y = 3 and z = 9. Since x=1 is fixed, we use the coordinate (1,3,9) as the next point on our tangent line. We use this to graph the line in geogebra which connects the point (1,2,5) to the point (1,3,9).