Problem 2 - Calculus Early Transcendentals

Question

Describe the usual appearance of a smooth surface at a saddle point.

Step-by-Step Solution

Answer

A: A smooth surface at a saddle point appears like a hyperbolic paraboloid, resembling the shape of a saddle. It rises along one axis while falling along the orthogonal axis. The tangent plane at the saddle point is horizontal, and the surface curves away from the tangent plane in the orthogonal directions, with positive curvature in one direction and negative curvature in the other.

Step 1: Define a saddle point

A saddle point is a point on a surface where the surface appears to be curved upwards along one axis and curved downwards along another axis, resembling the shape of a saddle. Mathematically, at a saddle point, the value of the partial derivatives of the function that defines the surface with respect to each axis is equal to zero, but the second derivative test is inconclusive.

Step 2: Consider the properties of a saddle point

At a saddle point, the tangent plane is horizontal. However, the curvature of the surface is different in different orthogonal directions. One direction has positive curvature (concave up), and the perpendicular direction has negative curvature (concave down). This means that at a saddle point, the surface has a local minimum along one axis and a local maximum along the orthogonal axis.

Step 3: Describe the appearance of a smooth surface at a saddle point

The smooth surface at a saddle point has the shape of a hyperbolic paraboloid. It looks like the surface is rising and falling at the same time - it rises in a direction along one axis and falls in the direction of the orthogonal axis. It resembles a saddle, where the middle part is sinking down, and the edges curve upwards. The tangent plane at the saddle point is horizontal, and the surface curves away from the tangent plane in the orthogonal directions.