Labware - MA35 Multivariable Calculus - Two Variable Calculus



Constrained Maxima & Minima


A special case of the chain rule occurs when z(t) = f(x(t),y(t)) = c is constant.

We then have

\begin{array}{rcl} 0 = (c)' = z'(t) & = & f_x(x(t),y(t))x'(t) + f_y(x(t),y(t))y'(t) \\ & = & \nabla f(x(t),y(t)) \cdot (x'(t),y'(t)) . \end{array}

It follows that the gradient of a function of two variables at a point (x(t),y(t)) is perpendicular to the tangent vector to a level curve through the point.

The highest point on the Massachusetts Turnpike is indicated by a sign somewhere in the hilly region of Western Massachusetts. This point will occur when the gradient vector of the turnpike's path is either the zero vector or when it is perpendicular to the turnpike's tangent vector. In the first case, the turnpike's maximum height occurs at a critical point of the surface of Western Massachuetts. In the second case, the turnpike's maximum height occurs when it is tangent to a contour line.



  • 1. Use the tape deck for the variable c to determine the highway's highest point. How are the path's tangent vector (light grey) and the surfaces gradient vector (yellow) related, as seen in the "Domain" window? At what angle is the gradient slice curve (yellow) cutting the curve (red) on the surface, as seen in the "Scenic Highway" window?
  • 2. Try entering w(t)=(t,t) and determine the highest point on the highway for this new path. What happens to the gradient vector (yellow) at the highest point? What happens to the gradient vector on either side of the highest point? What kind of critical point is this? Why?