Practice Problems on Gradients and Directional Derivatives 1) Consider the function f ( x,y ) = ln( p x 2 + y 2 ). Find its gradient at the point ( x,y ) = (1 ,-1). At this point, what is the directional derivative of f in the direction (3 / 5 , 4 / 5)? 2) Pikabo Street is skiing on a Colorado mountain. Colorado is a large square state with a surface of many high bumps. Surveyers have mapped out the state into “sections” which are 1 mile by 1 mile squares. The “origin” for each section is its Southwest corner. Each point within a section is assigned a number ( x,y ) where x feet is the number of feet that this point lies East from the origin and y is the number of feet that it lies North of the origin is denoted by ( x,y ). The altitude of the earth’s surface at the point ( x,y ) is given by the formula z = f ( x,y ). In the region of interest to us, f is a continuously diﬀerentiable function. Suppose that
This is the end of the preview. Sign up
access the rest of the document.