MATH 200 Lecture Notes - Lecture 4: Implicit Function, Partial Derivative, Canter And Gallop

• School: Drexel University
• Department: Mathematics
• Course: MATH 200
• Professor: unknown

Document Summary

3 space surface is similar to a 2d shape that is twisted and warped. A topographical map of the curve on an xy plane. Shape=cone with its point at (0,0,0) increasing in radius as z moves further to the left on the number line. Tip: try rewriting the equations with k=f(x,y) to make graphing easier f(x,y,z)=k becomes a 3-space surface. In general, for f(x,y), if we fix (cid:1877)=(cid:1877)then (cid:1858)(cid:4666)(cid:1876),(cid:1877)(cid:4667)is just a function of x. (cid:1856)/(cid:1856)(cid:1876) [(cid:1858)(cid:4666)(cid:1876),(cid:1877)(cid:4667)]=the instantaneous rate of change of f with respect to y. If z=f(x,y) and (cid:4666)(cid:1876),(cid:1877)(cid:4667) is a point in the domain of f, the partial derivative of f with respect to x at the point (cid:4666)(cid:1876),(cid:1877)(cid:4667), denoted (cid:1858)(cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667) is given by (cid:1858)(cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:1856)/(cid:1856)(cid:1876)[(cid:1858)(cid:3051)(cid:4666)(cid:1876) ,(cid:1877)(cid:4667)] | Similarly, the partial derivative of f with respect to y at the point (cid:4666)(cid:1876),(cid:1877)(cid:4667), denoted by (cid:1858)(cid:3052)(cid:4666)(cid:1876),(cid:1877)(cid:4667) is given by (cid:1858)(cid:3052)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:1856)/(cid:1856)(cid:1876)[(cid:1858)(cid:3052)(cid:4666)(cid:1876),(cid:1877) (cid:4667)] | (cid:3052)=(cid:3052) (cid:3051)=(cid:3051) The dashed line is a slice of the surface z=f(x,y) yo.