MATH 241 Study Guide - Midterm Guide: Partial Derivative, Cube Root, Tangent Space

• School: University of Maryland
• Department: Mathematics
• Course: MATH 241
• Professor: Roohollah Ebrahimian

Document Summary

Calculus 3: limits, partial derivatives and its applications review. This compilation of notes covers the exam on limits, partial derivatives and their applications. Determine if the following limits exist and if so, find the limit (cid:4666)(cid:3051),(cid:3052),(cid:3053)(cid:4667)=(cid:4666)(cid:2870),(cid:2871),(cid:2874)(cid:4667)(cid:1876)(cid:1877)(cid:4666)(cid:1878)(cid:4667)+(cid:887) lim. Just plug in the values since the function is defined everywhere. (cid:888)cos(cid:4666)(cid:888)(cid:4667)+(cid:887)=(cid:883)(cid:883) So the limit exists and is 11. lim(cid:4666)(cid:3051),(cid:3052)(cid:4667)=(cid:4666)(cid:2870),(cid:2871)(cid:4667)(cid:2871)(cid:3051)+(cid:3052) (cid:3117)(cid:3118)(cid:3051) (cid:3052) But this is no problem on the bottom since we don"t have zero when we plug in the numbers. We can evaluate the limit directly. (cid:885)(cid:4666)(cid:884)(cid:4667)+(cid:885) (cid:883)(cid:884)(cid:4666)(cid:884)(cid:4667) (cid:885)=(cid:891)/ (cid:884) So the limit exists and is -9/2. lim(cid:4666)(cid:3051),(cid:3052)(cid:4667)=(cid:4666)(cid:2868),(cid:2868),(cid:4667) (cid:3051)(cid:3052) (cid:2870)(cid:3051)(cid:3118)+(cid:2873)(cid:3052)(cid:3119) We have a problem here if we just plug in the values. We will get zero in the denominator and that is a big no no. However, that doesn"t mean the limit does not exist. We can reduce the limit to one variable if we simply focus on the vertical lines x=0 and y=0. (cid:2870)(cid:4666)(cid:2868)(cid:4667)+(cid:2873)(cid:3052)(cid:3118)=(cid:882) for x=0 (cid:2868)(cid:4666)(cid:3052)(cid:4667) (cid:2870)(cid:3051)(cid:3118)+(cid:2868)=(cid:882) for y=0 (cid:3051)(cid:4666)(cid:2868)(cid:4667)