Mathematics Exam: Limits, Gradients, and Integrals, Exams of Calculus

Download Mathematics Exam: Limits, Gradients, and Integrals and more Exams Calculus in PDF only on Docsity! Final 253 You have 150 min to solve 7 problems. Please note • Write your name/student id clearly • Write only on one side of the paper in the space design for it. Use the last few pages of the exam for calculations. Good luck! Name: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Student ID: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 1 1. Find the limit or show that it does not exist lim (x,y,z)→(0,0,0) xy + yz2 + xz2 x2 + y2 + z4 2 . 5 3. If z = f(x, y) where x = s + t and y = s− t show that (a) (zx) 2 − (zy)2 = (zs) (zt) (b) 4(zx)(zy) = (zs) 2 − (zt)2 6 . 7 5. a Find the critical points of the function f(x, y) = −(x2 − 1)2 − (x2y − x− 1)2 b Explore the points and check if they are minima/maxima or saddle points c In 3 lines or less, explain why the result you obtain does not have an equivalent when considering a function of a single variable. 10 . 11 6. Evaluate the integral ∫ 4 0 ∫ 2 √ x 1 y3 + 1 dy dx 12 . 15 Scrap Paper (we will not look here) 16 Scrap Paper (we will not look here) 17