Problem 1. Find the limits below or explain why they do not exist: (i) limx,y0
(x2 +y 2 )2 2x2 +3y 2
(2x2 + 3y 2 )2 (x2 + y 2 )2 = 2x2 + 3y 2 0. 2x2 + 3y 2 2x2 + 3y 2 Therefore, the original limit equals 0 as well. 0 (ii) limx,y0
x3 y x4 +y 4
We note that
Math 31BH - Winter 2011 - Midterm II
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PRACTICE PROBLEMS FOR MIDTERM II Problem 1. Find the critical points of the function f (x, y ) = 2x3 3x2 y 12x2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Problem 2. Determine the global max and min
Problem 1. Find the critical points of the function f (x, y ) = 2x3 3x2 y 12x2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Solution: Partial derivatives fx = 6x2 6xy 24x, fy = 3x2 6y. To nd the critic
Problem 1. Consider the function f (x, y ) = 3y 2 2y 3 3x2 + 6xy. Find the critical points of the function and determine their nature. We calculate fx = 6x + 6y = 0 = x = y fy = 6y 6y 2 + 6x = 0 = x + y = y 2 = 2y = y 2 = y = 0 or y = 2. We nd the second
PRACTICE PROBLEMS FOR MIDTERM I Problem 1. Find the limits (i) limx,y0 (ii) (iii)
x2 +y 6 x6 +y 2 ; 6 6 6 limx,y,z0 x2 +y2 +z2 . x +y +z 2 2 2 limx,y0 sin(x +y 2) x +2y
Problem 2. Consider the function f (x, y ) = (x 1)2 + (y 1)2 . (i) Draw the level diag
Math 31BH - Winter 2011 - Midterm I
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Instructions: Please print your name and student ID. During the test, you may not use books or notes. Read each question carefully, and show all your work. Answers with no explanation will receive no c
RECOMMENDED PROBLEMS - FINAL EXAM 1. Point Set Topology Prove rigourously that the set of matrices where A + A2 is invertible is open in Matnn . 2. Limits Problem 1.19, page 158. 3. Continuity Using and s, show that if f and g are continuous real valued f
Problem 1. Find the critical points of the function f (x, y ) = and determine their nature. Solution: We have fx = x y 3 = 0 = x = y 3 fy = 3y 3xy 2 = 0 = y = xy 2 = y = y 5 = y = 0, y = 1, or y = 1. We nd the critical points (0, 0), (1, 1) (1, 1). We cal
Math 31BH - Homework 1. Due January 11.
Part I. From the textbook solve the following problems: 1. (Monday) Problem 1.5.1, 1.5.2, 1.5.3. 2. (Wednesday-Friday) Problem 1.5.14(a) (b), 1.5.19, 1.5.21(a) (c) (e). Part II. EC0. Problem 1.5.24. EC1. (Monday. Ex
Math 31BH - Homework 2. Due January 25.
Part I. 1. (Wednesday, Jan 12.) Using the denition, show that the sequence xn = converges and nd its limit. 2. (Friday, Jan 14.) For the two functions (a) f (x, y ) = 2 x 2y (b) f (x, y ) = 2 (x 1)2 y 2 answer the f
Math 31BH - Homework 3. Due Wednesday, February 2.
Part I. 1. (Wednesday, January 26.) Laplacian and harmonic functions. (i) The temperature T (x, y ) in a long thin plane at the point (x, y ) satises Laplaces equation Txx + Tyy = 0. Does the function T (
Math 31BH - Homework 4. Due Tuesday, February 8.
1. (Wednesday, January 26.) Chain rule. From the textbook, solve 1.8.2, 1.8.10(a), 1.8.11. 2. (Wednesday, February 3.) Eulers identity. A function f : Rn R is said to be homogeneous of degree d if f (tx1 ,
Math 31BH - Homework 5. Due Tuesday, February 15.
1. (Wednesday, February 9.) Pathological functions. From the textbook, solve 1.9.1 and 1.9.2. 2. (Wednesday, February 9.) Pathological functions and second order derivatives. Put f (x, y ) = xy 0
x2 y 2 x2
Math 31BH - Homework 6. Due Thursday, February 24.
1. (Wednesday - Friday) From the textbook, solve 3.7.1, 3.7.3(a), 3.7.4(b), 3.7.5, 3.7.6, 3.7.7(a), 3.7.13, 3.7.14. 2. (Wednesday - Friday) Using Lagrange multipliers, optimize the function f (x, y ) = x2
Math 31BH - Homework 7. Due Thursday, March 3.
1. From the textbook, solve 2.10.2, 2.10.5, 2.10.9, 2.10.12. 2. Problems 3.1.2, 3.1.5(a), 3.1.6, 3.1.7, 3.1.10, 3.1.11, 3.1.19(a). 3. Recall the Taylor expansion ez = 1 + z + z2 z3 + + . 2! 3!
(i) Formally su
Math 31BH - Winter 2011 - Final Exam
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Instructions: During the test, you may not use books or notes. Read each question carefully, and show all your work. Answers with no explanation will receive no credit, even if they are correct. There