# termtest - as possible Which point or points on the island...

This preview shows pages 1–2. Sign up to view the full content.

MAT237Y – Multivariable Calculus Summer 2009 Term Test Thursday, June 25 from 6:10pm to 8:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] DeFne the directional derivative . (b) [5 marks] Give the deFnition of a convex set. 2. [10 marks] Is the set { ( x, y, z ) R 3 : y = e x , z = sin 2 ( x ) } connected? Explain. 3. (a) [5 marks] Consider the set C R 3 which is the solution of the system of equations arctan( y ) + ( x + 1) 5 = x 4 ( z 1) 3 cos( y ) + x 2 = 1 + ( y 3) z. Show that in a neighbourhood of the origin, y and z can be rep- resented as functions of x . (b) [5 marks] ±ind the line tangent to C at the origin. Give it as a subset of R 3 . 4. [10 marks] You are vacationing on the South PaciFc island of Laga Ranga with a coastline given by the set { ( x, y ) R 2 : 4 x 2 + y 2 = 16 } . Just as you are about to order another pi˜na colada from your resort’s swim-up bar, the ground violently shakes. Mt. Leibniz, located at ( 1 , 0), has erupted and will soon cover the entire island with molten lava. As you have no wish to see your hedonistic life come to an end, you decide to a head to a point on the island as far from the volcano

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: as possible. Which point or points on the island qualify? 5. [10 marks] Classify the critical points of the function f : R 2 → R , ( x, y ) m→ ( x − 2)( x 2 − y 2 ) . 1 6. [10 marks] Suppose that U is the non-convex set U = { ( x, y ) ∈ R 2 : y < x 2 + 1 } and f : U → R is a C 1 function such that b∇ f ( x ) b ≤ 3 for x ∈ U . Using the Mean Value Theorem, show that f (3 , 4) ≤ f ( − 3 , 0) + 30. 7. [10 marks] DeFne g : R 2 → R , ( x, y ) m→ sin( x 2 ) cos( y ) . ±ind ∂ α g (0 , 0) for all multi-indices α of order six. Note: Once you have a numerical answer, you need not simplify it. That is, 4! / 2! is just as acceptable an answer as 12. 8. Bonus Question: [5 marks] Let A be an n × n positive deFnite matrix and B j an n × n positive semi-deFnite matrix for j = 1 , . . . , ℓ . Show that the sum A + ℓ s j =0 B j is a positive deFnite matrix. 2...
View Full Document

## This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto.

### Page1 / 2

termtest - as possible Which point or points on the island...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online