Math 23b Theoretical Linear Algebra and Multivariable Calculus II MIDTERM EXAM 2 April 17 2006 Your name Problem 1 2 3 4 5 Total Points 21 20 20 20 20 101 Score In the following problems you can use any of the results we have proved in class if you state them clearly before using them Please show all your work on this exam paper You must show your work and clearly indicate your line of reasoning in order to get full credit If you have work on the back of a page indicate that on the exam cover 1 2 Problem 1 Decide whether the following statements are True or False Note There is no need to justify your answers You get 3 for every correct answer and 1 for every wrong answer T or F If f Rn Rm is differentiable at x a then all its directional derivatives are continuous at x a T or F If f R2 Rm is differentiable at a b then 2f x y a b 2f y x a b T or F A smooth function defined in the closed unit disk x y x2 y 2 1 has a maximum in this disk T or F A smooth function is equal to its Taylor series T or F Let R Rn be a closed rectangle If f R R is integrable on R then f is continuous in the interior of R T or F Every finite set S Rn is Riemann measurable T or F If f Rn R is continuous then f is integrable on every bounded Riemann measurable set A Rn 3 Problem 2 Prove or disprove each of the following statements In order to prove a statement just provide a brief justification while in order to disprove you need to present a counterexample no explanation is necessary as long as the example is correct a Suppose the function f R2 R is differentiable at 1 1 and Dh f 1 1 Dk f 1 1 0 where h 1 1 and k 1 2 Then 1 1 is a critical point for f b Let f R2 R be a smooth function Then the functions f x y and g x y f x y 2 have the same critical points c Let f g R2 R be smooth functions and consider a max min problem for f with the constraint g x y 0 Suppose f x y has a maximum value at a point a relative to the constraint g x y 0 If the Lagrange multiplier if 0 then a is also a critical point for f without the constraint d Let A Rn be bounded and Riemann measurable Suppose that f Rn R is bounded Then f is integrable on A Answer Proof or counterexample a T or F b T or F c T or F d T or F 4 Problem 3 Let f R4 R be given by f x y z exy xyz x sin z and let a 2 0 2 a Find the derivative of f at a b Find the directional derivative Dh f a in the direction of h 3 1 2 c Find the tangent plane to the graph of f at a Answer a f 0 a b Dh f a c 5 Problem 4 Let f R2 R be a smooth function Recall that the Laplacian of f is defined as 2 f 2f 2f 2 2 x y and that f is said to be harmonic if 2 f x 0 for every x R2 We now say that f is subharmonic if 2 f x 0 x R2 Let D be the unit disk centered at the origin o n D x y R2 x2 y 2 1 Prove that if f is subharmonic in D then f does not have a maximum in the interior of D Argument in short 6 Problem 5 Let R R3 be a closed rectangle and let f R R be a continuous non negative function i e f x y z 0 x R y z R Prove or disprove if there is some a R such that f a 0 then R f 0 Answer in short Proof or counterexample T or F 7 page intentionally left blank 8 page intentionally left blank