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Exam Questions: Multivariable Calculus - Maxima, Minima, Gradients, Jacobians, Exams of Mathematics

Birla Institute of Technology and ScienceMathematics

The examination questions for a multivariable calculus course, focusing on finding maximum and minimum values, gradients, jacobian matrices, tangent planes, and applying the chain rule.

Typology: Exams

2012/2013

Uploaded on 03/07/2013

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NAME_______________________________________
I______II______III______IV______V______VI______VII______ TOTAL ___________
October 24 Mathematics 206a Mr. Haines
2003 Multivariable Calculus
Examination #2
(11) I. Find the maximum and minimum values of the function
yxyxf โˆ’+= 1),( on the set [0,1] ร— [0,1] .
(21) II. Suppose yzxyyxzyxf 3),,( 32 โˆ’โˆ’+= and a = (1, 1, 1) .
A. โˆ‡f(x,y,z) =
B. โˆ‡f(a) =
C. The directional derivative of f at a in the direction parallel to x = (1, 2, 3) is
pf3
pf4

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Download Exam Questions: Multivariable Calculus - Maxima, Minima, Gradients, Jacobians and more Exams Mathematics in PDF only on Docsity!

NAME_______________________________________

I______II______III______IV______V______VI______VII______ TOTAL ___________

October 24 Mathematics 206a Mr. Haines 2003 Multivariable Calculus Examination #

(11) I. Find the maximum and minimum values of the function

f ( x , y )= 1 + x โˆ’ y on the set [0,1] ร— [0,1].

(21) II. Suppose f ( x , y , z )= x^2 y^3 + xy โˆ’ z โˆ’ 3 y and a = (1, 1, 1).

A. โˆ‡ f(x,y,z) =

B. โˆ‡ f ( a ) =

C. The directional derivative of f at a in the direction parallel to x = (1, 2, 3) is

(28) III. Suppose f^ :^ โ„œ^3 โ†’โ„œ^3 such that

f ( x , y , z )=( y , x^2 , z ) for all ( x , y , z )โˆˆโ„œ^3 and that a = (1, 2, 3).

A. The Jacobian matrix of f at a ,

J f ( a ) =

B. The (total) derivative of f at a ,

( Df ( a ))(x, y, z) =

C. curl f ( a ) =

D. div f ( a ) =

(10) VI. The point (0, 2) is a critical point of f (^ x , y )= 2 x^2 + x^2 y + y^2 โˆ’^4 y. Use the Second

Derivative Test to determine whether (0, 2) is a local minimum, a local maximum, or neither.

(10) VII. Suppose f ( x , y , z )= x + y^2 โˆ’ z^3 and g ( s , t )= ( st , s โˆ’ t , s + t ) Use the chain rule to

find the derivative of f ฮฟ g at the point (1, 1).