Fall 2007 - Cioaba's Class - Exam 2 (Version 2)

# Fall 2007 - Cioaba's Class - Exam 2 (Version 2) - Name...

This preview shows pages 1–3. Sign up to view the full content.

Name: PID: Discussion Section - No: Time: TA’s name: Midterm 2, Math 20C - Lecture D (Fall 2007) Duration: 50 minutes Please close your books, turn of your phones and put them away. You can use one page oF handwritten notes. To get Full credit you should explain your answers. Don’t Forget to write your name, ID number and your TA’s name. 1.a. (3 points) Find all second order partial derivatives of the function f ( x, y ) = 1 x 3 + y . Solution. We write f ( x, y ) = ( x 3 + y ) 1 . The ±rst order partial derivatives are f x = 3 x 2 · ( 1) · ( x 3 + y ) 1 1 = 3 x 2 ( x 3 + y ) 2 f y = 1 · ( 1) · ( x 3 + y ) 1 1 = ( x 3 + y ) 2 The second order partial derivatives are f xx = ( 6 x )( x 3 + y ) 2 + ( 3 x 2 )( 2)(3 x 2 )( x 3 + y ) 2 1 = 6 x ( x 3 + y ) 2 + 18 x 4 ( x 3 + y ) 2 f xy = ( 3 x 2 )( 2)( x 3 + y ) 2 1 = 6 x 2 ( x 3 + y ) 3 f yy = ( 1)( 2)( x 3 + y ) 2 1 = 2( x 3 + y ) 3 b. (2 points) Find the linear approximation of the function g ( x, y ) = e x +3 y at (0 , 0) and use it to approximate f (0 . 01 , 0 . 01). Solution. The linear approximation at (0 , 0) is f ( x, y ) f (0 , 0) + f x (0 , 0)( x 0) + f y (0 , 1)( y 0) We have f x ( x, y ) = e x +3 y and f y ( x, y ) = 3 e x +3 y which implies f x (0 , 0) = 1 and f y (0 , 0) = 3. Thus, the linear approximation near (0 , 0) is f ( x, y ) 1 + x + 3 y which implies that f (0 . 01 , 0 . 01) 1 + 0 . 01 + 3(0 . 01) = 1 . 04.

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

View Full Document
2. Suppose you are descending a valley whose shape is given by the equation z = 0 . 02 x 2 + 0 . 01 y 2 , where x, y and z are measured in meters, and you are standing at a point with coordinates (200 , 100 , 900). The positive
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

Fall 2007 - Cioaba's Class - Exam 2 (Version 2) - Name...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online