{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sample_exam3

# sample_exam3 - Exam 3 Least Squres Fitting Numerical...

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

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: Exam 3: Least Squres Fitting, Numerical Integration, and Root Finding MATH 3315 / CSE 3365 : 801C – Spring Semester 2007 Total points: 100 Thursday 26 April The SMU honor code applies. Don’t forget to write and sign your name. An answer should include necessary steps, unless it requires only one step. Question 1 (one- or two-step problems, 4 points each, 20 points) (1) Write down the midpoint rule R M ( f ) for approximating the definite integral I ( f ) = R 2 e x d x , then calculate the value of the approximation. Answer: R M ( f ) = (2- 0) f 2 + 0 2 = 2 f (1) = 2 e 1 = 2 e. (2) Write down the trapezoidal rule R T ( f ) for approximating the definite integral I ( f ) = R 1 x 2 d x , then calculate the value of the approximation. Answer: R T ( f ) = 1- 2 [ f (0) + f (1)] = 1 2 [0 2 + 1 2 ] = 1 2 . (3) Write down Simpson’s rule R S ( f ) for approximating the definite integral I ( f ) = R π sin( x )d x , then calculate the value of the approximation. Answer: R S ( f ) = π- 6 [ f (0) + 4 f 0 + π 2 + f ( π )] = π 6 [sin(0) + 4 sin π 2 + sin( π )] = 2 π 3 . (4) Write down the Newton iteration for finding a simple root of the function f ( x ) = x 2- x- 2. ( Note: No calculations are needed.) Answer: g ( x ) = x- f ( x ) f ( x ) = x- x 2- x- 2 2 x- 1 = x 2 + 2 2 x- 1 , x n +1 = g ( x n ) = x 2 n + 2 2 x n- 1 , n = 0 , 1 , 2 , ··· . 1 (5) Write down the values of the points in the closed 4-point Newton-Cotes rule for approximating the definite integral I ( f ) = R 1- 1 f ( x )d x . ( Hint: Think about the meaning of “closed”, “4-point”, and how the points in Newton-Cotes rules are distributed.) Answer: x =- 1 , x 1 =- 1 3 , x 2 = 1 3 , x 3 = 1 . Question 2 (15 points) Find the least squares linear polynomial fit p 1 ( x ) = a + a 1 x to the data set i x i f i- 4- 3 1- 2- 2 2 2 3 4 1 Answer: The normal equations are " ∑ N i =0 1 ∑ N i =0 x i ∑ N i =0 x i ∑ N i =0 x 2 i # a a 1 = " ∑ N i =0 f i ∑ N i =0 f i x i # ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

sample_exam3 - Exam 3 Least Squres Fitting Numerical...

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

View Full Document
Ask a homework question - tutors are online