Assignment 1 - solution - University of Texas at Dallas...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: University of Texas at Dallas Assignment #1 Last Name: First Name and Initial: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Course Name: Number: Mathematical Analysis 2 MATH 4302 Instructor: Due Date: Wieslaw Krawcewicz January 31, 2012 E-mail Address: Student’s Signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Question Weight Your Score Comments 1. 10 2. 10 3. 10 4. 10 5. 10 6. 10 7. 10 8. 10 9. 10 10. 10 Total: 100 2 Problem 1. Consider the function f : [ − a, a ] → R , a > 0, defined by f ( x ) = cosh (3 x ) , x ∈ [ − a, a ] . (a): Compute the n-th derivative of the function f . Solution: Notice that we have (by simple induction) cosh(3 x ) = e 3 x + e − 3 x 2 , d dx cosh(3 x ) = 3 1 e 3 x + ( − 1) 1 e − 3 x 2 d 2 dx 2 cosh(3 x ) = 3 2 e 3 x + ( − 1) 2 e − 3 x 2 , d n dx n cosh(3 x ) = 3 n e 3 x + ( − 1) n e − 3 x 2 . (b): For a given n ∈ N , write the Taylor polynomial T n ( x ) of f (centered at x o = 0). Solution: Then we have T n ( x ) = ⌊ n 2 ⌋ ∑ k =0 3 k (2 k )! x 2 k . (c): For a given ε > 0, find n ∈ N such that sup {| f ( x ) − T n ( x ) | : x ∈ [ − a, a ] } < ε. Solution: Notice that by the Lagrange formula for the remainder of the Taylor approximation, we have for some c ∈ ( − a, a ) r n ( x ) = f ( x ) − T n ( x ) = 3 n +1 e 3 c + ( − 1) n +1 e − 3 c 2( n + 1)! x n +1 Put t o = 3 ae − 2. Put M := e 3 a +1 2 . Then the inequality ( n + 1)! ≥ ( n +2) n +1 e n +1 implies | r n ( x ) | ≤ (3 a ) n +1 e 3 a + 1 2( n + 1)! ≤ M (3 a ) n +1 ( n + 1)! ≤ M (3 ae ) n +1 ( n + 2) n +1 ≤ M 3 ae n + 2 < ε, whenever n ≥ max { t o , 3 aeM ε − 2 } . 3 Problem 2. Consider the function f : [ − 1 , 1] → R , defined by f ( x ) = 1 √ 1 + 1 2 x , x ∈ [ − 1 , 1] . (a): Compute the n-th derivative of the function f . Solution: Consider first the function g ( x ) = (1 + x ) − 1 2 . Clearly, by applying principle of mathematical induction, we obtain g ′ ( x ) = ( − 1 2 ) (1 + x ) − 3 2 , g ′′ ( x ) = ( − 1 2 )( − 3 2 ) (1 + x ) − 5 2 , g ′′′ ( a ) = ( − 1 2 )( − 3 2 )( − 5 2 ) (1 + x ) − 7 2 g ( n ) ( x ) = ( − 1 2 )( − 3 2 ) . . . ( − 2 n − 1 2 ) (1 + x ) − 2 n +1 2 = ( − 1) n (2 n )!...
View Full Document

Page1 / 17

Assignment 1 - solution - University of Texas at Dallas...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online