{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

A3 - f x ≥ g x ≥ for x ≥ a • If R ∞ a f x dx is...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 128: Assignment 3 due Wed, Oct. 5 by 11:59 pm Koray Karabina ([email protected]) 1. Read “Section 7.5 Strategy for Integration” in the course textbook. 2. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. (a) R 1 x x 2 +4 dx (b) R 0 x e x dx (c) R -∞ cos ( πx ) dx (d) R -∞ x - 1 x 3 dx 3. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. (a) R π/ 2 π/ 4 sin x cos x dx (b) R 1 0 ln x x dx (c) R 1 0 1 4 x - 1 dx (d) R 1 0 1 1 - x 2 dx 4. The purpose of this question is to teach how to apply the Comparison Theorem to determine whether an improper integral is convergent or divergent. The statement of the Comparison Theorem is as follows (we will omit the proof for now): Theorem 1 (Comparison Theorem) . Suppose that f and g are continuous functions with
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ( x ) ≥ g ( x ) ≥ for x ≥ a . • If R ∞ a f ( x ) dx is convergent then R ∞ a g ( x ) dx is convergent. • If R ∞ a g ( x ) dx is divergent then R ∞ a f ( x ) dx is divergent. Use the Comparison Theorem to determine whether the integral is convergent or divergent. You do not have to evaluate the integrals. You might have a look at Example 9 and Example 10 in Section 7.8 in the course textbook. (a) R ∞ 1 2+ e-x x dx Hint: Try to find some relation between 2+ e-x x and 1 x for x ≥ 1. (b) R ∞ ( x 3 e-2 x ) dx Hint: Show that x 3 e-2 x ≤ e-x for sufficiently large x . You might use the fact that x ≥ 3 ln x for all x ≥ 5....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online