Homework 3

# Homework 3 - Rehman (aar638) – HW03 – sachse –...

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: Rehman (aar638) – HW03 – sachse – (56620) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Decide which of the following regions has area = lim n →∞ n summationdisplay i = 1 π 3 n tan iπ 3 n without evaluating the limit. 1. braceleftBig ( x, y ) : 0 ≤ y ≤ tan 2 x, ≤ x ≤ π 3 bracerightBig 2. braceleftBig ( x, y ) : 0 ≤ y ≤ tan x, ≤ x ≤ π 3 bracerightBig correct 3. braceleftBig ( x, y ) : 0 ≤ y ≤ tan 3 x, ≤ x ≤ π 3 bracerightBig 4. braceleftBig ( x, y ) : 0 ≤ y ≤ tan x, ≤ x ≤ π 6 bracerightBig 5. braceleftBig ( x, y ) : 0 ≤ y ≤ tan 3 x, ≤ x ≤ π 6 bracerightBig 6. braceleftBig ( x, y ) : 0 ≤ y ≤ tan 2 x, ≤ x ≤ π 6 bracerightBig Explanation: The area under the graph of y = f ( x ) on an interval [ a, b ] is given by the limit lim n →∞ n summationdisplay i = 1 f ( x i ) Δ x when [ a, b ] is partitioned into n equal subin- tervals [ a, x 1 ] , [ x 1 , x 2 ] , . . . , [ x n − 1 , b ] each of length Δ x = ( b- a ) /n . If A = lim n →∞ n summationdisplay i = 1 π 3 n tan iπ 3 n , therefore, we see that f ( x i ) = tan iπ 3 n , Δ x = π 3 n . But in this case x i = iπ 3 n , f ( x ) = tan x, [ a, b ] = bracketleftBig , π 3 bracketrightBig . Consequently, the area is that of the region under the graph of y = tan x on the interval [0 , π/ 3]. In set-builder notation this is the region braceleftBig ( x, y ) : 0 ≤ y ≤ tan x, ≤ x ≤ π 3 bracerightBig . 002 10.0 points Estimate the area under the graph of f ( x ) = 4 sin x between x = 0 and x = π 4 using five approx- imating rectangles of equal widths and right endpoints. 1. area ≈ 1 . 431 2. area ≈ 1 . 451 3. area ≈ 1 . 411 4. area ≈ 1 . 391 correct 5. area ≈ 1 . 471 Explanation: An estimate for the area under the graph of f on [0 , b ] with [0 , b ] partitioned in n equal subintervals [ x i − 1 , x i ] = bracketleftBig ( i- 1) b n , ib n bracketrightBig and right endpoints x i as sample points is A ≈ braceleftBig f ( x 1 ) + f ( x 2 ) + . . . + f ( x n ) bracerightBig b n . For the given area, f ( x ) = 4 sin x, b = π 4 , n = 5 , Rehman (aar638) – HW03 – sachse – (56620) 2 and x 1 = 1 20 π, x 2 = 1 10 π, x 3 = 3 20 π, x 4 = 1 5 π, x 5 = 1 4 π . Thus A ≈ 4 braceleftBig sin parenleftBig 1 20 π parenrightBig + . . . + sin parenleftBig 1 4 π parenrightBigbracerightBig π 20 . After calculating these values we obtain the estimate area ≈ 1 . 391 for the area under the graph. keywords: area, sin function, estimate area, numerical calculation, 003 10.0 points Rewrite the sum 4 n parenleftBig 2 + 5 n parenrightBig 2 + 4 n parenleftBig 2 + 10 n parenrightBig 2 + . . . + 4 n parenleftBig 2 + 5 n n parenrightBig 2 using sigma notation....
View Full Document

## This note was uploaded on 02/01/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas.

### Page1 / 11

Homework 3 - Rehman (aar638) – HW03 – sachse –...

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

View Full Document
Ask a homework question - tutors are online