Homework 1 - Version 106 Homework 01 Radin (58415) 1 This...

Info iconThis preview shows pages 1–3. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Version 106 Homework 01 Radin (58415) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Welcome to Quest Learning and As- sessment. Best of luck this semester. 001 10.0 points Find the value of f (0) when f ( t ) = 4(3 t- 1) and f (1) = 5 , f (1) = 2 . 1. f (0) =- 2 2. f (0) =- 5 3. f (0) =- 1 correct 4. f (0) =- 4 5. f (0) =- 3 Explanation: The most general anti-derivative of f has the form f ( t ) = 6 t 2- 4 t + C where C is an arbitrary constant. But if f (1) = 5, then f (1) = 6- 4 + C = 5 , i.e., C = 3 . From this it follows that f ( t ) = 6 t 2- 4 t + 3 , and the most general anti-derivative of the latter is f ( t ) = 2 t 3- 2 t 2 + 3 t + D , where D is an arbitrary constant. But if f (1) = 2, then f (1) = 2- 2 + 3 + D = 2 , i.e., D =- 1 . Consequently, f ( t ) = 2 t 3- 2 t 2 + 3 t- 1 . At x = 0, therefore, f (0) =- 1 . 002 10.0 points Consider the following functions: ( A ) F 1 ( x ) = cos 2 x 4 , ( B ) F 2 ( x ) = sin 2 x 2 , ( C ) F 3 ( x ) =- cos 2 x 2 . Which are anti-derivatives of f ( x ) = sin x cos x ? 1. F 2 only 2. F 1 and F 3 only 3. F 3 only 4. all of them 5. none of them 6. F 1 and F 2 only 7. F 1 only 8. F 2 and F 3 only correct Explanation: By trig identities, cos 2 x = 2 cos 2 x- 1 = 1- 2 sin 2 x , while sin 2 x = 2 sin x cos x . But d dx sin x = cos x, d dx cos x =- sin x . Consequently, by the Chain Rule, ( A ) Not anti-derivative. Version 106 Homework 01 Radin (58415) 2 ( B ) Anti-derivative. ( C ) Anti-derivative. 003 10.0 points Find f ( x ) on (- 2 , 2 ) when f ( x ) = 2 cos x + 4 sec 2 x and f ( 4 ) = 2. 1. f ( x ) = 5- 4 tan x + 2 sin x 2. f ( x ) = 7- 4 tan x- 2 cos x 3. f ( x ) = 4 tan x + 2 sin x- 3 correct 4. f ( x ) = 4 tan x- 2 sin x- 1 5. f ( x ) = 4 tan x- 2 cos x- 3 Explanation: The most general anti-derivative of f ( x ) = 2 cos x + 4 sec 2 x is f ( x ) = 2 sin x + 4 tan x + C with C an arbitrary constant. But if f parenleftBig 4 parenrightBig = 2, then f parenleftBig 4 parenrightBig = 1 + 4 + C = 2 , so C =- 3 . Consequently, f ( x ) = 4 tan x + 2 sin x- 3 . 004 10.0 points Find the unique anti-derivative F of f ( x ) = e 5 x- 4 e 2 x + 3 e 3 x e 2 x for which F (0) = 0. 1. F ( x ) = 1 3 e 3 x- 4 x- 3 5 e 5 x + 4 15 correct 2. F ( x ) = 1 5 e 5 x + 4 x + 1 3 e 3 x- 2 5 3. F ( x ) = 1 3 e 3 x- 4 x- 1 3 e 3 x 4. F ( x ) = 1 5 e 5 x- 4 x- 3 5 e 5 x- 2 5 5. F ( x ) = 1 3 e 3 x + 4 x + 3 5 e 5 x- 14 15 6. F ( x ) = 1 3 e 3 x + 4 x + 3 5 e 3 x- 4 15 Explanation: After division, e 5 x- 4 e 2 x + 3 e 3 x e 2 x = e 3 x- 4 + 3 e 5 x ....
View Full Document

This note was uploaded on 04/29/2008 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

Page1 / 8

Homework 1 - Version 106 Homework 01 Radin (58415) 1 This...

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

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