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
Unformatted text preview: Tran, Tony Homework 9 Due: Oct 24 2007, 3:00 am Inst: Samuels 1 This printout should have 20 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find the linearization of f ( x ) = 1 5 + x at x = 0. 1. L ( x ) = 1 5 1 1 5 x 2. L ( x ) = 1 5 1 + 1 10 x 3. L ( x ) = 1 5 + 1 5 x 4. L ( x ) = 1 5 1 + 1 10 x 5. L ( x ) = 1 5 1 5 x 6. L ( x ) = 1 5 1 1 10 x correct Explanation: The linearization of f is the function L ( x ) = f (0) + f (0) x. But for the function f ( x ) = 1 5 + x = (5 + x ) 1 / 2 , the Chain Rule ensures that f ( x ) = 1 2 (5 + x ) 3 / 2 . Consequently, f (0) = 1 5 , f (0) = 1 10 5 , and so L ( x ) = 1 5 1 1 10 x . keywords: linearization, square root function, differentials 002 (part 1 of 1) 10 points Use linear approximation with a = 4 to estimate the number 3 . 7 as a fraction. 1. 3 . 7 1 7 8 2. 3 . 7 1 9 10 3. 3 . 7 1 17 20 4. 3 . 7 1 33 40 5. 3 . 7 1 37 40 correct Explanation: For a general function f , its linear approxi mation at x = a is defined by L ( x ) = f ( a ) + f ( a )( x a ) and for values of x near a f ( x ) L ( x ) = f ( a ) + f ( a )( x a ) provides a reasonable approximation for f ( x ). Now set f ( x ) = x, f ( x ) = 1 2 x . Then, if we can calculate a easily, the linear approximation a + h a + h 2 a provides a very simple method via calculus for computing a good estimate of the value of a + h for small values of h . In the given example we can thus set a = 4 , h = 3 10 . Tran, Tony Homework 9 Due: Oct 24 2007, 3:00 am Inst: Samuels 2 For then 3 . 7 1 37 40 . keywords: linear approximation, square roots 003 (part 1 of 1) 10 points A cube with sides 3 inches long is covered with a fiberglass coating . 01 inches thick. Es timate the volume of the fiberglass shell. 1. fiberglass vol 23 50 cu.ins. 2. fiberglass vol 12 25 cu.ins. 3. fiberglass vol 27 50 cu.ins. correct 4. fiberglass vol 1 2 cu.ins. 5. fiberglass vol 13 25 cu.ins. Explanation: A cube with side length x has volume V ( x ) = x 3 . If the length of each side is changed by an amount x , then the approxi mate change, V , in volume is given by V V ( x ) x = 3 x 2 x. Now a . 01 inch thick fiberglass coating on each face will increase the side length by 1 50 = (2 . 01) inches . When the side length of the cube is 3 inches, therefore, the volume of this fiberglass shell will be approximately V = fiberglass vol 27 50 cu. ins. ....
View Full
Document
 Fall '08
 schultz

Click to edit the document details