CalcHW9Answers - Villarreal Natalie – Homework 9 – Due 3:00 am – Inst Louiza Fouli 1 This print-out should have 20 questions Multiple-choice

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Unformatted text preview: Villarreal, Natalie – Homework 9 – Due: Oct 24 2007, 3:00 am – Inst: Louiza Fouli 1 This print-out should have 20 questions. Multiple-choice 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 √ 3 + x at x = 0. 1. L ( x ) = 1 √ 3 ‡ 1 + 1 6 x · 2. L ( x ) = 1 √ 3 + 1 3 x 3. L ( x ) = 1 3 ‡ 1 + 1 6 x · 4. L ( x ) = 1 3 ‡ 1- 1 3 x · 5. L ( x ) = 1 √ 3- 1 3 x 6. L ( x ) = 1 √ 3 ‡ 1- 1 6 x · correct Explanation: The linearization of f is the function L ( x ) = f (0) + f (0) x. But for the function f ( x ) = 1 √ 3 + x = (3 + x )- 1 / 2 , the Chain Rule ensures that f ( x ) =- 1 2 (3 + x )- 3 / 2 . Consequently, f (0) = 1 √ 3 , f (0) =- 1 6 √ 3 , and so L ( x ) = 1 √ 3 ‡ 1- 1 6 x · . keywords: linearization, square root function, differentials 002 (part 1 of 1) 10 points Use linear approximation with a = 25 to estimate the number √ 24 . 8 as a fraction. 1. √ 24 . 8 ≈ 4 19 20 2. √ 24 . 8 ≈ 4 24 25 3. √ 24 . 8 ≈ 4 49 50 correct 4. √ 24 . 8 ≈ 4 97 100 5. √ 24 . 8 ≈ 4 47 50 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 = 25 , h =- 2 10 . Villarreal, Natalie – Homework 9 – Due: Oct 24 2007, 3:00 am – Inst: Louiza Fouli 2 For then √ 24 . 8 ≈ 4 49 50 . keywords: linear approximation, square roots 003 (part 1 of 1) 10 points A cube with sides 4 inches long is covered with a fiberglass coating . 02 inches thick. Es- timate the volume of the fiberglass shell. 1. fiberglass vol ≈ 49 25 cu.ins. 2. fiberglass vol ≈ 48 25 cu.ins. correct 3. fiberglass vol ≈ 99 50 cu.ins. 4. fiberglass vol ≈ 97 50 cu.ins. 5. fiberglass vol ≈ 19 10 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 Δ x , then the approxi- mate change, Δ V , in volume is given by Δ V ≈ V ( x )Δ x = 3 x 2 Δ x. Now a . 02 inch thick fiberglass coating on each face will increase the side length by 1 25 = (2 × . 02) inches . When the side length of the cube is 4 inches, therefore, the volume of this fiberglass shell will be approximately Δ V = fiberglass vol ≈ 48 25 cu. ins. ....
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This note was uploaded on 04/15/2011 for the course MATH 408K taught by Professor Gualdani during the Fall '09 term at University of Texas at Austin.

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CalcHW9Answers - Villarreal Natalie – Homework 9 – Due 3:00 am – Inst Louiza Fouli 1 This print-out should have 20 questions Multiple-choice

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