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Homework 9

# Homework 9 - Garcia Ilse Homework 9 Due 3:00 am Inst Fonken...

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Garcia, Ilse – Homework 9 – Due: Oct 24 2007, 3:00 am – Inst: Fonken 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 7 + x at x = 0. 1. L ( x ) = 1 7 1 + 1 14 x · 2. L ( x ) = 1 7 - 1 7 x 3. L ( x ) = 1 7 1 + 1 14 x · 4. L ( x ) = 1 7 1 - 1 7 x · 5. L ( x ) = 1 7 + 1 7 x 6. L ( x ) = 1 7 1 - 1 14 x · correct Explanation: The linearization of f is the function L ( x ) = f (0) + f 0 (0) x . But for the function f ( x ) = 1 7 + x = (7 + x ) - 1 / 2 , the Chain Rule ensures that f 0 ( x ) = - 1 2 (7 + x ) - 3 / 2 . Consequently, f (0) = 1 7 , f 0 (0) = - 1 14 7 , and so L ( x ) = 1 7 1 - 1 14 x · . keywords: linearization, square root function, differentials 002 (part 1 of 1) 10 points Use linear approximation with a = 4 to estimate the number 4 . 5 as a fraction. 1. 4 . 5 2 3 20 2. 4 . 5 2 3 40 3. 4 . 5 2 1 8 correct 4. 4 . 5 2 7 40 5. 4 . 5 2 1 10 Explanation: For a general function f , its linear approxi- mation at x = a is defined by L ( x ) = f ( a ) + f 0 ( a )( x - a ) and for values of x near a f ( x ) L ( x ) = f ( a ) + f 0 ( a )( x - a ) provides a reasonable approximation for f ( x ). Now set f ( x ) = x, f 0 ( 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 = 5 10 .

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Garcia, Ilse – Homework 9 – Due: Oct 24 2007, 3:00 am – Inst: Fonken 2 For then 4 . 5 2 1 8 . keywords: linear approximation, square roots 003 (part 1 of 1) 10 points A cube with sides 6 inches long is covered with a fiberglass coating . 04 inches thick. Es- timate the volume of the fiberglass shell. 1. fiberglass vol 431 50 cu.ins. 2. fiberglass vol 216 25 cu.ins. correct 3. fiberglass vol 43 5 cu.ins. 4. fiberglass vol 214 25 cu.ins. 5. fiberglass vol 429 50 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 0 ( x x = 3 x 2 Δ x . Now a . 04 inch thick fiberglass coating on each face will increase the side length by 2 25 = (2 × . 04) inches . When the side length of the cube is 6 inches, therefore, the volume of this fiberglass shell will be approximately Δ V = fiberglass vol 216 25 cu. ins. . keywords: cube, volume, estimate 004 (part 1 of 1) 10 points The radius of a circle is estimated to be 14 inches, with a maximum error in measure- ment of ± 0 . 06 inches. Use differentials to estimate the maximum error in calculating the area of the circle using this estimate. 1. Max error ≈ ± 5 . 2778 sq.ins correct 2. Max error ≈ ± 5 . 2938 sq.ins 3. Max error ≈ ± 5 . 3018 sq.ins 4. Max error ≈ ± 5 . 2698 sq.ins 5. Max error ≈ ± 5 . 2858 sq.ins Explanation: The area of a circle having radius x is given by A = πx 2 . By differentials, therefore, Δ A Δ x dA dx = 2 πx Δ x.
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Homework 9 - Garcia Ilse Homework 9 Due 3:00 am Inst Fonken...

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