HW 8 Calc - garcia(jjg2564 – HW 08 – gualdani –(56410...

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Unformatted text preview: garcia (jjg2564) – HW 08 – gualdani – (56410) 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 After spending $ x in advertising per day, a McDonalds restaurant finds that it sells N hamburgers where N = 1400 + 230 x- 5 x 2 . Estimate using differentials how many more hamburgers the restaurant will sell if it in- creases its daily spending on advertising from $11 to $11 . 40. 1. 45 more hamburgers 2. 47 more hamburgers 3. 46 more hamburgers 4. 48 more hamburgers correct 5. 49 more hamburgers Explanation: After differentiation, N ′ ( x ) = 230- 10 x. If x is increased by $Δ x , then the approximate increase, Δ N , in N will thus be given by Δ N ≈ N ′ ( x ) Δ x = (230- 10 x ) Δ x. When x = 11 and Δ x = . 4, therefore, the McDonalds restaurant can expect to sell approximately 48 more hamburgers. 002 10.0 points Estimate the value of 15 1 / 4 using differen- tials. 1. 15 1 / 4 ≈ 31 16 2. 15 1 / 4 ≈ 65 32 3. 15 1 / 4 ≈ 63 32 correct 4. 15 1 / 4 ≈ 61 32 5. 15 1 / 4 ≈ 2 Explanation: Set f ( x ) = x 1 / 4 . Then df dx = 1 4 x 3 / 4 . By differentials, therefore, we see that f ( a + Δ x )- f ( a ) ≈ df dx vextendsingle vextendsingle vextendsingle x = a Δ x = Δ x 4 a 3 / 4 . Thus, with a = 16 and Δ x =- 1, 15 1 / 4- 2 ≈ - 1 32 . Consequently, 15 1 / 4 ≈ 63 32 . 003 10.0 points Find the differential, dy , of y = f ( x ) = tan(4 x 2 ) . 1. dy = 8 x sec 2 (4 x ) + dx 2. dy = 8 x sec 2 (4 x 2 ) 3. dy = 4 sec 2 (4 x 2 ) tan(4 x 2 ) 4. dy = 4 sec 2 (4 x 2 ) tan(4 x 2 ) + dx 5. dy = 4 sec 2 (4 x 2 ) tan(4 x 2 ) dx garcia (jjg2564) – HW 08 – gualdani – (56410) 2 6. dy = 8 x sec 2 (4 x 2 ) dx correct Explanation: The differential, dy , of y = f ( x ) is given by dy = f ′ ( x ) dx . On the other hand, d dx tan x = sec 2 x . Thus by the Chain Rule, f ′ ( x ) = 8 x sec 2 (4 x 2 ) . Consequently, dy = 8 x sec 2 (4 x 2 ) dx . keywords: differential, trig function, tan func- tion, Chain Rule, 004 10.0 points Find the linearization of f ( x ) = 1 √ 3 + x at x = 0. 1. L ( x ) = 1 √ 3 parenleftBig 1 + 1 6 x parenrightBig 2. L ( x ) = 1 √ 3 parenleftBig 1- 1 6 x parenrightBig correct 3. L ( x ) = 1 3 parenleftBig 1 + 1 6 x parenrightBig 4. L ( x ) = 1 3 parenleftBig 1- 1 3 x parenrightBig 5. L ( x ) = 1 √ 3 + 1 3 x 6. L ( x ) = 1 √ 3- 1 3 x 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 parenleftBig 1- 1 6 x parenrightBig ....
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This note was uploaded on 10/08/2010 for the course MATH 408K taught by Professor Gualdani during the Fall '09 term at University of Texas.

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HW 8 Calc - garcia(jjg2564 – HW 08 – gualdani –(56410...

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