HW07-solutions

HW07-solutions - qazi(kaq87 – HW07 – berg –(55290 1...

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Unformatted text preview: qazi (kaq87) – HW07 – berg – (55290) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the derivative of r ( t ) = a + t b + t 8 c . 1. r ′ ( t ) = a + b + 8 t 7 c 2. r ′ ( t ) = a + b − t 7 c 3. r ′ ( t ) = b + 8 t 7 c correct 4. r ′ ( t ) = a + b − 8 t 7 c 5. r ′ ( t ) = b − t 7 c Explanation: The derivative of r ( t ) = a + t b + t 8 c with respect to t is the sum of the derivatives of each of the terms. Now d dt ( a ) = 0 , d dt ( t b ) = b , while d dt ( t 8 c ) = 8 t 8 − 1 c = 8 t 7 c . Consequently, r ′ ( t ) = b + 8 t 7 c . keywords: vector function, derivative, 002 10.0 points Determine the vector I = integraldisplay 1 r ( t ) dt when r ( t ) = (Big 8 1 + t 2 , 2 t 1 + t 2 , 6 (1 + t ) 2 )Big . 1. I = ( 2 ln2 , π, 3 ) 2. I = ( 2 π, ln2 , 3 ) correct 3. I = ( 8 ln2 , 2 , 6 π ) 4. I = ( 8 , 2 π, 6 ln2 ) 5. I = ( 8 , ln 2 , 6 π ) 6. I = ( 2 π, 2 , 3 ln2 ) Explanation: For a vector function r ( t ) = ( f ( t ) , g ( t ) , h ( t ) ) , the components of the vector I = integraldisplay 1 r ( t ) dt are given by integraldisplay 1 f ( t ) dt, integraldisplay 1 g ( t ) dt, integraldisplay 1 h ( t ) dt , respectively. But when r ( t ) = (Big 8 1 + t 2 , 2 t 1 + t 2 , 6 (1 + t ) 2 )Big , we see that integraldisplay 1 f ( t ) dt = integraldisplay 1 8 1 + t 2 dt = bracketleftBig 8 tan − 1 t bracketrightBig 1 = 2 π , while integraldisplay 1 g ( t ) dt = integraldisplay 1 2 t 1 + t 2 dt = bracketleftBig ln(1 + t 2 ) bracketrightBig 1 = ln 2 , and integraldisplay 1 h ( t ) dt = integraldisplay 1 6 (1 + t ) 2 dt = bracketleftBig − 6 1 + t bracketrightBig 1 = 3 . qazi (kaq87) – HW07 – berg – (55290) 2 Consequently, I = ( 2 π, ln2 , 3 ) . 003 10.0 points If r ( t ) = (big t, t 5 , t 7 )big , find r ′′ ( t ). 1. r ′′ ( t ) = (big , 5 t 3 , 7 t 5 )big 2. r ′′ ( t ) = (big 1 , 20 t 3 , 42 t 5 )big 3. r ′′ ( t ) = (big , 20 t 3 , 42 t 5 )big correct 4. r ′′ ( t ) = (big 1 , 5 t 3 , 7 t 5 )big 5. r ′′ ( t ) = (big , 20 t 4 , 42 t 6 )big 6. r ′′ ( t ) = (big 1 , 5 t 4 , 7 t 6 )big Explanation: For a vector function r ( t ) = ( f ( t ) , g ( t ) , h ( t ) ) , the components of the vector function r ′′ ( t ) are given by f ′′ ( t ) , g ′′ ( t ) , h ′′ ( t ) , respectively. But, when r ( t ) = (big t, t 5 , t 7 )big , we see that f ′ ( t ) = 1 , f ′′ ( t ) = 0 , while g ′ ( t ) = 5 t 4 , g ′′ ( t ) = 20 t 3 , and h ′ ( t ) = 7 t 6 , h ′′ ( t ) = 42 t 5 . Consequently, r ′′ ( t ) = (big , 20 t 3 , 42 t 5 )big . keywords: 004 10.0 points Find an equation in parametric form for the tangent line to the graph of r ( s ) = ( s 3 , s 4 , s 5 ) at the point (1 , 1 , 1)....
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HW07-solutions - qazi(kaq87 – HW07 – berg –(55290 1...

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