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Unformatted text preview: Villarreal, Natalie Homework 5 Due: Sep 25 2007, 3:00 am Inst: Louiza Fouli 1 This printout should have 24 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 slope of the secant line passing through the points ( 1 , f ( 1)) , ( 1 + h, f ( 1 + h )) when f ( x ) = 2 x 2 + x 3 . 1. slope = 2 h 3 correct 2. slope = 2 h + 3 3. slope = 4 h + 5 4. slope = 4 h + 3 5. slope = 4 h 5 6. slope = 2 h 5 Explanation: Since the secant line passes through the points ( 1 , f ( 1)) , ( 1 + h, f ( 1 + h )) , its slope is given by f ( 1 + h ) f ( 1) h = { 2( 1 + h ) 2 + ( 1 + h ) 3 } + 2 h = 2 h 2 3 h h = 2 h 3 . keywords: slope, secant line 002 (part 1 of 1) 10 points If P ( a, f ( a )) is the point on the graph of f ( x ) = x 2 + 6 x + 1 at which the tangent line is parallel to the line y = 5 x + 3 , determine a . 1. a = 1 2 2. a = 0 3. a = 1 4. a = 1 5. a = 1 2 correct Explanation: The slope of the tangent line at the point P ( a, f ( a )) on the graph of f is the value f ( a ) = lim h f ( a + h ) f ( a ) h of the derivative of f at x = a . To compute the value of f ( a ), note that f ( a + h ) = ( a + h ) 2 + 6( a + h ) + 1 = a 2 + h (2 a + 6) + h 2 + 6 a + 1 , while f ( a ) = a 2 + 6 a + 1 . Thus f ( a + h ) f ( a ) = h { (2 a + 6) + h } , in which case f ( a ) = lim h { (2 a + 6) + h } = 2 a + 6 . If the tangent line at P is parallel to the line y = 5 x + 3 , Villarreal, Natalie Homework 5 Due: Sep 25 2007, 3:00 am Inst: Louiza Fouli 2 then they have the same slopes, so f ( a ) = 2 a + 6 = 5 . Consequently, a = 1 2 . keywords: tangent line, parallel, slope, deriva tive 003 (part 1 of 1) 10 points Find the xintercept of the tangent line at the point P ( 2 , f ( 2)) on the graph of f when f is defined by f ( x ) = 3 x 2 4 x + 3 . 1. xintercept = 16 9 2. xintercept = 9 16 3. xintercept = 9 4. xintercept = 16 9 5. xintercept = 9 16 correct 6. xintercept = 9 Explanation: The slope, m , of the tangent line at the point P ( 2 , f ( 2)) on the graph of f is the value of the derivative f ( x ) = 6 x 4 at x = 2, i.e. , m = 16. On the other hand, f ( 2) = 23. Thus by the pointslope formula an equation for the tangent line at P ( 2 , f ( 2)) is y 23 = 16( x + 2) , i . e ., y = 16 x 9 . Consequently, xintercept = 9 16 . keywords: tangent line, xintercept, slope 004 (part 1 of 1) 10 points Find an equation for the tangent line to the graph of g at the point P (1 , g (1)) when g ( x ) = 3 x 3 . 1. y + 3 x = 5 correct 2. y + 3 x + 5 = 0 3. y = 3 x + 5 4. y = 5 x 3 5. y + 5 x + 3 = 0 Explanation: If x = 1, then g (1) = 2. Thus the Newto nian different quotient for g ( x ) = 3 x 3 at the point (1 , 2) becomes g (1 + h ) g (1) h = h 3 (1 + h ) 3 i 2 h = 3 h 3 3 h 2 3 h 3 h ....
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