Calculus 408K Exam 1 Review

Calculus 408K Exam 1 Review - Barkley Thane – Review 1...

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Unformatted text preview: Barkley, Thane – Review 1 – Due: Dec 11 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 derivative of g when g ( x ) = x 3 cos x . 1. g ( x ) = x 2 (3 sin x- x cos x ) 2. g ( x ) = x 3 (2 cos x- sin x ) 3. g ( x ) = x 2 (3 cos x + x sin x ) 4. g ( x ) = x 3 (2 sin x- cos x ) 5. g ( x ) = x 2 (3 sin x + x cos x ) 6. g ( x ) = x 2 (3 cos x- x sin x ) correct Explanation: By the Product rule, g ( x ) = x 3 (- sin x ) + (cos x ) · 3 x 2 . Consequently, g ( x ) = x 2 (3 cos x- x sin x ) . keywords: derivative, product rule, trigono- metric function 002 (part 1 of 1) 10 points Find the derivative of the function f ( x ) = 4- 3 x x . 1. f ( x ) = 8- 3 x x 2 2. f ( x ) = 6- 4 x x 2 3. f ( x ) = 6 + 4 x x 3 4. f ( x ) = 6- 4 x x 3 correct 5. f ( x ) = 8- 3 x x 3 Explanation: After simplification, 4- 3 x x = 4 x- 3 x 2 . Thus by the Quotient Rule, f ( x ) = 4 x 2- 2 x (4 x- 3) x 4 . Consequently, f ( x ) = 6- 4 x x 3 . keywords: derivatives, quotient rule 003 (part 1 of 1) 10 points Find the derivative of f when f ( x ) = sin x 1- 2 cos x . 1. f ( x ) = sin x- 2 cos 2 x (1- 2 cos x ) 2 2. f ( x ) = sin x- 2 1- 2 cos x 3. f ( x ) = cos x- 2 (1- 2 cos x ) 2 correct 4. f ( x ) = cos x + 2 1- 2 cos x 5. f ( x ) = cos x + 2 (1- 2 cos x ) 2 Barkley, Thane – Review 1 – Due: Dec 11 2007, 3:00 am – Inst: Louiza Fouli 2 6. f ( x ) = cos x- 2 sin 2 x (1- 2 cos x ) 2 Explanation: By the Quotient Rule, f ( x ) = cos x (1- 2 cos x )- 2 sin x sin x (1- 2 cos x ) 2 = cos x- 2 (cos 2 x + sin 2 x ) (1- 2 cos x ) 2 . But cos 2 x + sin 2 x = 1, so f ( x ) = cos x- 2 (1- 2 cos x ) 2 . keywords: derivative, trigonometric function, quotient rule 004 (part 1 of 1) 10 points There is one point in the first quadrant at which the tangent line to the graph of y = 4 + x + x 2- x 3 is horizontal. Find the y-coordinate of this point. 1. y = 6 2. y = 5 correct 3. y = 7 4. y = 4 5. y = 8 Explanation: The tangent line to the graph will be hori- zontal when dy dx = 1 + 2 x- 3 x 2 = (3 x + 1)(1- x ) = 0 . The only solution of this for which x > 0 oc- curs at x = 1. But at x = 1 the corresponding value of y is y = 5. Since this value of y is positive, the only point in the first quadrant at which the tangent line is horizontal is the point P = (1 , 5) . keywords: horizontal tangent line, derivative, extrema, polynomial 005 (part 1 of 1) 10 points If f is a function defined on (- 2 , 2) whose graph is 1 2- 1- 2 1 2- 1- 2 which of the following is the graph of its derivative f ? 1. 1 2- 1- 2 1 2- 1- 2 Barkley, Thane – Review 1 – Due: Dec 11 2007, 3:00 am – Inst: Louiza Fouli 3 2....
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Calculus 408K Exam 1 Review - Barkley Thane – Review 1...

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