6 - silva (jrs4378) HW 06 gualdani (56455) 1 This print-out...

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Unformatted text preview: silva (jrs4378) HW 06 gualdani (56455) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find the x-intercept of the tangent line to the graph of f ( x ) = 3 sin x- 2 cos x at the point (0 , f (0)). 1. x-intercept = 3 2 2. x-intercept =- 2 3. x-intercept = 2 3 correct 4. x-intercept =- 2 3 5. x-intercept =- 3 2 6. x-intercept =- 3 Explanation: When f ( x ) = 3 sin x- 2 cos x , then f ( x ) = 3 cos x + 2 sin x . Thus at x = 0, f (0) =- 2 , f (0) = 3 . So by the Point-Slope formula, an equation for the tangent line at (0 , f (0)) is y + 2 = 3( x- 0) , which after rearranging become y = 3 x- 2 . Consequently, the tangent line to the graph of f at (0 , f (0)) has x-intercept = 2 3 . keywords: tangent, trig function, sin, cos, trig derivative, intercept, point-slope formula, 002 10.0 points Find the derivative of f when f ( x ) = sin x 1 + 3 cos x . 1. f ( x ) = cos x + 3 1 + 3 cos x 2. f ( x ) = cos x + 3 sin 2 x (1 + 3 cos x ) 2 3. f ( x ) = cos x + 3 (1 + 3 cos x ) 2 correct 4. f ( x ) = cos x- 3 1 + 3 cos x 5. f ( x ) = cos x- 3 (1 + 3 cos x ) 2 Explanation: By the Quotient Rule, f ( x ) = cos x (1 + 3 cos x ) + 3 sin x sin x (1 + 3 cos x ) 2 = cos x + 3(cos 2 x + sin 2 x ) (1 + 3 cos x ) 2 . But cos 2 x + sin 2 x = 1, so f ( x ) = cos x + 3 (1 + 3 cos x ) 2 . 003 10.0 points Determine the derivative of f ( x ) = tan x 4 + sec x . silva (jrs4378) HW 06 gualdani (56455) 2 1. f ( x ) = 4 sec 2 x (4 + sec x ) 2 2. f ( x ) = tan x (4 sec x + 1) 4 + sec x 3. f ( x ) = tan x (4 sec x- 1) 4 + sec x 4. f ( x ) = 4 sec x tan x 4 + sec x 5. f ( x ) = sec x (4 sec x + 1) (4 + sec x ) 2 correct 6. f ( x ) = sec x (4 sec x- 1) (4 + sec x ) 2 Explanation: Since (tan x ) = sec 2 x, (sec x ) = sec x tan x , the Quotient Rule ensures that f ( x ) = sec 2 x (4 + sec x )- sec x tan 2 x (4 + sec x ) 2 = 4 sec 2 x + sec x (sec 2 x- tan 2 x ) (4 + sec x ) 2 . But sec 2 x = 1 + tan 2 x . Consequently, f ( x ) = sec x (4 sec x + 1) (4 + sec x ) 2 . 004 10.0 points Find an equation for the tangent line to the graph of f at the point P ( 4 , f ( 4 )) when f ( x ) = 6 tan x . 1. y = 18 x + 2 parenleftBig 1- 2 parenrightBig 2. y = 3 x + 14 parenleftBig 1- 2 parenrightBig 3. y = 21 x + 9 parenleftBig 1- 4 parenrightBig 4. y = 12 x + 6 parenleftBig 1- 2 parenrightBig correct 5. y = 5 x + 15 parenleftBig 1- 4 parenrightBig Explanation: When x = 4 , then f ( x ) = 6, so P = ( 4 , 6)....
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6 - silva (jrs4378) HW 06 gualdani (56455) 1 This print-out...

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