1024worksheet8

# 1024worksheet8 - horizontal. (a) y = sin 2 x (b) y = tan...

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Polytechnic Institute of NYU MA 1024/1324 Worksheet 8 Print Name: Signature: ID #: Instructor/Section: Directions: Show all your work for every problem. Problem Possible Points 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1

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2 (1) Find the derivatives of the given functions. (a) y = cos 4 ( x ) (b) y = tan (cos x ) (c) y = tan ( x ) · e sin( x 2 ) (d) y = sin 2 (2 x ) + 2 cos 3 (3 x ) + 3
3 (2) Find the derivatives of the given functions. (a) y = p (cos(3 x )) 3 (b) y = tan x 2 + e 2 x (c) y = sin 2 x + sin ( x 2 ) (d) y = cos 4 (4 x ) · sin 7 (7 x )

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4 (3) Verify the following diﬀerentiation formulas. (a) d dx (cot x ) = - csc 2 x (Hint: cot x = cos x sin x ) (b) d dx (sec x ) = sec x tan x (Hint: sec x = 1 cos x ) (c) d dx (csc x ) = - csc x cot x (Hint: csc x = 1 sin x )
5 (4) Find an equation for the tangent line to the curve at x = a . (a) y = csc x ; a = π/ 3 (b) y = tan x ; a = π/ 6 (c) y = cot x ; a = π/ 6

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6 (5) An object moves along the x -axis, its position at each time t being given by x ( t ). Determine those times from t = 0 to t = 2 π when the object is moving to the right ( x 0 ( t ) > 0) with increasing speed ( x 00 ( t ) > 0). (a) x ( t ) = sin t + cos t (b) x ( t ) = t - 2 sin t
7 (6) Determine the number x between 0 and 2 π where the tangent line to the curve is

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Unformatted text preview: horizontal. (a) y = sin 2 x (b) y = tan x-2 x (c) y = cot x-2 csc x 8 (7) Let f ( x ) = cos x, x ≥ ax + b, x < (a) Determine a and b so that f is diﬀerentiable at 0. (b) Using the values found above, sketch the graph of f . 9 (8) Let f ( x ) = sin x, x ≤ 2 π/ 3 ax + b, x > 2 π/ 3 (a) Determine a and b so that f is diﬀerentiable at 2 π/ 3. (b) Using the values found above, sketch the graph of f . 10 (9) Let f ( x ) = 1 + a cos x, x ≤ π/ 3 b + sin ± x 2 ² , x > π/ 3 (a) Determine a and b so that f is diﬀerentiable at π/ 3. (b) Using the values found above, sketch the graph of f . 11 (10) Let y = A sin ( ωt ) + B cos ( ωt ), where A , B , and ω are constants. Show that y satisﬁes the equation d 2 y dt 2 + ω 2 y = 0...
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## This note was uploaded on 12/09/2009 for the course MA 1024/1324 taught by Professor N/a during the Spring '09 term at NYU Poly.

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1024worksheet8 - horizontal. (a) y = sin 2 x (b) y = tan...

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