test3ps - Practice Midterm #3 Solutions 1. Trig Functions...

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1. Trig Functions (a) Find the derivatives of the following functions. i. y = sec x y = (sec x ) 1 2 , so by the chain rule, y 0 = ± 1 2 (sec x ) - 1 2 ²± d dx sec x ² = (sec x ) - 1 2 2 (sec x tan x ) = sec x tan x 2 . ii. T = 5 - t cos(4 πt ) T 0 = 0 - d dt ( t cos(4 πt )) = ( - cos(4 πt ))(1) + ( - t )( - sin(4 πt )) d dt (4 πt ) = 4 πt sin(4 πt ) - cos(4 πt ) . (b) If a 10 inch pendulum is swinging so that the angle θ of the pendulum with the vertical axis is given by θ = . 25 cos(8 πt ), where t is the time in seconds, what is the rate of change of θ when t = 3? dt = . 25[( - sin(8 πt ))(8 π )] = - 2 π sin(8 πt ) , so when t = 3, dt (3) = - 2 π · sin(24 π ) = 0 . (c) Verify y = cos 2 x + sin 2 x satisfies the equation y 00 + 4 y = 0. Since y = cos 2 x + sin 2 x , y 0 = - 2 sin 2 x + 2 cos 2 x , and so y 00 = - 4 cos 2 x - 4 sin 2 x = ( - 4)(cos 2 x + sin 2 x ) = ( - 4) y. Thus,
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This note was uploaded on 05/22/2011 for the course MATH 16A taught by Professor Sabalka during the Fall '08 term at UC Davis.

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test3ps - Practice Midterm #3 Solutions 1. Trig Functions...

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