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test3ps

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

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Practice Midterm #3 Solutions 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 d dx sec x = (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, y 00 + 4 y = ( - 4) y + 4 y = 0.

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2. Chain Rule (a) State the General Power Rule. d dx ( f ( x ) n ) = nf ( x ) n - 1 f 0 ( x ) . (b) Find the derivatives of the following functions. i. f ( x ) = (2 x - 7) 3 By the General Power Rule, f 0 ( x ) = (3)(2 x - 7) 3 - 1 d dx (2 x - 7) = 6(2 x - 7) 2 .
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test3ps - Practice Midterm#3 Solutions 1 Trig Functions(a...

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