Chap3_Sec3

Chap3_Sec3 - the motion of the object Example 3(3.3...

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Differentiate For what values of x does the graph of f have a horizontal tangent? In simplifying the answer, we need to use the identity tan 2 x + 1 = sec 2 x . sec ( ) 1 tan x f x x = + Example 2 (3.3) DERIVS. OF TRIG. FUNCTIONS
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Since sec x is never 0, we see that f’ ( x ) = 0 when tan x = 1. s This occurs when x = n π + π /4 , where n is an integer. Example 2 DERIVS. OF TRIG. FUNCTIONS
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Trigonometric functions are often used in modeling real-world phenomena. s In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. s In the following example, we discuss an instance of simple harmonic motion. APPLICATIONS
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An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0. s In the figure, note that the downward direction is positive. s Its position at time t is s = f ( t ) = 4 cos t s Find the velocity and acceleration at time t and use them to analyze
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Unformatted text preview: the motion of the object. Example 3 (3.3) APPLICATIONS The object oscillates from the lowest point ( s = 4 cm) to the highest point ( s = -4 cm). The period of the oscillation is 2 π , the period of cos t . Example 3 APPLICATIONS The speed is | v | = 4|sin t |, which is greatest when |sin t | = 1, that is, when cos t = 0. s So, the object moves fastest as it passes through its equilibrium position ( s = 0). s Its speed is 0 when sin t = 0, that is, at the high and low points. Example 3 APPLICATIONS The acceleration a = -4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points. Example 3 APPLICATIONS Find the 27th derivative of cos x . Example 4 (3.3) DERIVS. OF TRIG. FUNCTIONS Find Use this equation: sin 7 lim 4 x x x → Example 5 (3.3) DERIVS. OF TRIG. FUNCTIONS sin lim 1 θ → = Calculate . lim cot x x x → Example 6 (3.3) DERIVS. OF TRIG. FUNCTIONS...
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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Chap3_Sec3 - the motion of the object Example 3(3.3...

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