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lecture13 - position and released at time t = 0 Its...

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3.4 Derivatives of Trig Functions Angle Sum Identity for the Sine sin( x + h ) = sin x cos h + cos x sin h Angle Sum Identity for the Cosine cos( x + h ) = cos x cos h - sin x sin h Derivative of the Sine Function If f ( x ) = sin x then f ( x ) = 1
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Derivative of the Cosine Function If f ( x ) = cos x then f ( x ) = The derivative ot the sine function is the cosine function: d dx (sin x ) = cos x The derivative ot the cosine function is the negative sine function: d dx (cos x ) = - sin x Examples Find the derivative of the following: 1. y = e x sin x 2
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2. y = sin x cos x 3. y = cos x 1 - sin x 4. y = sin x x Simple Harmonic Motion The motion of an object bobbing freely up/down on a spring or a bungee cord is called harmonic motion. Consider the next example where there are no opposing forces to slow the motion. Example Suppose a body hanging from a spring is stretched down five units from its rest
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Unformatted text preview: position and released at time t = 0. Its position at any time t ≥ 0 is given by s ( t ) = 5 cos t . What are its velocity and acceleration at time t ? 3 Derivatives of other Basic Trig Functions Because sin x, cos x are differentiable functions of x , the related functions tan x = sin x cos x , cot x = cos x sin x , sec x = 1 cos x , csc x = 1 sin x are differentiable at every value of x for which they are defined. We have d dx (tan x ) = d dx (cot x ) = d dx (sec x ) = d dx (csc x ) = Example 5 Show d dx (tan x ) = sec 2 x . Example 6 Find y 00 if y = sec x . Example 7 lim x → √ 2+sec x cos( π-tan x ) 4...
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