3.5 derivatives of trigonometric functions. revisiting the differentiation rules find the...
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3.5 Derivatives of Trigonometric Functions
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Revisiting the Differentiation Rules• Find the derivatives of (a) y = x²sinx and
(b) y = cosx / (1 – sinx).
(a)
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Revisiting the Differentiation Rules(b)
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Simple Harmonic Motion
• The motion of a weight bobbing up and down on the end of a spring is an example of simple harmonic motion.– The following example describes a case in which
there are no opposing forces like friction or buoyancy to slow down the motion.
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The Motion of a Weight on a Spring• A weight hanging from a spring is stretched 5 units beyond
its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is s = 5 cos t. What are its velocity and acceleration at time t? Describe its motion.
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The Motion of a Weight on a Spring• We have:– Position:
– Velocity:
– Acceleration:
• See observations on p. 143 and 144 # 1 – 4.
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Jerk• A sudden change in acceleration is called a “jerk.” When a
ride in a car or a bus is jerky, it is not that the accelerations involved are necessarily large, but that the changes in acceleration are abrupt.
• The derivative responsible for jerk is the THIRD derivative of position.
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A Couple of Jerks(a) The jerk caused by the constant acceleration of
gravity (g = -32 ft/sec²) is zero:This explains why we don’t experience motion
sickness while just sitting around.(b) The jerk of the simple harmonic motion in Example
2 is:
It has its greatest magnitude when sin t = ± 1. This does not occur at the extremes of the displacement, but at the rest position, where the acceleration changes direction and sign.
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Derivatives of the Other Basic Trigonometric Functions
• Because sin x and cos x are differentiable functions of x, the related functions
are differentiable at every value of x for which they are defined.
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Derivatives of the Other Basic Trigonometric Functions
• Their derivatives are given by the following formulas:
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A Trigonometric Second Derivative• Find y” if y = sec x.
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More Practice!!!!!
• Homework – Textbook p. 146 # 1 – 10 ALL, 17 – 23 ALL.