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3.5 Derivatives of Trigonometric Functions

3.5 Derivatives of Trigonometric Functions. Revisiting the Differentiation Rules. Find the derivatives of (a) y = x²sinx and (b) y = cosx / (1 – sinx). (a). Revisiting the Differentiation Rules. (b). Simple Harmonic Motion.

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3.5 Derivatives of Trigonometric Functions

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  1. 3.5 Derivatives of Trigonometric Functions

  2. Revisiting the Differentiation Rules • Find the derivatives of (a) y = x²sinx and (b) y = cosx / (1 – sinx). (a)

  3. Revisiting the Differentiation Rules (b)

  4. 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.

  5. 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.

  6. The Motion of a Weight on a Spring • We have: • Position: • Velocity: • Acceleration: • See observations on p. 143 and 144 # 1 – 4.

  7. 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.

  8. A Couple of Jerks • 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. • 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.

  9. 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.

  10. Derivatives of the Other Basic Trigonometric Functions • Their derivatives are given by the following formulas:

  11. A Trigonometric Second Derivative • Find y” if y = sec x.

  12. More Practice!!!!! • Homework – Textbook p. 146 # 1 – 10 ALL, 17 – 23 ALL.

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