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3.5 Derivatives of Trigonometric Functions What you’ll learn about…. Derivatives of the Sine and Cosine Functions Simple Harmonic Motion Jerk Derivatives.

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Presentation on theme: "3.5 Derivatives of Trigonometric Functions What you’ll learn about…. Derivatives of the Sine and Cosine Functions Simple Harmonic Motion Jerk Derivatives."— Presentation transcript:

1 3.5 Derivatives of Trigonometric Functions What you’ll learn about…. Derivatives of the Sine and Cosine Functions Simple Harmonic Motion Jerk Derivatives of the Tangent, Cotangent, Secant and Cosecant functions

2 Warm Up: Exploration 1 Page 141 Graph sin x and nderiv(sin (x), x, x) in a trigonometric window Make a conjecture, what function do you believe is the derivative of sin x? If f(x) = sin x, then f’(x) =

3 Derivatives of Trigonometric Functions

4 Find

5 Example 1: Use the product & quotient rules to find the following derivatives

6 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. If a weight hanging from a spring is stretched 5 units beyond its resting 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 is its velocity and acceleration at time t? v(t) = a(t) = Describe its motion. Starts at Amplitude Period Greatest Velocity Greatest Acceleration

7 What about JERK? A sudden change in acceleration is called a “jerk”. The derivative responsible for jerk is the third derivative of position. Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is

8 Example 3: A Couple of Jerks a)FYI The jerk caused by the constant acceleration of gravity (g = -32 ft/sec 2 ) is zero. This explains why we don’t experience motion sickness sitting around! b) Find the jerk of the simple harmonic motion in Example 2.

9 Finding Tangent and Normal Lines Find equations for the lines that are tangent and normal to the graph of Slope of tangent Tangent line Normal line

10 Example 5 A Trigonometric Second Derivative Find y” if y = csc x.

11 Find more examples & Practice http://archives.math.utk.edu/visual.calculus/index.html Select Derivatives Derivatives of the Trigonometric Functions Tutorial Drill with Power Rule Drill with Quotient Rule

12 Homework Page 146 Quick Review 1-10 Exercises 1-11 odds

13 Agenda for Today Opener Page 155 Exercises 70-73 No Calculator! (turn in when completed) Present homework under document camera Class / Homework Page 146 Exercises 13-31 odds, 37, 40, 42


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