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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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

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) =

Derivatives of Trigonometric Functions

Find

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

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

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

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.

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

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

Find more examples & Practice Select Derivatives Derivatives of the Trigonometric Functions Tutorial Drill with Power Rule Drill with Quotient Rule

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

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