Lesson 3-5: Derivatives of Trig Functions AP Calculus Mrs. Mongold

Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

We can do the same thing for slope The resulting curve is a sine curve that has been reflected about the x-axis.

We can find the derivative of tangent x by using the quotient rule.

Derivatives of the remaining trig functions can be determined the same way.

Example A wight 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. It’s position at any later time t is s=5cost What are the velocity and acceleration? Describe its motion.

Jerk The derivative of acceleration is jerk. If a body’s position at time t is s(t) the body’s jerk at time t is When a ride in a car or bus is jerky, it is not that the accelerations are large but the changes in acceleration are abrupt

Example: A couple of jerks A) the jerk caused by the constant acceleration of gravity (g = -32 ft/sec 2 ) is zero. –This is why we don’t get motion sickness when we are sitting around doing nothing B) The jerk of a simple harmonic motion from example 1 is what?

Example Find equations for the lines that are tangent and normal to the graph of at x = 2.

Example Find y’’ if y = secx

Homework Page 140/1-10, 12-14, 17, 20-22