Lesson 3-5: Derivatives of Trig Functions

1
Lesson 3-5 Derivatives of Trig Functions

• AP Calculus
• Mrs. Mongold

2
Consider the function
We could make a graph of the slope
Now we connect the dots!
The resulting curve is a cosine curve.
3
We can do the same thing for
The resulting curve is a sine curve that has been
reflected about the x-axis.
4
We can find the derivative of tangent x by using
the quotient rule.
5
Derivatives of the remaining trig functions can
be determined the same way.
6
Example

• A wight hanging from a spring is stretched 5
  units beyond its rest position (s0) and released
  at time t0 to bob up and down. Its position at
  any later time t is s5cost
• What are the velocity and acceleration? Describe
  its motion.

7
Jerk

• The derivative of acceleration is jerk. If a
  bodys position at time t is s(t) the bodys 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

8
Example A couple of jerks

• A) the jerk caused by the constant acceleration
  of gravity (g -32 ft/sec2) is zero.
• This is why we dont get motion sickness when we
  are sitting around doing nothing
• B) The jerk of a simple harmonic motion from
  example 1 is what?

9
Example

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

10
Example

• Find y if y secx

11
Homework

• Page 140/1-10, 12-14, 17, 20-22