MATH 116 ACTIVITY 2:
WHY:
Calculus (derivative and antiderivative) with trig functions
The trigonometric functions complete the set of "elementary functions" from which we build (by composititon,
multiplication, etc.) most of the functions used in applications. This activity is intended to allow us to revisit the
basic derivative and antiderivative procedures including all th eelementary functions before we move on to more
advanced topics.
LEARNING OBJECTIVES:
1.
Be able to extend the derivative and antiderivative techniques to include new functions
2.
Become familiar with the derivative and antiderivative rules for sine and cosine functions:
D sin u  (cos u )Du
D cos u  ( sin u )Du
 sin u d u   cos u  C  cos u d u  sin u  C
3.
Work as a team, using the team roles.
CRITERIA:
1.
Success in completing the exercises.
2.
Success in working as a team and in filling the team roles.
RESOURCES:
1.
Your text - especially sections 9.1-9.3
2.
The review sheet for material from the first semester
3.
The team role desk markers (handed out in class for use during the semester)
4.
40 minutes
PLAN:
1.
Select roles, if you have not already done so, and decide how you will carry out steps 2 through 5
2.
Work through the models given here
3.
Complete the exercises given here - be sure all members of the team understand and agree with all the results in the recorder's
report.
4.
Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade .
MODELS:
1.
Finding the derivative of f ( x )  e x cos
D (e x
cos x
2
) = [exponential rule] e x cos
x
x
2
2
[derivative of cosine, derivative of x) e x cos
e x cos
2.
x
2
D (x cos x 2) =[product rule] e x cos
x
2
x
2
( xD (cos x 2)  cos x 2(Dx )) =
( x ( sin x 2D (x 2))  cos x 2 (1)) =(power rule]
( x sin x 2(2x )  cos x 2) which would more often be written (2x 2 sin x 2  cos x 2)e x cos
Given that F (t )  t cos( 3t
2
2
2
2
 4) and that F(0) = 5 , we find the formula for F(t) and the value of F(3)
 F (t )dt  t cos( 3t  4)dt [Try  cos u d u  sin u  C - will require u  3t
1
1
F(t )  t cos( 3t  4)d t 
6t cos( 3t  4)d t  s in( 3t  4) C

6
6
F(t ) 
x
2
2
2
 4 so u  6t ]
5  F (0) 
So F(t ) 
1
6
1
6
s in( 3(0)  4)  C 
2
s in( 3t
2
 4)  5 
1
6
1
6
s in( 4)  C so C  5 
s in( 4) and F(3) 
1
6
1
6
s in( 4)  4.874 (to three places)
sin( 3(3)  4)  5 
2
1
6
sin( 4)  4.732
EXERCISES:
1.
Give each of these derivatives
a.) Dt sin 3( t  5)
b.) D x sin( e x ) cos( e x )
2.
Give each of these general antiderivatives
 3x )dx

sin( 3x )
b.)
 cos( 3x )  9 d x
c.) sin (5x  2) cos( 5x  2)d x

a.) (2x  1) sin( 3x
2
4
3.
If H (t )  sin(
t
2
)e
t
cos( )
2
and H ( )  4 , give the formula for H (t ) and give the value of H (2)
CRITICAL THINKING QUESTIONS:(answer individually in your journal)
1.
To extend our graphing techniques from last semester (intervals where f is increasing, inrevals where f is decreasing, etc.] to
use with sine & cosine, we'd need to be able to solve equations like 3sin(2x) = 0. How many solutions are there to the
equation sin(2x) = 0? What do they look like?
2.
Which of the exercises do you feel was the most difficult? Why?
3.
What periodic functions would you expect would be important in biology? What would the period be for each?
SKILL DEVELOPMENT EXERCISES (due next Thursday)
Text: p 577 #7-10, 21-23 Worksheet Some derivative/antiderivative exercises involving trig functions #2-5 (part c on each]
and #8-11(part c only)