MA111 CALCULUS I
Friday, 3/2/12
Today:
Reading:
Derivatives of Trigonometric Functions
3.3
Exercises:
3.3, p. 197: 1-15 odd, 21-22, 33-35
Friday, 3/2/12, Slide #1
Finding the derivative of sin x
1. Try to use graph of sin x to draw graph
of D[sin x].
2. Use this to make a guess about the
derivative of sin x.
3. Use definition of derivative to confirm
guess. Need two facts:
sin(a + b) = sin(a) cos(b) + cos(a) sin(b)
cos(x) < sin(x) / x < 1
Friday, 3/2/12, Slide #2
CQ #1
Theorem 1: D[sin x] = cos x
Theorem 2: D[cos x] = -sin x
Trig Identity: sin2x + cos2x = 1
Using these facts and the Quotient Rule, find:
D[tan x] = D[sin x / cos x]
A. tan2x
B. 1 / sin2x
C. -cot x
D. tan x
E. 1 / cos2x
Friday, 3/2/12, Slide #3
Derivatives of Trig Functions
d
(sin x ) = cos x
dx
d
(tan x ) = sec 2 x = 12
dx
cos x
d
(sec x ) = sec x tan x = sin2x
dx
cos x
d
(cos x ) = − sin x
dx
d
(cot x ) = − csc 2 x = − 12
dx
sin x
d
(csc x ) = − csc x cot x = − cos2 x
dx
sin x
The derivative of each “co” function is obtained by
switching “co” and not “co”, and multiplying by -1.
So can use derivative formulas for sin x, tan x, and
sec x to get derivative formulas for cos x, cot x, and
csc x
Friday, 3/2/12, Slide #4
CQ #2
FACTS: sin(p/6) = ½; cos(p/6) = ÷3 / 2
What are:
(i) the slope of the tangent line, and
(ii) the concavity, of the graph of
y = sin x at the point (p/6, ½)?
A. (i) ÷3 / 2
B. (i) -½
C. (i) ÷3 / 2
D. (i) ½
E. (i) ÷3 / 2
(ii)
(ii)
(ii)
(ii)
(ii)
÷3 / 2
÷3 / 2
-½
½
½
Friday, 3/2/12, Slide #5
Hours of Daylight Function
Equinoxes: 12 hours daylight/12 hours dark
Solstices: Maximum and minimum hours of
daylight
What are properties of 1st and 2nd derivatives
at equinoxes and
at solstices?
*Google “Daylight
Hours Explorer”
Friday, 3/2/12, Slide #6