drill convert 105 degrees to radians convert 5π/9 to radians what is the range of the equation y =...
TRANSCRIPT
Drill
• Convert 105 degrees to radians
• Convert 5π/9 to radians
• What is the range of the equation y = 2 + 4cos3x?
• 7π/12
• 100 degrees
• [-2, 6]
Derivatives of Trigonometric Functions
Lesson 3.5
Objectives
• Students will be able to– use the rules for differentiating the six basic
trigonometric functions.
Find the derivative of the sine function.
xy sin h
xfhxfxf
h
0lim'
H
xHxy
H
sinsinlim'
0
H
xHxHxy
H
sinsincoscossinlim'
0
H
HxxHxy
H
sincossincossinlim'
0
H
HxHxy
H
sincos1cossinlim'
0
Find the derivative of the sine function.
xy sin h
xfhxfxf
h
0lim'
H
xHxy
H
sinsinlim'
0
H
xHxHxy
H
sinsincoscossinlim'
0
H
HxxHxy
H
sincossincossinlim'
0
H
HxHxy
H
sincos1cossinlim'
0
H
Hx
H
Hxy
HH
sincoslim
1cossinlim'
00
H
Hx
H
Hxy
HH
sinlimcos
1coslimsin'
00
1cos0sin' xxy
xy cos'
Find the derivative of the cosine function.xy cos
h
xfhxfxf
h
0lim'
H
xHxy
H
coscoslim'
0
H
xHxHxy
H
cossinsincoscoslim'
0
H
HxxHxy
H
sinsincoscoscoslim'
0
H
HxHxy
H
sinsin1coscoslim'
0
Find the derivative of the cosine function.xy cos
H
HxHxy
H
sinsin1coscoslim'
0
H
Hx
H
Hxy
HH
sinsinlim
1coscoslim'
00
H
Hx
H
Hxy
HH
sinlimsin
1coslimcos'
00
1sin0cos' xxy
xy sin'
Derivatives of Trigonometric Functions
xxdx
dcossin
xxdx
dsincos
Example 1 Differentiating with Sine and Cosine
Find the derivative.
xxy cos3
33 coscos xdx
dxx
dx
dx
dx
dy
23 3cossin xxxxdx
dy
xxxxdx
dysincos3 32
Example 1 Differentiating with Sine and Cosine
Find the derivative.
x
xy
cos2
sin
2cos2
cos2sinsincos2
x
xdx
dxx
dx
dx
dx
dy
2cos2
sin0sincoscos2
x
xxxx
dx
dy
Example 1 Differentiating with Sine and Cosine
Find the derivative.
x
xy
cos2
sin
2cos2
sin0sincoscos2
x
xxxx
dx
dy
2cos2
sinsincoscos2
x
xxxx
dx
dy
Example 1 Differentiating with Sine and Cosine
Find the derivative.
x
xy
cos2
sin
2cos2
sinsincoscos2
x
xxxx
dx
dy
2
22
cos2
sincoscos2
x
xxx
dx
dy
Example 1 Differentiating with Sine and Cosine
Find the derivative.
x
xy
cos2
sin
2
22
cos2
sincoscos2
x
xxx
dx
dy
2
22
cos2
cos1coscos2
x
xxx
dx
dy
Remember that cos2 x + sin2 x = 1So sin x = 1 – cos 2x
Example 1 Differentiating with Sine and Cosine
Find the derivative.
x
xy
cos2
sin
2
22
cos2
cos1coscos2
x
xxx
dx
dy
2cos2
1cos2
x
x
dx
dy
Homework, day #1
• Page 146: 1-3, 5, 7, 8, 10• On 13 – 16
Velocity is the 1st derivative Speed is the absolute value of velocity Acceleration is the 2nd derivative Look at the original function to determine
motion
Find the derivative of the tangent function.xy tan
x
xy
cos
sin
x
xdx
dxx
dx
dx
y2cos
cossinsincos'
x
xxxxy
2cos
sinsincoscos'
x
xxy
2
22
cos
sincos'
Find the derivative of the tangent function.xy tan
x
xxy
2
22
cos
sincos'
xy
2cos
1'
2
cos
1'
xy
xy 2sec'
Derivatives of Trigonometric Functions
xxdx
dcossin
xxdx
dsincos
xxdx
d 2sectan
Derivatives of Trigonometric Functions
xxdx
dcossin
xxdx
dsincos
xxdx
d 2sectan xxdx
d 2csccot
xxxdx
dtansecsec
xxxdx
dcotcsccsc
More Examples with Trigonometric FunctionsFind the derivative of y.
xxy cot11sin
1sincot1cot11sin xdx
dxx
dx
dx
dx
dy
xxxxdx
dycoscot1csc1sin 2
xx
x
xx
dx
dycos
sin
cos1
sin
11sin
2
x
xx
xxdx
dy
sin
coscos
sin
1
sin
1 2
2
xxxxxdx
dycsccoscoscsccsc 22
xxxxxdx
dycsccoscoscsccsc 22
xxxxxdx
dy 22 csccoscsccoscsc
xxxxdx
dy 22 csccos)cos1(csc
xxxxdx
dy 22 csccos)(sincsc
xxxxdx
dy 22 csccos)(sinsin
1
xxxdx
dy 2csccossin
More Examples with Trigonometric Functions
Find the derivative of y.
3
3
sec
2tan
xx
xxy
23
3333
sec
sec2tan2tansec
xx
xxdx
dxxxx
dx
dxx
dx
dy
23
23223
sec
3tansec2tan6secsec
xx
xxxxxxxxx
dx
dy
52323
223
6secsec6sec
6secsec
xxxxxx
xxxx
5322
23
6tansec2tan3tansec
3tansec2tan
xxxxxxxx
xxxxx
)6tansec2tan3tan(sec6secsec6sec 532252323 xxxxxxxxxxxxxx
xxxxxxxxxxxx tansec2tan3tansecsecsec6sec 3222323
23
3222323
sec
tansec2tan3tansecsecsec6sec'
xx
xxxxxxxxxxxxy
Whatta Jerk!
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
.3
3
dt
sd
dt
datj
Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:
s1(t) = 3cos t
s2(t) = 2sin t – cos t
Find the jerks of the bodies at time t.
tts cos31
tdt
dssin31 tsin3
tdt
sdcos3
21
2
velocity
acceleration
Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:
s1(t) = 3cos t
s2(t) = 2sin t – cos t
Find the jerks of the bodies at time t.
tts cos31
tdt
dssin31 tsin3
tdt
sdcos3
21
2
velocity
acceleration
tdt
sdsin3
31
3
tsin3jerk
Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:
s1(t) = 3cos t
s2(t) = 2sin t – cos t
Find the jerks of the bodies at time t.
ttts cossin22
ttttdt
dssincos2sincos21
ttttdt
sdcossin2cossin2
21
2
velocity
acceleration
ttttdt
sdsincos2sincos2
31
3
jerk
Homework, day #2
• Page 146: 4, 6, 9, 11, 12, 17-20, 22 28, 32