7.6 Graphs of the Sine and Cosine Functions 7.8 Phase shift; Sinusoidal Curve Fitting
Consider the function We could make a graph of the slope: slope Now we connect the dots! The...
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Transcript of Consider the function We could make a graph of the slope: slope Now we connect the dots! The...
2
0
2
Consider the function siny
We could make a graph of the slope: slope
1
0
1
0
1Now we connect the dots!
The resulting curve is a cosine curve.
sin cosd
x xdx
2
0
2
We can do the same thing for cosy slope
0
1
0
1
0The resulting curve is a sine curve that has been reflected about the x-axis.
cos sind
x xdx
Derivative of y=sinx
• Use the definition of the derivative
To prove the derivative of y=sinx is y’=cosx.
lim ( ) ( )'
0
f x h f xy
h h
Hints:
sin( ) sin cosh cos sinh
lim cosh 10
0
x h x x
h h
Derivative of y=sinx
lim ( ) ( )'
0
f x h f xy
h h
lim sin( ) sin
0
x h x
h h
sin 0 cos 1x x lim sin cosh cos sinh sin
0
x x x
h h
lim sin (cosh 1) cos sinh
0
x x
h h
lim lim lim limcosh 1 sinhsin cos
0 0 0 0x x
h h h hh h
lim cosh 1 sinhsin cos
0x x
h h h
Shortcut: y’=cosx
The proof of the d(cosx) = -sinx is almost identical
product rule:
d dv duuv u v
dx dx dx Notice that this is not just the
product of two derivatives.
This is sometimes memorized as: d uv u dv v du
2 33 2 5d
x x xdx
5 3 32 5 6 15d
x x x xdx
5 32 11 15d
x x xdx
4 210 33 15x x
2 3x 26 5x 32 5x x 2x
4 2 2 4 26 5 18 15 4 10x x x x x
4 210 33 15x x
Example:
2 cosdx x
dx
22 cos
cosdx d x
x xdx dx
2 cos2 cos
d xx x x
dx
22 cos sinx x x x
Try this:
21. sind
xdx
2sin sin sind d
x x xdx dx
sin sin sin sind d
x x x xdx dx
2sin cosx x
Try this:
2 22. sin cosd
x xdx
1 0d
dx
2 2
Or,
sin cosd d
x xdx dx
2sin cos
cos cos cos cos
x x
d dx x x x
dx dx
2sin cos
sin cos cos sin
x x
x x x x
2sin cos 2sin cos x x x x 0
quotient rule:
2
du dvv ud u dx dx
dx v v
or 2
u v du u dvdv v
3
2
2 5
3
d x x
dx x
2 2 3
22
3 6 5 2 5 2
3
x x x x x
x
Quotient Rule
2
1 sin
5
d x
dx x
2 2
22
1 sin 5 1 sin 5
5
x x x x
x
2
22
5 cos 1 sin 10
5
x x x x
x
3
cos 2sin 2
5
x x x
x
1
cos
d
dx x
2
1 cos 1 cos
cos
d dx x
dx dxx
2
sin
cos
x
x
1 sin
cos cos
x
x x
sec tanx x