higher-order derivatives and power series
DESCRIPTION
Higher-order derivatives and power series. ACME General Store Sales receipt. Higher-order derivatives and power series. 2 nd derivative and curvature. Power series. 1. 1/2. Power series for sine and cosine. Approximation of p. 2 nd derivative and curvature. upward slant. horizontal. - PowerPoint PPT PresentationTRANSCRIPT
Higher-order derivatives and power series
1
ACME General StoreSales receipt
Item A $10.7892
Item B $2.4934
Item C $3.4435
Total $16.7261
π (π₯ )=π0+π1 (π₯β π₯πΈ )+π2 (π₯β π₯πΈ )2+π3 (π₯βπ₯πΈ )3+β―
2nd derivative and curvature Power series
Power series for sine and cosine
sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )=1β π2
2 !+π44 !β π
6
6 !+β―
Approximation of p
β32
1/21
π6
π β 3+ π 3
216β π 5
155520
2
Higher-order derivatives and power series
2nd derivative and curvature
3
π (π₯ )
π₯0
π π /ππ₯
π₯00
π2 π /ππ₯2
π₯0
π2 ππ π₯2
βπ (π ππ π₯ )ππ₯
downward slantupward slant
negative slope
positive slope
horizontal
zero slopeincreasing slope
positive second derivative
4
Power series
Power series for sine and cosine Approximation of p
2nd derivative and curvature
sin (π )β πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )β 1β π2
2 !+π44 !β π
6
6 !+β― β3
2
1/21
π6
π β 3+ π 3
216β π 5
155520
Higher-order derivatives and power series
π (π₯ )=π0+π1 (π₯β π₯πΈ )+π2 (π₯β π₯πΈ )2+π3 (π₯βπ₯πΈ )3+β―
π (π₯ )
π₯0
5
π₯πΈ
π (π₯πΈ )=π0+π1 (π₯πΈβπ₯πΈ )+π2 (π₯πΈβπ₯πΈ )2+β―
π ππ π₯ |π₯=π1+2π2 (π₯β π₯πΈ )+3π3 (π₯βπ₯πΈ )2+β―π₯πΈπ₯πΈ
π ππ π₯ |π₯=π1+2π2 (π₯β π₯πΈ )+3π3 (π₯βπ₯πΈ )2+β―
π₯πΈ
π2 ππ π₯2|π₯=2π2+3 β2π3 (π₯βπ₯πΈ )+β―π₯πΈ
π₯πΈ
π2 ππ π₯2|π₯=2π2+3 β2π3 (π₯βπ₯πΈ )+β―
π3 ππ π₯3|π₯=3 β2π3+ powersof (π₯β π₯πΈ )β―π₯πΈ
π₯πΈ
ππ ππ π₯π|
π₯πΈ
=π!ππ
β
Constructing power series representations
π (π₯ )
π₯0
Constructing power series representations
6
π (π₯ )=π0+π1 (π₯β π₯πΈ )+π2 (π₯β π₯πΈ )2+π3 (π₯βπ₯πΈ )3+β―
π₯πΈ
π (π₯ )=π (π₯πΈ )0 !
+ 11 !
π ππ π₯|
π₯πΈ
(π₯βπ₯πΈ )
+12 !
π2 ππ π₯2|π₯πΈ
(π₯β π₯πΈ )2
+13 !
π3 ππ π₯3|π₯πΈ
(π₯βπ₯πΈ )3+β―
π (π₯ )=βπ=0
β 1π!
ππ ππ π₯π|
π₯πΈ
(π₯βπ₯πΈ )πSubtract
Multiply coefficient and powers of binomial
Add terms
1π!
ππ ππ π₯π|
π₯ πΈ
=π!ππ
π (π₯ )
π₯0
Construction of accurate power series representation can fail
7
π₯πΈ
π (π₯ )=βπ=0
β 1π!
ππ ππ π₯π|
π₯πΈ
(π₯βπ₯πΈ )π
π (π₯ )
π₯0 π₯πΈ
π (π₯ )
π₯0 π₯πΈ
Too jagged
Too straight
Higher-order derivatives and power series
8
Power series2nd derivative and curvature
Power series for sine and cosine
sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )=1β π2
2 !+π44 !β π
6
6 !+β―
Approximation of p
β32
1/21
π6
π β 3+ π 3
216β π 5
155520
2ππ0
-1
1
ππ4
π2
3π2
12
β1 /2
β2/2β3 /2
ββ3 /2ββ2 /2
3π4
5π4
7π4
π6
π3
sin (π )
Qualitative expectations for power series for sine
9
π (π₯ )=βπ=0
β 1π!
ππ ππ π₯π|
π₯πΈ
(π₯βπ₯πΈ )ππ (π )=sin (π )Expand around qE = 0
sin (π )β stuff πβother stuff farther away
k Expression Value at qE = 0 k!
0 0 0! 0
1 1 1!
2 0 2! 0
3 -1 3!
4 0 4! 0
10
π (π₯ )=βπ=0
β 1π!
ππ ππ π₯π|
π₯πΈ
(π₯βπ₯πΈ )ππ (π )=sin (π )Expand around qE = 0
1
Power series for sine
k Expression Value at qE = 0 k!
0 0 0! 0
1 1 1!
2 0 2! 0
3 -1 3!
4 0 4! 0
Power series for sine
11
π (π₯ )=βπ=0
β 1π!
ππ ππ π₯π|
π₯πΈ
(π₯βπ₯πΈ )ππ (π )=sin (π )Expand around qE = 0
π (π )=sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
12
2ππ
0
-1
1
ππ4
π2
3π2
12
β1 /2
β2/2β3 /2
ββ3 /2ββ2 /2
3π4
5π4
7π4
π6
π3
2π35π6
7π6
4π3
5π311π6
sin (π )
cos (π )
π (π )=sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )=1β π2
2 !+π44 !β π
6
6 !+β―
Power series for sine is as expected
sin (π )β stuff πβother stuff farther away
Higher-order derivatives and power series
13
Power series for sine and cosine
2nd derivative and curvature Power series
sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )=1β π2
2 !+π44 !β π
6
6 !+β―
Approximation of p
β32
1/21
π6
π β 3+ π 3
216β π 5
155520
14
sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
Using a power series and iteration to approximate p
β32
1/21
π6
sin (π6 )=(π6 )β (π6 )3
3 !+( π6 )
5
5 !ββ―
12=( π6 )β ( π6 )
3
3 !+( π6 )
5
5 !ββ―
π β 3+ π 3
216β π 5
155520
Guess for p RHS calculation Output
3.0000 3.1234
3.1234 3.1392
3.1392 3.1413
3.1413 3.1415
3.1415 3.1416 (stable)
Approximation of p
2nd derivative and curvature Power series
Power series for sine and cosine
sin (π )=πβ π3
3 !+π55 !β π
7
7 !+β―
cos (π )=1β π2
2 !+π44 !β π
6
6 !+β― β3
2
1/21
π6
π β 3+ π 3
216β π 5
155520
15
ACME General StoreSales receipt
Item A $10.7892
Item B $2.4934
Item C $3.4435
Total $16.7261
Higher-order derivatives and power series