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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 Presentation
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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
𝑓 (𝑥 )=𝑎0+𝑎1 (𝑥− 𝑥𝐸 )+𝑎2 (𝑥− 𝑥𝐸 )2+𝑎3 (𝑥−𝑥𝐸 )3+⋯
2nd derivative and curvature Power series
Power series for sine and cosine
sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )=1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯
Approximation of p
√32
1/21
𝜋6
𝜋 ≅ 3+ 𝜋 3
216− 𝜋 5
155520
2
Higher-order derivatives and power series
2nd derivative and curvature
3
𝑓 (𝑥 )
𝑥0
𝑑 𝑓 /𝑑𝑥
𝑥00
𝑑2 𝑓 /𝑑𝑥2
𝑥0
𝑑2 𝑓𝑑 𝑥2
≔𝑑 (𝑑 𝑓𝑑 𝑥 )𝑑𝑥
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 (𝜃 )≅ 𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )≅ 1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯ √3
2
1/21
𝜋6
𝜋 ≅ 3+ 𝜋 3
216− 𝜋 5
155520
Higher-order derivatives and power series
𝑓 (𝑥 )=𝑎0+𝑎1 (𝑥− 𝑥𝐸 )+𝑎2 (𝑥− 𝑥𝐸 )2+𝑎3 (𝑥−𝑥𝐸 )3+⋯
𝑓 (𝑥 )
𝑥0
5
𝑥𝐸
𝑓 (𝑥𝐸 )=𝑎0+𝑎1 (𝑥𝐸−𝑥𝐸 )+𝑎2 (𝑥𝐸−𝑥𝐸 )2+⋯
𝑑 𝑓𝑑 𝑥 |𝑥=𝑎1+2𝑎2 (𝑥− 𝑥𝐸 )+3𝑎3 (𝑥−𝑥𝐸 )2+⋯𝑥𝐸𝑥𝐸
𝑑 𝑓𝑑 𝑥 |𝑥=𝑎1+2𝑎2 (𝑥− 𝑥𝐸 )+3𝑎3 (𝑥−𝑥𝐸 )2+⋯
𝑥𝐸
𝑑2 𝑓𝑑 𝑥2|𝑥=2𝑎2+3 ∙2𝑎3 (𝑥−𝑥𝐸 )+⋯𝑥𝐸
𝑥𝐸
𝑑2 𝑓𝑑 𝑥2|𝑥=2𝑎2+3 ∙2𝑎3 (𝑥−𝑥𝐸 )+⋯
𝑑3 𝑓𝑑 𝑥3|𝑥=3 ∙2𝑎3+ powersof (𝑥− 𝑥𝐸 )⋯𝑥𝐸
𝑥𝐸
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
=𝑘!𝑎𝑘
≅
Constructing power series representations
𝑓 (𝑥 )
𝑥0
Constructing power series representations
6
𝑓 (𝑥 )=𝑎0+𝑎1 (𝑥− 𝑥𝐸 )+𝑎2 (𝑥− 𝑥𝐸 )2+𝑎3 (𝑥−𝑥𝐸 )3+⋯
𝑥𝐸
𝑓 (𝑥 )=𝑓 (𝑥𝐸 )0 !
+ 11 !
𝑑 𝑓𝑑 𝑥|
𝑥𝐸
(𝑥−𝑥𝐸 )
+12 !
𝑑2 𝑓𝑑 𝑥2|𝑥𝐸
(𝑥− 𝑥𝐸 )2
+13 !
𝑑3 𝑓𝑑 𝑥3|𝑥𝐸
(𝑥−𝑥𝐸 )3+⋯
𝑓 (𝑥 )=∑𝑘=0
∞ 1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
(𝑥−𝑥𝐸 )𝑘Subtract
Multiply coefficient and powers of binomial
Add terms
1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥 𝐸
=𝑘!𝑎𝑘
𝑓 (𝑥 )
𝑥0
Construction of accurate power series representation can fail
7
𝑥𝐸
𝑓 (𝑥 )=∑𝑘=0
∞ 1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
(𝑥−𝑥𝐸 )𝑘
𝑓 (𝑥 )
𝑥0 𝑥𝐸
𝑓 (𝑥 )
𝑥0 𝑥𝐸
Too jagged
Too straight
Higher-order derivatives and power series
8
Power series2nd derivative and curvature
Power series for sine and cosine
sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )=1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯
Approximation of p
√32
1/21
𝜋6
𝜋 ≅ 3+ 𝜋 3
216− 𝜋 5
155520
2𝜋𝜃0
-1
1
𝜋𝜋4
𝜋2
3𝜋2
12
−1 /2
√2/2√3 /2
−√3 /2−√2 /2
3𝜋4
5𝜋4
7𝜋4
𝜋6
𝜋3
sin (𝜃 )
Qualitative expectations for power series for sine
9
𝑓 (𝑥 )=∑𝑘=0
∞ 1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
(𝑥−𝑥𝐸 )𝑘𝑓 (𝜃 )=sin (𝜃 )Expand around qE = 0
sin (𝜃 )≅ stuff 𝜃−other 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
𝑓 (𝑥 )=∑𝑘=0
∞ 1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
(𝑥−𝑥𝐸 )𝑘𝑓 (𝜃 )=sin (𝜃 )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
𝑓 (𝑥 )=∑𝑘=0
∞ 1𝑘!
𝑑𝑘 𝑓𝑑 𝑥𝑘|
𝑥𝐸
(𝑥−𝑥𝐸 )𝑘𝑓 (𝜃 )=sin (𝜃 )Expand around qE = 0
𝑓 (𝜃 )=sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
12
2𝜋𝜃
0
-1
1
𝜋𝜋4
𝜋2
3𝜋2
12
−1 /2
√2/2√3 /2
−√3 /2−√2 /2
3𝜋4
5𝜋4
7𝜋4
𝜋6
𝜋3
2𝜋35𝜋6
7𝜋6
4𝜋3
5𝜋311𝜋6
sin (𝜃 )
cos (𝜃 )
𝑓 (𝜃 )=sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )=1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯
Power series for sine is as expected
sin (𝜃 )≅ stuff 𝜃−other stuff farther away
Higher-order derivatives and power series
13
Power series for sine and cosine
2nd derivative and curvature Power series
sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )=1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯
Approximation of p
√32
1/21
𝜋6
𝜋 ≅ 3+ 𝜋 3
216− 𝜋 5
155520
14
sin (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
Using a power series and iteration to approximate p
√32
1/21
𝜋6
sin (𝜋6 )=(𝜋6 )− (𝜋6 )3
3 !+( 𝜋6 )
5
5 !−⋯
12=( 𝜋6 )− ( 𝜋6 )
3
3 !+( 𝜋6 )
5
5 !−⋯
𝜋 ≅ 3+ 𝜋 3
216− 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 (𝜃 )=𝜃− 𝜃3
3 !+𝜃55 !− 𝜃
7
7 !+⋯
cos (𝜃 )=1− 𝜃2
2 !+𝜃44 !− 𝜃
6
6 !+⋯ √3
2
1/21
𝜋6
𝜋 ≅ 3+ 𝜋 3
216− 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
