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Chem 302 - Math 252
Chapter 4Differentiation & Integration
Differentiation & Integration
• Experimental data at discrete points
• Need to know the rate of change of the dependent variable with respect to the independent variable
• Need to know area under curve
• Need to integrate an analytic function that is too complicated to do analytically
• Can do interpolation/curvefitting to get an analytic function
Linear Differentiation
0 00 0lim x
f x x f xf x
x
0 00 small (Eqn 1)
f x x f xf x x
x
0 00 small (Eqn 2)
2
f x x f x xf x x
x
Eqn (1) Eqn (2)
x % error % error
Exact value 4.481689 4.481689
0.1 4.713434 5.0 4.489162 0.170.01 4.504172 0.50 4.481764 1.7×10-3
0.001 4.483931 5.0×10-2 4.481690 1.7×10-5
1×10-4 4.481913 5.0×10-3 4.481689 1.7×10-7
1×10-5 4.481711 5.0×10-4 4.481689 2.2×10-9
1×10-6 4.481691 5.0×10-5 4.481689 5.8×10-9
1×10-7 4.481689 5.2×10-6 4.481689 6.4×10-8
1×10-8 4.481689 1.3×10-7 4.481689 1.1×10-6
1×10-9 4.481690 1.4×10-5 4.481689 3.8×10-6
1×10-10 4.481695 1.3×10-4 4.481691 3.4×10-5
1×10-11 4.481748 1.3×10-3 4.481704 3.3×10-4
1×10-12 4.482636 2.1×10-2 4.482192 1.1×10-2
1×10-13 4.485301 8.1×10-2 4.480860 1.8×10-2
1×10-14 4.529710 1.1 4.485301 8.1×10-2
1×10-15 5.329071 19 4.884981 9.0
Linear Differentiation x xf x e f x e 5.1'f 5.1'f
0 00 (Eqn 1)
f x x f xf x
x
1 1
1 1
k kk
k k
f x f xf x
x x
Smaller spacing not necessarily better
0 00 (Eqn 2)
2
f x x f x xf x
x
3 point Differentiation
• Linear differentiation ignores actual point
0 1 0 1 0 0 1 0 2f x p f x h p f x p f x h
1
2 0
2
3 0
1f x
f x x x
f x x x
• Make exact for
2 2 2 21 0 2 2 1 0 2 0 1
01 2 1 2
h f x h h h f x h f x hf x
h h h h
Maple Sheet
0 00 2
f x h f x hf x
h
Multi-point Differentiation
• Formulae only derived for equal spacing
• Non equal spacing solve equations numerically
0 0 0 0 0
12 8 8 2
2f x f x h f x h f x h f x h
h
Multi-point Differentiation
Coefficient
Demominator -4h -3h -2h -h 0 h 2h 3h 4h Exact to
1st derivative
2h -1 0 1 Quadratic
12h 1 -8 0 8 -1 Quartic
60h -1 9 -45 0 45 -9 1 6th order
840h 3 -32 168 -672 0 672 -168 32 -3 8th order
Multi-point Differentiation
Coefficient
Demominator -4h -3h -2h -h 0 h 2h 3h 4h Exact to
2nd derivative
h2 1 2 1 Quadratic
12h2 -1 16 -30 16 -1 Quartic
540h2 6 -81 810 -1470 810 -81 6 6th order
5040h2 -9 128 -1008 8064 -14350 8086 -1008 128 -9 8th order
Multi-point Differentiation
Coefficient
Demominator -4h -3h -2h -h 0 h 2h 3h 4h Exact to
3rd derivative
2h3 -1 2 0 -2 1 Quartic
48h3 6 -48 78 0 -78 48 -6 6th order
240h3 -7 72 -338 488 0 -488 338 -72 7 8th order
Technique% error % error % error
3-point -0.35635222 0.001 -0.18966826 0.0004
5-point -0.35635566 0 -0.18966907 0 0.20689002 0.002
7-point -0.35635566 0 -0.18966907 0 0.20689365 0
9-point -0.35635566 0 -0.18966907 0 0.20689365 0
Exact value -0.35635566 -0.18966907 0.20689365
Example
sin
0
1 0
xx
f x xx
xf xf xf
01.0
1.256637064.0
h
x
Numerical Integration
Midpoint Formula
• Uses value of function and slope at midpoint of interval
1 2 0w b a w
2 2
1 22 2
b a a bf x x w w b
11f x b a w
1 22 2
ba b a b
a
f x dx w f w f
• Determine w1 & w2
2
ba b
a
f x dx b a f
Composite Midpoint Formula
• n subintervals (equal spacing)
b ah
n
3 5 2 112 2 2 2
1
2
2 1
2
bn
a
n
i
f x dx h f a h f a h f a h f a h f a h f a h
ih f a h
Trapezoidal Integration
• Approximate f(x) by a linear function over interval [a,b]
f b f af x f a x a
b a
2 212
12
12
b b b
a a a
f b f af x dx f a dx x a dx
b a
f b f af a b a b a a b a
b a
f a b a f b f a b a a
f b f a b a
Trapezoidal Integration
• Alternate derivation• Linear combination of endpoints that give best estimate of
integral
1 2
b
a
f x dx w f a w f b
• Determine w1 & w2
1 21f x b a w w
2 2
1 22
b af x x w a w b
1 2 2
b aw w
Composite Trapezoidal Integration
• n subintervals (equal spacing)
b ah
n
1 12 2
2 2 32
2 3
b
a
hf x dx f a f a h f a h f a h f a h f a h f b
h f a f a h f a h f a h f b
Simpson’s Rule• Combines Trapezoidal and Midpoint
• Also referred to as 3 - point
1 3 2
2
6 3
b ab aw w w
2 2
1 2 32 2
b a a bf x x w a w w b
1 2 31f x b a w w w
1 2 32
ba b
a
f x dx w f a w f w f b
• Determine w1 w2 & w3
246
4 23
ba b
a
b af x dx f a f f b
hf a f a h f a h
23 32 2
1 2 33 2
b a a bf x x w a w w b
Package
Composite Simpson’s Rule
• 2n subintervals (equal spacing)
2
b ah
n
1
24 2 2 4 3 4 2
6
2 2 2 2 1 23
b
a
n
i
hf x dx f a f a h f a h f a h f a h f a h f a nh
hf a f a nh f a i h f a ih
Newton-Cotes Formula
• Generalization to use more than 3 points – Trapezoidal exact up to linear – (1st order NC)– Simpson’s exact up to quadratic (by definition but turns out to be
exact for up to cubic) – (2nd order NC)– Equivalent to integration of Lagrangian interpolation functions– 3rd order NC
• Use 4 points and functions up to cubic
– Higher orders can give larger errors
Newton-Cotes Formula
rd
3
3 order
33 3 2 3
8
a h
a
hf x dx f a f a h f a h f a h
th
4
4 order
27 32 12 2 32 3 7 4
45
a h
a
hf x dx f a f a h f a h f a h f a h
Package
Gaussian Quadratures
• So far evaluated function at fixed points & optimized coefficients
• Optimize locations also
1
11
n
i ii
z dz w z
• Optimize wi & zi
1
1
2 ( )
b
a
x dx z dz
x b az
b a
Gaussian Quadratures
• 1-point
1
1 1
1
z dz w z
• Need two equations
• Make exact for (z) = 1, & (z) = z
1
1 1
1
1
For 1
2
z
z dz w z
w
1
1 1
1
11 2
1 1
1 1
1
For
02
0
z z
z dz w z
zzdz w z
z
1
1
2 0z dz
Gaussian Quadratures
• 2-point
1
1 1 2 2
1
z dz w z w z
• Need four equations
• Make exact for (z) = 1, (z) = z, (z) = z2, (z) = z3
• Does not give unique solution
• Make symmetric about 0
1
1 1 1
1
z dz w z z
Gaussian Quadrature.mws
Gaussian Quadratures
1
1 1 1
1
1
1
1
1
1
For 1
1 2
2 2
1
z
z dz w z z
dz w
w
w
1
1 1
1
1
1 1
1
For
0 0
z z
z dz z z
zdz z z
2
1
1 1
1
12 2
1
1
1321
1
1
For
2
22
3 3
1
3
z z
z dz z z
z dz z
zz
z
Gaussian Quadratures
1
11
( ) ( )n
i ii
z dz w z
Roots (zi) Weight Factors (wi)
Two-Point Formula
±0.57735 02691 89626 1.00000 00000 00000
Three-Point Formula
0 0.88888 88888 88889
±0.77459 66692 41483 0.55555 55555 55556
Four-Point Formula
±0.33998 10435 84856 0.65214 51548 62546
±0.86113 63115 94053 0.34785 48451 37454
Gaussian Quadratures
1
11
( ) ( )n
i ii
z dz w z
Roots (zi) Weight Factors (wi)
Five-Point Formula
0±0.53846 93101 05683 ±0.90617 98459 38664
0.56888 88888 88889 0.47862 86704 99366 0.23692 68850 56189
Six-Point Formula
±0.23861 91860 83197±0.66120 93864 66265±0.93246 95142 03152
0.46791 39345 726910.36076 15730 481390.17132 44923 79170
Gaussian Quadratures
1
11
( ) ( )n
i ii
z dz w z
Roots (zi) Weight Factors (wi)
Ten-Point Formula
±0.14887 43389 81631±0.43339 53941 29247±0.67940 95682 99024±0.86506 33666 88985±0.97390 65285 17172
0.29552 42247 147530.26926 67193 099960.21908 63625 159820.14945 13491 505810.06667 13443 08688
Fifteen-Point Formula
0±0.20119 40939 97435±0.39415 13470 77563±0.57097 21726 08539±0.72441 77313 60170±0.84820 65834 10427±0.93727 33924 00706±0.98799 25180 20485
0.20257 82419 255610.19843 14853 271110.18616 10001 155620.16626 92058 169940.13957 06779 261540.10715 92204 671720.07036 60474 881080.03075 32419 96117
Gaussian Quadratures
• Other forms
0
ze z dz
2
8
ze z dz
Gaussian Quadratures - Example
21
0
1
2xe dx
• Simpson’s Rule– Use 100 intervals
• Gaussian Quadrature– 3 and 15 point
Simpsons Gaussian Quadrature Example.mws