The Cooper Union for the Advancement of Science and Art 
ChE352 
Numerical Techniques for Chemical Engineers 
Professor Stevenson 
Lecture 4 The Cooper Union for the Advancement of Science and Art 
Real Analysis 
●Properties of functions and series over 
the real numbers 
●Formalize ideas 
about calculus 
●Vocabulary and 
notation we'll use 
for the rest of the 
course The Cooper Union for the Advancement of Science and Art 
F&B 1.2: We 💜
 Continuous Functions 
Sufficient condition for f to be a continuous function :
If f is a continuous function of one independent variable 
on the interval from a to b, we say “ f is in C A B ”:
 Why are continuous functions useful? The Cooper Union for the Advancement of Science and Art 
Sequences and Convergence 
   More "numerical" definition of continuity: 
If a function f(x) is continuous, a sequence of 
trial values xn such that xn → x0 will make f( xn) 
approach f( x0).
   This means questions about continuous 
functions can be solved iteratively .
f is a function taking one real input & giving 
one real output, and X is also real. The Cooper Union for the Advancement of Science and Art 
Example: Convergence to a Root 
A series of guesses xi all have valid f( xi)
What else is defined for all these 
guesses xi besides f( xi)?The Cooper Union for the Advancement of Science and Art 
Derivatives and Differentiability 
For a scalar function f(x): 
•If the limit above exists for all x in [a,b], then we 
say the function f is differentiable  over [a,b] 
•A function which is differentiable over [a, b] is 
continuous over [a, b] too ( sufficient condition )
•f’(x0) is the slope of, aka tangent to, f(x) at x0Lagrange Leibniz Newton Euler The Cooper Union for the Advancement of Science and Art 
We Like Differentiable Functions 
•Polynomials  have derivatives of all orders (but most 
of them are zero) 
•Sine, cosine, exponential functions have nonzero 
derivatives of all orders 
•Rational  (polynomial divided by polynomial) and 
logarithmic functions have all continuous derivatives 
too, but careful about their domain 
•The sum & difference & product of differentiable & 
continuous functions is differentiable & continuous, 
but not always the quotient (division). Why? 
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Is f(Z) continuous? Is it differentiable? 
Over what domain of Z is f(Z) physically  
meaningful? Activity: Taking Derivatives 
Find the derivative of the SRK equation of state 
with respect to the compressibility factor Z: 
The Cooper Union for the Advancement of Science and Art 
Answer: Taking Derivatives 
Compare to the standard form of the 
Soave-Redlich-Kwong equation of state. All polynomials are differentiable. f(Z) is 
physically meaningful for positive real values of 
Z, because Z is compressibility factor (PV/nRT). 
Is this differentiable in terms of molar volume Vm?
'The Cooper Union for the Advancement of Science and Art 
Numerical Definitions of Integrals 
The mean of f(x) over [a, b] Area under the 
curve of f(x) 
between a and b Discretize x 
between a and b 
into n equal slices 
The Cooper Union for the Advancement of Science and Art 
Extreme Value Theorem 
Extreme Value Theorem: if f(x) 
is continuous on [a, b], there 
exist points c, d in [a, b] such 
that f(c) is the maximum of f(x) 
and f(d) is the minimum. 
EVT
We build models (like neural networks) from 
continuous functions partly to take advantage 
of the Extreme Value Theorem, which shows 
such functions can be optimized. The Cooper Union for the Advancement of Science and Art 
Mean Value Theorem 
Mean Value Theorem: if f(x) is 
differentiable on [a, b], point c 
exists within [a, b] such that f'(c) 
is exactly the slope of the line 
between a, f(a) and b, f(b). 
MVT 
Used to define the derivative and 
integral (Fundamental Theorem of 
Calculus) and to represent any 
differentiable function in terms of 
derivatives (Taylor's Theorem) The Cooper Union for the Advancement of Science and Art 
Taylor’s Theorem 
•Rn(x) is the “remainder”, ξ(x) is a value on [x0, x]•Estimate f(x) near x0 using only info at point x0
Linear approximation 
Quadratic Everything 
The Cooper Union for the Advancement of Science and Art 
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0, np .pi/2, 0.01) 
sin_x = np .sin(x)
plt.plot(x, sin_x)
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
What is the first nonzero 
term of the Taylor Series 
for sin(x)? Example: approx sin(x)? The Cooper Union for the Advancement of Science and Art 
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0, np .pi/2, 0.01) 
sin_x = np .sin(x)
plt.plot(x, sin_x)
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
Open Colab and plot the 
first-order (linear) term of 
the Taylor Series for sin(x) Example: approx sin(x) up to 𝝅/2The Cooper Union for the Advancement of Science and Art 
Example: approx sin(x) as x 
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0,  np.pi/2, 0.01)
sin_x = np .sin(x)
sinish_x = x 
plt.plot(x, sin_x)
plt.plot(x, sinish_x) 
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
The Cooper Union for the Advancement of Science and Art 
Example: approx sin(x) better 
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0,  np.pi/2, 0.01)
sin_x = np .sin(x)
sinish_x = x - x**3/(3*2) + x**5/(5*4*3*2) 
plt.plot(x, sin_x)
plt.plot(x, sinish_x) 
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
The Cooper Union for the Advancement of Science and Art 
Example: approx sin(x) up to 𝝅
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0,  np.pi, 0.01)
sin_x = np .sin(x)
sinish_x = x - x**3/(3*2) + x**5/(5*4*3*2) 
plt.plot(x, sin_x)
plt.plot(x, sinish_x) 
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
The Cooper Union for the Advancement of Science and Art 
Example: approx sin(x) up to 2 𝝅
import numpy as np
import matplotlib.pyplot  as plt
x = np.arange(0,  np.pi*2, 0.01)
sin_x = np .sin(x)
sinish_x = x - x**3/(3*2) + x**5/(5*4*3*2) 
plt.plot(x, sin_x)
plt.plot(x, sinish_x) 
plt.xlabel('x')
plt.ylabel('sin(x)' )
plt.show() 
The Cooper Union for the Advancement of Science and Art 
Taylor series error 
●Like any infinite series 
approximation, Taylor 
series suffer from: 
○Truncation error 
○Round-off error 
●Using lots of terms will 
decrease truncation 
error, but increase 
round-off error 
●Best to keep (x - a) small 
For any Taylor series of order n, if 
we know that the (n+1)th derivative 
f(n+1)(x) ≤ M, we know that the 
truncation error is bounded by: The Cooper Union for the Advancement of Science and Art 
Exercise: error bounds 
For any Taylor series of order n, if 
we know that the (n+1)th derivative 
f(n+1)(x) ≤ M, we know that the 
truncation error is bounded by: 
Hardest part is finding a bound M on the 
(n+1)th derivative of our true function f. 
Sometimes impossible, sometimes easy. 
What is M in this case? For our 5th-order Taylor 
approximation of sin(x), 
x - x3/6 + x5/120, find a 
bound xb such that truncation 
error | E5(xb)| ≤ 0.01 The Cooper Union for the Advancement of Science and Art 
Exercise: error bounds 
For any Taylor series of order n, if 
we know that the (n+1)th derivative 
f(n+1)(x) ≤ M, we know that the 
truncation error is bounded by: 
Answer: max of all 
derivatives of sine & cosine 
is 1. Here, a = 0 (this Taylor 
series is centered at zero). 
Plug in x = xb and solve. 
For our 5th-order Taylor 
approximation of sin(x), 
x - x3/6 + x5/120, find a 
bound xb such that truncation 
error | E5(xb)| ≤ 0.01 The Cooper Union for the Advancement of Science and Art 
Exercise: error bounds 
For any Taylor series of order n, if 
we know that the (n+1)th derivative 
f(n+1)(x) ≤ M, we know that the 
truncation error is bounded by: 
Does this still work if 
we use n = 6? Why? 
For our 5th-order Taylor 
approximation of sin(x), 
x - x3/6 + x5/120, find a 
bound xb such that truncation 
error | E5(xb)| ≤ 0.01 The Cooper Union for the Advancement of Science and Art 
Test the answer in Python 
x = 1.39  # 5th-order bound xb 
sinish_x  = x - x**3/(3*2) + x**5/(5*4*3*2)
sin_x = np.sin( x)
print(f'True error = {sinish_x  - sin_x:.4f}')
Open Colab and try this! The Cooper Union for the Advancement of Science and Art 
Test the answer in Python 
x = 1.39  # 5th-order bound xb 
sinish_x  = x - x**3/(3*2) + x**5/(5*4*3*2)
sin_x = np.sin( x)
print(f'True error = {sinish_x  - sin_x:.4f}')
True error = 0.0019 
(Well below our bound of 0.01) The Cooper Union for the Advancement of Science and Art 
Test the answer in Python 
x = 1.75  # 6th-order bound xb 
sinish_x  = x - x**3/(3*2) + x**5/(5*4*3*2)
sin_x = np.sin( x)
print(f'True error = {sinish_x  - sin_x:.4f}')
True error = 0. 0096
(Right below our bound of 0.01) The Cooper Union for the Advancement of Science and Art 
Stability 
•How can you tell if an approximation is stable ?
•Try small  perturbations in input (“small” 
depends on the problem at hand) 
•If the output changes significantly  (as defined 
by the problem at hand), you have instability 
Ariane 5 algorithm was 
conditionally stable The Cooper Union for the Advancement of Science and Art 
Exercise: stability bounds 
sinish_x = x - x**3/(3*2) + x**5/(5*4*3*2) 
We can still find stability bounds for sinish (x) 
even if we don't know any bound M on the 
derivatives of the true function sin( x).
We only need the change in the known  function 
sinish (x) with respect to small changes in x.
The Cooper Union for the Advancement of Science and Art 
Exercise: stability bounds 
sinish_x = x - x**3/(3*2) + x**5/(5*4*3*2) 
Find bounds xb such that, within these bounds, 
a given change in x always produces an equal 
or smaller change in sinish (x):
The Cooper Union for the Advancement of Science and Art 
?Exercise: stability bounds 
The Cooper Union for the Advancement of Science and Art 
Exercise: stability bounds 
The Cooper Union for the Advancement of Science and Art 
Exercise: stability bounds 
Substitute xb for x 
and the bound 1 
for dsinish (x)/dx The Cooper Union for the Advancement of Science and Art 
Safe to divide 
by xb2 since it is 
always positive. 
Within these bounds, 
the approximation is 
stable, but is it 
accurate? (~3.46) Exercise: stability bounds 
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Safe to divide 
by xb2 since it is 
always positive. 
(~3.46) Exercise: stability bounds 
Already bad for 
x > 3.0, but we 
avoid disaster The Cooper Union for the Advancement of Science and Art 
Summary and Problems 
●Open Python Numerical Methods, go to 
18.4: Summary and Problems 
●Do problems 2, 4, & 5. For reference: 
Definition of a 
Taylor series 
All reading for next week: writing fast 
code (PNM 8.1-8.3), root finding (PNM 
19.1-19.5), convergence (F&B 1.4)