SIT Internal
PHE1021
Engineering Mathematics 2
Trimester 2
AY 2023-24
SIT Internal
Instructor
▪ A/Prof Fung Ho-Ki
Email: Hoki.Fung@SingaporeTech.edu.sg
▪ Email me for questions / schedule a
consultation
2
SIT Internal
Tentative Schedule
Week Lecture Topic(s)
1
2
3
Revision on Differentiation (pre-recorded
lecture)
Revision on Integration
Partial Derivatives, Multiple Integrals
6
7
Introduction to ODEs
Analytical Methods of Linear ODEs
Analytical Methods of Linear ODEs
(cont’d)
Linear Optimisation
RECESS WEEK
8
Unconstrained Optimisation
9
10
11
12
Constrained Optimisation
Basic Concepts of PDEs
Analytical Methods of PDEs
Application of ODEs/PDEs in Pharm Eng
4
5
Written
Others
Assignment
No tutorial this week
WA 1
WA 1 due on Jan 28
WA 2
WA 2 due on Feb 11
Online Quiz (available Feb 9-10)
WA 3
WA 3 due on Mar 3
In-class Midterm Quiz on Feb 29
WA 4
WA 4 due on Mar 17
WA 5
WA 5 due on Mar 31
- tutorials are conducted every week except Week 1 and Recess Week (Week 7)
3
SIT Internal
Module Delivery
Face-to-face (except week 1 lecture which is pre-recorded)
Lectures
▪ I will follow the notes that I posted on the LMS
Tutorials
▪ Weeks 2 – 6 & Weeks 8 – 13
▪
I will post the materials at least a few days before the actual tutorial
timeslot. They are usually questions covering the lecture topic in the
previous week, and I will go through the answers during the tutorial.
4
SIT Internal
Textbook
Recommended Main Textbooks
▪ Modern Engineering Mathematics,
6th Edition, Glyn James & Phil Dyke
ISBN-13: 978-1292253497
▪ Advanced Engineering Mathematics,
International Student Version, 10th Edition,
Erwin Kreyszig
ISBN-13: 9780470646137
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SIT Internal
Assessment
Assessment
Weighting
5 Written Assignments
20%
Online Quiz (Week 5)
(1-hr, all MCQs, at home)
15%
Midterm Exam (Week 8)
(1-hr, written, in-class)
Final Written Exam
(2-hr, written @SIT Dover)
30%
35%
Total = 100%
6
SIT Internal
Revision on
Differentiation
James & Dyke Ch.8
SIT Internal
Definition of a derivative
▪ Formally we define the derivative of a function 𝑓(𝑥) at
the point 𝑥𝑜 to be:
𝑓 𝑥𝑜 + Δ𝑥 − 𝑓(𝑥𝑜 )
Δ𝑓(𝑥𝑜 )
lim
= lim
Δ𝑥→0
Δ𝑥→0 Δ𝑥
Δ𝑥
where Δ𝑓(𝑥𝑜 ) = 𝑓 𝑥𝑜 + Δ𝑥 − 𝑓(𝑥𝑜 ) is the change in
𝑓(𝑥) corresponding to a change Δ𝑥 at the point 𝑥 = 𝑥𝑜 .
▪ Two notations are used for the derivative, either
𝑑𝑓
𝑓 𝑥+Δ𝑥 −𝑓(𝑥)
or 𝑓′(𝑥), which is lim
. As such,
𝑑𝑥
Δ𝑥
Δ𝑥→0
𝑑𝑓
𝑓 𝑥𝑜 +Δ𝑥 −𝑓(𝑥𝑜 )
Δ𝑓(𝑥𝑜 )
= 𝑓 ′ 𝑥𝑜 = lim
= lim
𝑑𝑥 𝑥=𝑥𝑜
Δ𝑥
Δ𝑥→0
Δ𝑥→0 Δ𝑥
.
8
SIT Internal
Graphical representation of a derivative
𝑓 𝑥𝑜 +Δ𝑥 −𝑓(𝑥𝑜 )
Δ𝑥
Δ𝑥→0
lim
Δ𝑓(𝑥𝑜 )
Δ𝑥→0 Δ𝑥
= lim
▪ 𝑓′(𝑥𝑜 ) is the slope of the tangent line (i.e., line AC
above) to the function 𝑓(𝑥) at the point 𝑥 = 𝑥𝑜
= 𝑓 ′ 𝑥𝑜
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SIT Internal
Differentiable Functions
▪ The formal definition of the derivative of 𝑓 𝑥 implies
that the limits from below and above are equal. In some
cases this does not happen. For example, the function
𝑓 𝑥 = 1 + sin 𝑥 is such that its two limits are unequal.
3𝜋
3𝜋
𝑓 2 +Δ𝑥 −𝑓( 2 )
lim
Δ𝑥
Δ𝑥→0−
3𝜋
=
3𝜋
𝑓 2 +Δ𝑥 −𝑓( 2 )
lim
Δ𝑥
Δ𝑥→0+
=
−1
(‘left-hand’
2
1
2
derivative)
(‘right-hand’ derivative)
(knowledge on taking limits is not required for this module)
Clearly the derivative of the function is not defined
at 𝑥 = 3𝜋/2.
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SIT Internal
Differentiable Functions
▪ At 𝑥 = 3𝜋/2, a unique tangent cannot be drawn to the
graph of the function 𝑓 𝑥 = 1 + sin 𝑥.
Figure 8.6
The graph of y = (1 + sin x)
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SIT Internal
Differentiable Functions
▪ For a function 𝑓(𝑥) to be differentiable at a point 𝑥 = 𝑎, the
graph of 𝑓(𝑥) must have a unique, non-vertical well-defined
tangent at 𝑥 = 𝑎. Otherwise the limit
𝑓 𝑎+Δ𝑥 −𝑓(𝑎)
Δ𝑥
Δ𝑥→0
lim
▪
does not exist.
We say that a function is differentiable if it is
differentiable at all points in its domain. For practical
purposes it is sufficient to interpret a differentiable
function as one having a smooth continuous graph with
no sharp corners.
Figure 8.7
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SIT Internal
Basic Rules of Differentiation
13
SIT Internal
Differentiation Rules
Inverse-function rule
▪ If 𝑦 = 𝑓 −1 (𝑥), then 𝑥 = 𝑓(𝑦) and
𝑑𝑦
1
1
=
=
𝑑𝑥 𝑑𝑥/𝑑𝑦 𝑓′(𝑦)
Parametric differentiation rule
▪ If 𝑦 = 𝑓(𝑥) where 𝑥 = 𝑔(𝑡) and 𝑦 = ℎ 𝑡 and 𝑡 is a
parameter, then
𝑑𝑦
𝑑𝑦 𝑑𝑡
= ൙𝑑𝑥
𝑑𝑥
𝑑𝑡
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SIT Internal
Examples
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
a) 8𝑥 4 − 4𝑥 2
b) 2𝑥 2 + 5 𝑥 2 + 3𝑥 + 1
c) 4𝑥 7 𝑥 2 − 3𝑥
d) 𝑥 + 1 𝑥
e)
f)
𝑥
𝑥+1
𝑥 3 +2𝑥+1
𝑥 2 +1
15
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
a) 8𝑥 4 − 4𝑥 2
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
b) 2𝑥 2 + 5 𝑥 2 + 3𝑥 + 1
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
c) 4𝑥 7 𝑥 2 − 3𝑥
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
d) 𝑥 + 1 𝑥
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
e)
𝑥
𝑥+1
SIT Internal
Examples
Solution
Find 𝑓′(𝑥) where 𝑓(𝑥) is:
f)
𝑥 3 +2𝑥+1
𝑥 2 +1
SIT Internal
Differentiation of Composite Functions
SIT Internal
Differentiation of Composite Functions
SIT Internal
Differentiation of Composite Functions
SIT Internal
Differentiation of Composite Functions
SIT Internal
Differentiation of Composite Functions
SIT Internal
Differentiation of Trigonometric Functions
Example: y =
SIT Internal
-1
sin
x
▪ Taking 𝑦 = 𝑠𝑖𝑛−1 𝑥 we have 𝑥 = 𝑠𝑖𝑛𝑦, so that
dx
= cos y
dy
▪ Then from the inverse-function rule:
dy
1
=
dx cos y
▪ Using the identity 𝑐𝑜𝑠 2 y = 1 − 𝑠𝑖𝑛2 y, this simplifies to
d
1
−1
(sin x) =
, | x | 1
dx
1 − x2
SIT Internal
Differentiation of Trigonometric Functions
𝑑𝑦
Find where 𝑦 is given by:
𝑑𝑥
a) 𝑥 2 cos 𝑥
b) 𝑥 tan 2𝑥
c) sin−1 6𝑥
d)
2𝑥
−1
tan
1+𝑥 2
SIT Internal
Differentiation of Trigonometric Functions
Solution
𝑑𝑦
Find where
𝑑𝑥
a) 𝑥 2 cos 𝑥
𝑦 is given by:
SIT Internal
Differentiation of Trigonometric Functions
Solutions
𝑑𝑦
𝑑𝑥
Find where 𝑦 is given by:
b) 𝑥 tan 2𝑥
SIT Internal
Differentiation of Trigonometric Functions
Solutions
𝑑𝑦
Find where
𝑑𝑥
c) sin−1 6𝑥
𝑦 is given by:
SIT Internal
Differentiation of Trigonometric Functions
Solutions
Find
𝑑𝑦
𝑑𝑥
where 𝑦 is given by:
d) tan−1
2𝑥
1+𝑥 2
Differentiation of Exponential and
Logarithmic Functions
SIT Internal
Differentiation of Exponential and
Logarithmic Functions
SIT Internal
SIT Internal
Differentiation of Hyperbolic Functions
Hyperbolic functions
▪ The hyperbolic functions are closely related to the
exponential function:
𝑒 𝑥 − 𝑒 −𝑥
𝑒 𝑥 + 𝑒 −𝑥
sinh 𝑥 =
, cosh 𝑥 =
2
2
and as such, their derivatives can be readily deduced.
d
d  e x − e− x  1 x −x
(sinh x) =

 = (e + e ) = cosh x
dx
dx  2  2
d
d  e x + e−x  1 x −x
(cosh x) =

 = (e − e ) = sinh x
dx
dx  2  2
d
d  sinh x  (cosh x)(cosh x) − (sinh x)(sinh x)
(tanh x) =
=


dx
dx  cosh x 
cosh 2 x
1
2
=
=
sech
x
2
cosh x
SIT Internal
Differentiation of Hyperbolic Functions
SIT Internal
Implicit Differentiation
▪ The chain rule may also be used to differentiate functions
expressed in an implicit form. For example, for the equation
𝑦 3 = 𝑥 2 , to obtain the derivative 𝑑𝑦/𝑑𝑥, we use the method
known as implicit differentiation, by which we treat y as an
unknown function of x and differentiate both sides term by
term with respect to x. This gives
𝑑 3
𝑑 2
𝑦 =
𝑥
𝑑𝑥
𝑑𝑥
𝑑 3 𝑑𝑦
𝑦 ∙
= 2𝑥
𝑑𝑦
𝑑𝑥
𝑑𝑦
2
3𝑦
= 2𝑥
𝑑𝑥
𝑑𝑦
2𝑥
2𝑥
2
= 2 = 4/3 = 1/3
𝑑𝑥 3𝑦
3𝑥
3𝑥
SIT Internal
Implicit Differentiation
Example
Find
𝑑𝑦
𝑑𝑥
when 𝑥 2 + 𝑦 2 + 𝑥𝑦 = 1.
SIT Internal
Applications of Differentiation
▪ Differentiation is broadly applied to solve optimization problems in real
life. The basic idea is that the optimal value of a differentiable function
𝑓(𝑥) (i.e., its maximum or minimum value) generally occurs when its
derivative is zero; that is
𝑓′ 𝑥 = 0
▪ However, as the figure below suggests, the extremal values found
using 𝑓 ′ 𝑥 = 0 are generally only local maximum or minimum values,
which are not the best values (best values are absolute maximum or
absolute minimum). In seeking extremal values of a function, it is also
necessary to check the end points of the domain of the function.
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Applications of Differentiation
▪
Figure below gives another reason for caution; at some points of inflection
(points where the graph crosses it own tangent), the tangent may be horizontal
(i.e., 𝑓 ′ 𝑥 is also zero).
There are in general 3 kinds of critical or stationary points where 𝑓 ′ 𝑥 is zero:
1) If 𝑓 ′ 𝑥 changes from positive to negative (or 𝑓" 𝑥 < 0) as we pass from
left to right through a stationary point, the latter is a local maximum;
2) If 𝑓 ′ 𝑥 changes from negative to positive (or 𝑓" 𝑥 > 0) as we pass from
left to right through a stationary point, the latter is a local minimum;
3) If 𝑓 ′ 𝑥 does not change sign (or 𝑓" 𝑥 = 0) as we pass through a
stationary point, the latter is a point of inflection.
▪ If we are interested in getting absolute (or global) maximum or minimum points,
then we also need to check the end points of the domain of the function.
▪
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Example
Maximizing the volume of a box cut out from a piece of tin sheet
42
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Example
Maximizing the volume of a box cut out from a piece of tin sheet
▪ Solution:
12 − 2𝑥
𝑥
12 − 2𝑥
43
SIT Internal
Example
Maximizing the volume of a box cut out from a piece of tin sheet
▪ Solution:
12 − 2𝑥
𝑥
12 − 2𝑥
SIT Internal
Example
Maximizing the volume of a box cut out from a piece of tin sheet
▪ Solution:
12 − 2𝑥
𝑥
(2,128)
12 − 2𝑥
𝑉(𝑥)
(0,0)
(6,0)
SIT Internal
Example
Coffee is draining from a conical filter into a cylindrical coffeepot at the rate
of 10 in3/min.
a) How fast is the level in the pot rising when the coffee in the cone is 5
in deep?
b) How fast is the level in the cone falling then?
SIT Internal
Example
Solution:
Coffee is draining from a conical filter into a cylindrical coffeepot at the rate
of 10 in3/min.
a) How fast is the level in the pot rising when the coffee in the cone is 5
in deep?
ℎ(𝑡)
6"
SIT Internal
Example
Solution:
Coffee is draining from a conical filter into a cylindrical coffeepot at the rate
of 10 in3/min.
b) How fast is the level in the cone falling then?
SIT Internal
Example
Solution:
Coffee is draining from a conical filter into a cylindrical coffeepot at the rate
of 10 in3/min.
b) How fast is the level in the cone falling then?
6"
𝑟(𝑡)
6"
ℎ(𝑡)
SIT Internal
End of Lecture
for Week 1