The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about Rational functions and its zeroes. It is also comprised of some examples and exercises to be done for the said topic.
MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.
Work immersionย refers to the subject of the senior high school curriculum, which involves hands-on experience or work simulation in which learners can apply their competencies and acquired knowledge relevant to their track.
*Disclaimer: the pictures/information/media used in this ppt do not belong to me. Credits to the rightful owners.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Work immersionย refers to the subject of the senior high school curriculum, which involves hands-on experience or work simulation in which learners can apply their competencies and acquired knowledge relevant to their track.
*Disclaimer: the pictures/information/media used in this ppt do not belong to me. Credits to the rightful owners.
It is a powerpoint presentation that will help the students to enrich their knowledge about Senior High School subject of General Mathematics. It is comprised about functions and its operations such as addition, subtraction, multiplication, division and composition. It is also comprised of some examples and exercises to be done for the said topic.
Generalized Laplace - Mellin Integral TransformationIJERA Editor
ย
The main propose of this paper is to generalized Laplace-Mellin Integral Transformation in between the positive regions of real axis. We have derived some new properties and theorems .And give selected tables for Laplace-Mellin Integral Transformation.
Differential Geometry for Machine LearningSEMINARGROOT
ย
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincarรฉ Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
"Stochastic Optimal Control and Reinforcement Learning", invited to speak at the Nonlinear Dynamic Systems class taught by Prof. Frank Chong-woo Park, Seoul National University, December 4, 2019.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
ย
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Palestine last event orientationfvgnh .pptxRaedMohamed3
ย
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as โdistorted thinkingโ.
2. Exponents (Powers)
โข Given ๐ a positive integer and ๐ a real number,
๐ ๐
indicates that ๐ is multiplied by itself ๐ times:
๐ ๐
= ๐ ร ๐ ร โฏ ร ๐
๐ ๐๐๐๐๐
โข According to definition:
๐ ๐
= ๐ and ๐ ๐
= ๐
3. Exponents Rules
๏ง If ๐ and ๐ are positive integers and ๐ is a real
number, then:
๐ ๐
ร ๐ ๐
= ๐ ๐+๐
With this rule we can define the concept of negative
exponent (power):
๐0
= 1
๐ ๐โ๐
= 1
๐ ๐+(โ๐)
= 1
๐ ๐
ร ๐โ๐
= 1
๐โ๐
=
๐
๐ ๐
5. Algebraic Expressions, Equations and Identities
โข An algebraic expression is a combination of real numbers
and variables, such as:
๏ฑMonomials :
5๐ฅ3
, โ1.75
๐ฆ ,
3๐ฅ
4๐ง2
=
3
4
๐ฅ๐งโ2
๏ฑBinomials:
4๐ฅ3
+ 3๐ฅ2
,
3๐ฅ + 1
4๐ง2
=
3
4
๐ฅ๐งโ2
+
1
4
๐งโ2
๏ฑPolynomials:
๐ฅ2 โ 3๐ฅ โ 6 , ๐ฅ3 + ๐ฅ๐ฆ2 + 6๐ฅ๐ฆ๐ง
6. Algebraic Expressions, Equations and Identities
โข Equations can be made when two expressions are equal to
one another or an expression is equal to a number:
3๐ฅ โ 1 = ๐ฅ
4๐ฅ + 3๐ฆ = 2
5๐ฅ2 โ 2๐ฅ๐ฆ = ๐ฅ โ 6๐ฆ2
๐ฅ2 โ 3๐ฅ โ 6 = 0
The first and second equations are linear with one and two
variables respectively and the third equation is a quadratic in
terms of ๐ and ๐ and the forth equation is a quadratic
equation in terms of ๐ .
Note: Not all equations are solvable and many of them have
no unique solutions.
7. Algebraic Expressions, Equations and Identities
โข If two expressions are equal for all values of their
variable(s), the equation is called an identity.
โข For example;
๐ฅ + 3 2 = ๐ฅ2 + 6๐ฅ + 9
๏ง Some important identities are:
โข ๐ ยฑ ๐ ๐ = ๐ ๐ ยฑ ๐๐๐ + ๐ ๐
โข ๐ ยฑ ๐ ๐
= ๐ ๐
ยฑ ๐๐ ๐
๐ + ๐๐๐ ๐
ยฑ ๐ ๐
โข ๐ โ ๐ ๐ + ๐ = ๐ ๐ โ ๐ ๐
โข ๐ ยฑ ๐ ๐ ๐
โ ๐๐ + ๐ ๐
= ๐ ๐
ยฑ ๐ ๐
โข ๐ ยฑ ๐ ๐ ยฑ ๐ = ๐ ๐ ยฑ ๐ + ๐ ๐ + ๐๐
10. Functions
โข All equations represent a relationship between two or
more variables, e.g.:
๐ฅ๐ฆ = 1 ,
๐ฅ
2๐ฆ
+ ๐ง = 0
โข Given two variables in relation, there is a functional
relationship between them if for each value of one of
them there is one and only one value of another.
โข If the relationship between ๐ and ๐ can be shown by ๐ =
๐ ๐ and for each value of ๐ there is one and only one
value of ๐ , then there is a functional relationship
between them or alternatively it can be said that ๐ is a
function of ๐ , which means ๐ as a dependent variable
follows ๐ as an independent variable.
11. Functions
โข The idea of function is close to a processing (matching)
machine. It receives inputs (which are the values of ๐ and is
called domain of the function, ๐ซ ๐) and after the processing
them the output will be values of ๐ in correspondence with
๐โฒ
๐ (which is called range of the function, ๐น ๐).
โข There should be no element from ๐ซ ๐ without a match from
๐น ๐, but it might be found some free elements in ๐น ๐.
๐ = ๐ ๐, ๐ ๐ , ๐ ๐, ๐ ๐ , โฆ , ๐ ๐, ๐ ๐
๐
๐ ๐, ๐ ๐, โฆ , ๐ ๐ ๐ ๐, ๐ ๐, โฆ , ๐ ๐
12. Functions
โข Functions can be considered as correspondence
(matching) rules, which corresponds all elements of
๐ to some elements of ๐.*
โข For example, the correspondence rule (f), which
corresponds ๐ to each value of ๐, can be written
as:
Or ๐ฆ = ๐ฅ
xxf ๏ก: 1
2
4
15
1
๐
2
๐๐
20
x y
13. Functions
โข The correspondence rule, which corresponds
๐ ๐
โ ๐๐ to each value of ๐ can be shown as:
Or ๐ = ๐ ๐
โ ๐๐
10: 2
๏ญxxg ๏ก
-3
-2
0
2
3
-1
-6
-10
x y
14. Functions
โข Some correspondence rules indicate there is a relationship
between ๐ and ๐ but not a functional relationship, i.e.
the relationship cannot be considered as a function.
โข For example, ๐ = ยฑ ๐ (๐ ๐ = ๐) is
not a function (according to the
definition of function) because
for each value of ๐ there are
two symmetrical values of ๐ .
Adopted from http://www.education.com/study-help/article/trigonometry-help-inverses-circular/
15. Functions
โข Note that in the graphical representation of a
function, any parallel line with y-axis cross the graph
of a function at one and only one point. Why?
Adopted from http://mrhonner.com/archives/8599
16. Some Basic Functions
โข Power Function : ๐ = ๐ ๐
Adoptedfromhttp://mysite.verizon.net/bnapholtz/Math/powers.html
If n>0 they
all pass
through the
origin. If
n<0 the
function is
not defined
at x=0
๐ฆ = ๐ฅโ1
๐ฆ = ๐ฅโ1
17. Some Basic Functions
โข Exponential Function : ๐ = ๐ ๐
(๐ > ๐, โ ๐)
Adopted from http://www.softmath.com/tutorials-3/relations/exponential-functions-2.html
All exponential
functions passing
through the point
(0,1)
18. Some Basic Functions
โข Logarithmic Function : ๐ = ๐ฅ๐จ๐ ๐ ๐ (๐ > ๐, โ ๐)
Adopted fromhttp://mtc.tamu.edu/9-12/index_9-12.htm?9-12M2L2.htm
Adopted from
http://www.cliffsnotes.com/math/calculus/precalculus/exponential-and-
logarithmic-functions/logarithmic-functions
All logarithmic
Functions passing
through the point (1,0)
20. โข All trigonometric functions are periodic, i.e. after adding or
subtracting a constant, which is called principal periodic constant,
they repeat themselves. This periodic constant is ๐๐ for ๐๐๐๐
and ๐๐๐๐ but it is ๐ for ๐๐๐๐ and ๐๐๐๐ , i.e. :
(k is a positive integer)
๐ ๐๐๐ฅ = sin ๐ฅ ยฑ ๐๐ = sin ๐ฅ ยฑ 4๐ = โฏ = sin ๐ฅ ยฑ 2๐๐
๐๐๐ ๐ฅ = cos ๐ฅ ยฑ ๐๐ = cos ๐ฅ ยฑ 4๐ = โฏ = cos ๐ฅ ยฑ 2๐๐
๐ก๐๐๐ฅ = tan ๐ฅ ยฑ ๐ = tan ๐ฅ ยฑ 2๐ = โฏ = tan(๐ฅ ยฑ ๐๐)
๐๐๐ก๐ฅ = cot ๐ฅ ยฑ ๐ = cot ๐ฅ ยฑ 2๐ = โฏ = cot(๐ฅ ยฑ ๐๐)
Some Basic Functions
21. Elementary Functions
โข Elementary functions can be made by combining
basic functions through adding, subtracting,
multiplying, dividing and also composing these
basic functions.
โข For example:
๐ฆ = ๐ฅ2
+ 4๐ฅ โ 1
๐ฆ = ๐ฅ. ๐โ๐ฅ
=
๐ฅ
๐ ๐ฅ
๐ฆ = ๐ ๐ ๐๐๐ฅ
๐ฆ = ln ๐ฅ2 + 4
๐ฆ = ๐ ๐ฅ
(๐ ๐๐3๐ฅ โ ๐๐๐ 3๐ฅ)
22. Behaviour of a Function
โข After finding the relationship between two variables ๐
and ๐ in the functional form ๐ = ๐(๐) the first question is
how this function behaves.
โข Here we are interested in knowing about the magnitude
and the direction of the change of ๐ (๐. ๐. โ๐) when the
change of ๐ (๐. ๐. โ๐) is getting smaller and smaller around
a point in its domain. The technical term for this locality
around a point is neighbourhood. So, we are trying to find
the magnitude and the direction of the change of ๐ in the
neighbourhood of ๐.
โข Slope of a function is the concept which helps us to have
this information. The value of the slope shows the
magnitude of the change and the sign of slope shows the
direction of the change.
23. Slope of a Linear Function
โข Letโs start with one of the most used functions in
science , which is the linear function:
๐ = ๐๐ + ๐
Where ๐ shows the slope of the line (the average change
of ๐ in terms of a change in ๐). That is; ๐ =
๐ซ๐
๐ซ๐
= ๐ญ๐๐ง ๐ถ .
The value of intercept is ๐ which is the distance between the
intersection point of the graph and y-axis from the Origin.
The slope of a liner
function is constant in its
whole domain.
y
x
h
๐ = ๐๐ + ๐
โ๐
โ๐
๐ถ
๐ถ
24. Slope of a Function in its General Form
โข Imagine we want to find the slope of the function ๐ = ๐(๐)
at a specific point (for e.g. at ๐ ๐) in its domain.
โข Given a change of
๐ from ๐ ๐ to ๐ ๐ + โ๐
the change of ๐
Would be from ๐ ๐ ๐
to ๐(๐ ๐ + โ๐) .
โข This means a
movement along the
curve from A to B. Adopted from http://www.bymath.com/studyguide/ana/sec/ana3.htm
25. Slope of a Function in its General Form
โข The average change of ๐ in terms of a change in ๐
can be calculated by
๐ซ๐
๐ซ๐
= ๐ญ๐๐ง ๐ถ , which is the
slope of the line AB.
โข If the change in ๐ gradually disappear (โ๐ โ ๐)*,
point B moves toward point A and the slope line
(secant line) AB reaches to a limiting (marginal)
situation AC, which is a tangent line on the curve
of ๐ = ๐(๐) at point ๐จ(๐ ๐, ๐(๐ ๐)).
26. Slope of a Function in its General Form
โข The slope of this tangent line AC is what is called derivative
of ๐ in terms of ๐ at point ๐ฅ0 and it is shown by different
symbols such as
๐๐ฆ
๐๐ฅ ๐ฅ=๐ฅ0
, ๐โฒ
๐ฅ0 ,
๐๐
๐๐ฅ ๐ฅ=๐ฅ0
, ๐ฆโฒ
(๐ฅ0) , .
โข The slope of the tangent line at any point of the domain of
the function is denoted by:
๐๐ฆ
๐๐ฅ
, ๐โฒ ๐ฅ ,
๐๐
๐๐ฅ
, ๐ฆโฒ, ๐๐ฅ
โฒ
โข Definition: The process of finding a derivative of a function
is called differentiation .
'
0xf
27. Slope of a Function in its General Form
โข Therefore, the derivative of ๐ = ๐(๐)at any point in its
domain is:
๐โฒ =
๐ ๐
๐ ๐
= ๐๐๐
โ๐โ๐
โ๐
โ๐
= ๐๐๐
โ๐โ๐
๐ ๐+โ๐ โ๐(๐)
โ๐
And the derivative of ๐ = ๐(๐) at the specific point ๐ = ๐ ๐
is:
๐โฒ ๐ ๐ = ๐ฅ๐ข๐ฆ
โ๐โ๐
๐ ๐ ๐ + โ๐ โ ๐(๐ ๐)
โ๐
Where ๐ฅ๐ข๐ฆ stands for โlimitโ, showing limiting (marginal)
situation of the ratio
๐ซ๐
๐ซ๐
.
28. Slope of a Function in its General Form
โข Note: For non-linear functions, slope of the function at
any point depends on the value of that point and it is not
constant in the whole domain of the function. This
means that the derivative of a function is a function of
the same variable itself.
Adopted from http://www.columbia.edu/itc/sipa/math/slope_nonlinear.html
http://www.pleacher.com/mp/mlessons/calc2006/day21.html
29. Derivative of Fundamental Basic Functions
โข Find the derivative of ๐ฆ = 2๐ฅ โ 1 at any point in
its domain.
๐ ๐ฅ = 2๐ฅ โ 1
๐ ๐ฅ + โ๐ฅ = 2 ๐ฅ + โ๐ฅ โ 1 = 2๐ฅ + 2โ๐ฅ โ 1
โ๐ = ๐ ๐ + โ๐ โ ๐ ๐ = ๐โ๐
According to definition:
๐ฆโฒ =
๐๐ฆ
๐๐ฅ
= lim
โ๐ฅโ0
๐ ๐ฅ + โ๐ฅ โ ๐(๐ฅ)
โ๐ฅ
= lim
โ๐ฅโ0
2โ๐ฅ
โ๐ฅ
= 2
30. Derivative of the Fundamental Basic Functions
โข Applying the same method, the derivative of the
fundamental basic functions can be obtained as
following:
๏ฑ ๐ = ๐ ๐
โ ๐โฒ
= ๐๐ ๐โ๐
e.g. :
๐ฆ = 3 โ ๐ฆโฒ = 0
๐ฆ = ๐ฅ3 โ ๐ฆโฒ = 3๐ฅ2
๐ฆ = ๐ฅโ1 โ ๐ฆโฒ = โ๐ฅโ2
๐ฆ = 5
๐ฅ โ ๐ฆโฒ
=
1
5
๐ฅ
1
5
โ1
=
1
5
5
๐ฅ4
33. Differentiability of a Function
๏A function is differentiable at a point if despite any
side approach to the point in its domain (from left or
right) the derivative is the same and a finite number.
Sharp corner points and points of discontinuity* are
not differentiable.
Adopted from Ahttp://www-math.mit.edu/~djk/calculus_beginners/chapter09/section02.html
34. Rules of Differentiation
โข If ๐(๐) and ๐ ๐ are two differentiable functions in their
common domain, then:
๏ถ ๐(๐) ยฑ ๐(๐) โฒ = ๐โฒ(๐) ยฑ ๐โฒ(๐)
๏ถ ๐ ๐ . ๐(๐) โฒ = ๐โฒ ๐ . ๐ ๐ + ๐โฒ ๐ . ๐(๐)
๏ถ
๐(๐)
๐(๐)
โฒ
=
๐โฒ ๐ .๐ ๐ โ๐โฒ ๐ .๐(๐)
๐(๐) ๐ (Quotient Rule)
๏ถ ๐(๐ ๐ ) โฒ = ๐โฒ ๐ . ๐โฒ(๐ ๐ ) (Chain Rule)
(Summation & Sub. Rules. They
can be extended to n functions)
(Multiplication Rule
and can be extended
to n functions)
36. o ๐ฆ = ๐๐2 ๐ฅ โถ ๐โฒ =
๐
๐
. ๐. ๐๐๐ =
๐๐๐๐
๐
o ๐ฆ = 5 ๐ฅ2
+ tan 3๐ฅ โถ ๐โฒ = ๐ ๐ ๐
. ๐ฅ๐ง๐. (๐๐) + ๐(๐ + ๐๐๐ ๐ ๐๐)
โข The last rule(page 32) is called the chain rule which should
be applied for composite functions such as the above
functions, but it can be extended to include more
functions.
โข If ๐ = ๐ ๐ and ๐ = ๐ ๐ and ๐ = ๐ ๐ and ๐ = ๐(๐)
then ๐ depends on ๐ but through some other variables
๐ = ๐ ๐ ๐ ๐ ๐
Rules of Differentiation
37. โข Under such circumstances we can extend the chain rule
to cover all these functions, i.e.
๐ ๐
๐ ๐
=
๐ ๐
๐ ๐
.
๐ ๐
๐ ๐
.
๐ ๐
๐ ๐
.
๐ ๐
๐ ๐
o ๐ฆ = ๐๐๐ 3
2๐ฅ + 1 โถ
๐ฆ = ๐ข3
๐ข = ๐๐๐ ๐ง
๐ง = 2๐ฅ + 1
๐โฒ =
๐ ๐
๐ ๐
=
๐ ๐
๐ ๐
.
๐ ๐
๐ ๐
.
๐ ๐
๐ ๐
= ๐๐ ๐. โ๐๐๐๐ . ๐
= โ๐๐๐๐ ๐ ๐๐ + ๐ . ๐ฌ๐ข๐ง(๐๐ + ๐)
Rules of Differentiation
38. Implicit Differentiation
โข ๐ = ๐ ๐ is an explicit function because the dependent
variable ๐ is at one side and explicitly expressed by
independent variable ๐. Implicit form of this function can
be shown by ๐ญ ๐, ๐ = ๐ where both variables are in one
side:
o Explicit Functions: ๐ฆ = ๐ฅ2 โ 3๐ฅ , ๐ฆ = ๐ ๐ฅ. ๐๐๐ฅ , ๐ฆ =
๐ ๐๐๐ฅ
๐ฅ
o Implicit Functions: 2๐ฅ โ 7๐ฆ + 3 = 0 , 2 ๐ฅ๐ฆ โ ๐ฆ2 = 0
โข Many implicit functions can be easily transformed to an
explicit function but it cannot be done for all. In this case,
differentiation with respect to ๐ can be done part by part
and ๐ should be treated as a function of ๐.
39. o Find the derivative of ๐๐ โ ๐๐ + ๐ = ๐.
Differentiating both sides with respect to ๐, we have:
๐
๐๐ฅ
2๐ฅ โ 7๐ฆ + 3 =
๐
๐๐ฅ
0
2 โ 7๐ฆโฒ + 0 = 0 โ ๐โฒ =
๐
๐
o Find the derivative of ๐ ๐ โ ๐๐๐ + ๐ ๐ = ๐.
Using the same method, we have:
2๐ฅ โ 2๐ฆ โ 2๐ฅ๐ฆโฒ + 3๐ฆ2 ๐ฆโฒ = 0 โ ๐โฒ =
๐๐ โ ๐๐
๐๐ ๐ โ ๐๐
Implicit Differentiation
41. Higher Orders Derivatives
โข As ๐โฒ = ๐โฒ(๐) is itself a function of ๐ , in case it is differentiable,
we can think of second, third or even n-th derivatives:
โข Second Derivative:
๐โฒโฒ
,
๐ ๐ ๐
๐ ๐ ๐
,
๐ (
๐ ๐
๐ ๐
)
๐ ๐
,
๐
๐ ๐
๐โฒ
, ๐โฒโฒ
๐
โข Third Derivative:
๐โฒโฒโฒ ,
๐ ๐
๐
๐ ๐ ๐
,
๐ (
๐ ๐ ๐
๐ ๐ ๐)
๐ ๐
,
๐
๐ ๐
๐โฒโฒ , ๐โฒโฒโฒ ๐
โข N-th Derivative:
๐(๐) ,
๐ ๐ ๐
๐ ๐ ๐
,
๐ (
๐ (๐โ๐)
๐
๐ ๐(๐โ๐))
๐ ๐
,
๐
๐ ๐
๐(๐โ๐) , ๐(๐) ๐
42. o Find the second and third derivatives of ๐ = ๐โ๐.
๐ฆโฒ = โ๐โ๐ฅ
๐ฆโฒโฒ = ๐โ๐ฅ
๐ฆโฒโฒโฒ = โ๐โ๐ฅ
o If ๐ = ๐ ๐ฝ๐
show that the equation ๐โฒโฒโฒ
โ ๐โฒโฒ
= ๐ has
two roots.
๐ฆโฒ = ๐๐ ๐๐ฅ
๐ฆโฒโฒ
= ๐2
๐ ๐๐ฅ
๐ฆโฒโฒโฒ
= ๐3
๐ ๐๐ฅ
๐ฆโฒโฒโฒ โ ๐ฆโฒโฒ = ๐3 ๐ ๐๐ฅ โ ๐2 ๐ ๐๐ฅ = 0
๐2
๐ ๐๐ฅ
๐ โ 1 = 0
๐ ๐๐ฅ
โ 0 โ ๐ = 0, ๐ = 1
Higher Orders Derivatives
43. First & Second Order Differentials
โข If ๐ = ๐(๐) is differentiable on an interval then at any point of that
interval the derivative of ๐ can be defined as:
๐โฒ
= ๐โฒ
๐ =
๐ ๐
๐ ๐
= ๐ฅ๐ข๐ฆ
โ๐โ๐
๐ซ๐
๐ซ๐
โข This means when ๐ซ๐ becomes โinfinitesimalโ (getting smaller
infinitely; โ๐ โ ๐), the ratio
๐ซ๐
๐ซ๐
approaches to the derivative of the
function, i.e. the difference between
๐ซ๐
๐ซ๐
and ๐โฒ ๐ is infinitesimal
itself and ignorable:
๐ซ๐
๐ซ๐
โ ๐โฒ
๐ ๐๐ โ๐ โ ๐โฒ
๐ . โ๐
โข ๐โฒ ๐ . โ๐ is called โ differential of ๐ โ and is shown by ๐ ๐, so:
โ๐ โ ๐โฒ ๐ . โ๐ = ๐ ๐
As โ๐ is an independent increment of ๐ we can always assume that
๐ ๐ = โ๐; so we can re-write the above as โ๐ โ ๐โฒ
๐ . ๐ ๐ = ๐ ๐
44. โข The geometric interpretation of ๐ ๐ and โ๐ :
โ๐ represents the change in height of the curve and ๐ ๐ represents the
change in height of the tangent line when โ๐ changes (see the graph)
Adopted fromhttp://www.cliffsnotes.com/math/calculus/calculus/applications-of-the-
derivative/differentials
So: ๐ ๐ = ๐โฒ. ๐ ๐
Some rules:
If ๐ and ๐ are differentiable functions, then:
i. ๐ ๐๐ = ๐. ๐ ๐ (c is constant)
ii. ๐ ๐ ยฑ ๐ = ๐ ๐ ยฑ ๐ ๐ (can be extended
to more than two functions)
iii. ๐ ๐. ๐ = ๐. ๐ ๐ + ๐. ๐ ๐ (extendable)
iv. ๐
๐
๐
=
๐.๐ ๐โ๐.๐ ๐
๐ ๐
First & Second Order Differentials
45. โข Using the third rule of differentials, the second order differential of
๐ can be calculated, i.e. :
๐ ๐
๐ = ๐ ๐ ๐ = ๐ ๐โฒ
. ๐ ๐
= ๐ ๐โฒ. ๐ ๐ + ๐โฒ. ๐ ๐ ๐
= ๐โฒโฒ. ๐ ๐. ๐ ๐ + ๐โฒ. ๐ ๐ ๐
= ๐โฒโฒ. ๐ ๐ ๐ + ๐โฒ. ๐ ๐ ๐
As ๐ is not dependent on another variable and ๐ ๐ is a constant :
๐ ๐ ๐ = ๐ ๐ ๐ = ๐
So, ๐ ๐ ๐ = ๐โฒโฒ. ๐ ๐ ๐ = ๐โฒโฒ. ๐ ๐ ๐ (or in the familiar form ๐โฒโฒ =
๐ ๐ ๐
๐ ๐ ๐ )
Where ๐ ๐ ๐ = ๐ ๐ ๐ is always positive and the sign of ๐ ๐ ๐ depends
on the sign of ๐โฒโฒ.
โข Applying the same method we have ๐ ๐ ๐ = ๐(๐). ๐ ๐ ๐ .
First & Second Order Differentials
46. Derivative and Optimisation of Functions
โข Function ๐ = ๐ ๐ is said to be an increasing function at
๐ = ๐ if at any small neighbourhood (โ๐) of that point:
๐ + โ๐ฅ > ๐ โ ๐ ๐ + โ๐ฅ > ๐ ๐
From the above inequality we can conclude that:
๐ ๐+โ๐ฅ โ๐(๐)
โ๐ฅ
โ ๐โฒ(๐) > 0
So, the function is increasing
at ๐=๐ if ๐โฒ(๐)>๐ , and
decreasing if ๐โฒ(๐)<๐ .
Adopted from http://portal.tpu.ru/SHARED/k/KONVAL/Sites/English_sites/calculus/3_Geometric_f.htm
a a
47. โข More generally, the function ๐ = ๐(๐) is increasing
(decreasing) in an interval if at any point in that interval
๐โฒ ๐ > ๐ ( ๐โฒ ๐ < ๐ ).
Derivative and Optimisation of Functions
Adopted from http://www.webgraphing.com/polynomialdefs.jsp
48. Derivative and Optimisation of Functions
โข If the sign of ๐โฒ
(๐) is changing when passing a point such as ๐ =
๐ (from negative to positive or vice versa) and ๐ = ๐(๐) is
differentiable at that point, It is very logical to think that ๐โฒ
(๐)
at that point should be zero, i.e. : ๐โฒ ๐ = ๐. (in this case the
tangent line is horizontal)
โข This point is called local (relative) maximum or local (relative)
minimum. In some books it is called critical point or extremum point.
http://www-rohan.sdsu.edu/~jmahaffy/courses/s00a/math121/lectures/graph_deriv/diffgraph.html
Not an extremum or critical point
49. โข If ๐โฒ ๐ = ๐ but the sign of ๐โฒ(๐) does not change when passing
the point ๐ = ๐, the point (๐, ๐ ๐ ) is not a extremum or critical
point (point C in the previous slide).
โข For a function which is differentiable in its domain(or part of that),
a sign change of ๐โฒ
when passing a point is a sufficient evidence of
the point being a extremum point. Therefore, at that point ๐โฒ(๐)
will be necessarily zero.
Necessary and Sufficient Conditions
๐โฒ
๐ > ๐
๐โฒ
๐ < ๐
๐โฒ
๐ = 0
Adopted and altered from http://homepage.tinet.ie/~phabfys/maxim.htm/
๐โฒ
(๐) > ๐
๐โฒ
๐ = 0
๐โฒ
๐ < ๐
a
b
50. โข If a function is not differentiable at a point (see the graph, point
x=c) but the sign of ๐โฒ changes, it is sufficient to say the point is a
extremum point despite non-existence of ๐โฒ(๐) .
Necessary and Sufficient Conditions
Adopted from http://www.nabla.hr/Z_IntermediateAlgebraIntroductionToFunctCont_3.htm
๐โฒ
(๐) is not defined as it goes to infinity
These types of critical
points cannot be
obtained through
solving the equation
๐โฒ ๐ = ๐ as they are
not differentiable at
these points.
51. Second Derivative Test
โข Apart from the sign change of ๐โฒ
(๐) there is another test to
distinguish between extremums. This test is suitable for those
functions which are differentiable at least twice at the critical points.
โข Assume that ๐โฒ ๐ = ๐; so, the point (๐, ๐ ๐ ) is suspicious to be a
maximum or minimum. If ๐โฒโฒ ๐ > ๐, the point is a minimum point
and if ๐โฒโฒ ๐ < ๐, the point is a maximum point.
Adopted and altered from http://www.webgraphing.com/polynomialdefs.jsp
Inflection point
Concave Down
Concave up
๐โฒ
๐ฅ = 0
๐โฒ
๐ฅ = 0
๐โฒโฒ
๐ฅ = 0
52. Inflection Point & Concavity of Function
โข If ๐โฒ ๐ = ๐ and at the same time ๐โฒโฒ ๐ = ๐, we need other tests
to find out the nature of the point. It could be a extremum point
[e.g. ๐ = ๐ ๐
, which has minimum at ๐ = ๐]or just an inflection
point (where the tangent line crosses the graph of the function and
separate that to two parts; concave up and concave down)
Adopted and altered from http://www.ltcconline.net/greenl/courses/105/curvesketching/SECTST.HTM Adopted from http://www.sparkle.pro.br/tutorial/geometry
๐โฒโฒ ๐ฅ = 0
๐โฒ ๐ฅ > 0
Concave Down
Concave up
53. Some Examples
o Find extremums of ๐ = ๐ ๐ โ ๐๐ ๐ + ๐, if any.
To find the points which could be our extremums (critical points) we
need to find the roots of this equation: ๐โฒ ๐ = ๐,
So, ๐โฒ ๐ = ๐๐ ๐ โ ๐๐ = ๐ โ ๐๐ ๐ โ ๐ = ๐
โ ๐ = ๐, ๐ = ๐
Two points ๐จ(๐, ๐) and ๐ฉ(๐, โ๐) are possible extremums.
Sufficient condition(1st method): As the sign of ๐โฒ = ๐โฒ(๐) changes
while passing through the points there is a maximum and a minimum.
๐ โโ +โ
๐ฆโฒ + โ +
๐ฆ
0 2
2 -2
Max Min
54. Some Examples
โข Sufficient condition (2nd method): we need to find the sign of ๐โฒโฒ(๐)
at those critical points:
๐โฒโฒ ๐ = ๐๐ โ ๐
๐โฒโฒ ๐ = ๐ = โ๐ โ ๐จ ๐, ๐ ๐๐ ๐๐๐๐๐๐๐
๐โฒโฒ ๐ = ๐ = ๐ โ ๐ฉ ๐, โ๐ ๐๐ ๐๐๐๐๐๐๐
o Find the extremum(s) of ๐ = ๐ โ
๐
๐ ๐, if any.
๐โฒ =
โ๐
๐ ๐
๐
Although ๐โฒ cannot be zero but its sign changes when passing through
๐ = ๐, so the function has a maximum at point ๐จ(๐, ๐). The second
method of the sufficient condition cannot be used here. Why?
๐ โโ +โ
๐ฆโฒ +
๐ฆ
0
1
Max