The document discusses curve tracing through Cartesian equations. It defines important concepts like singular points, multiple points, points of inflection, and asymptotes. It outlines the standard method of tracing a curve by examining its symmetry, intersection with axes, regions where the curve does not exist, and tangents. Several examples are provided to demonstrate how to apply this method to trace specific curves like cissoids, parabolas and hyperbolas.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Lecture 5.1.5 graphs of quadratic equationsnarayana dash
Graphs of quadratic equations. The graphs of quadratic functions like y= ax^2 +bx+c or any variant of thereof may be cast into the graph of y = x^2 only. So this you may call parent graph.
Rai University provides high quality education for MSc, Law, Mechanical Engineering, BBA, MSc, Computer Science, Microbiology, Hospital Management, Health Management and IT Engineering.
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!
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
1. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 1
Unit-I: CURVE TRACING
Sr. No. Name of the Topic Page No.
1 Important Definitions 2
2 Method of Tracing Curve 3
3 Examples 7
4 Some Important Curves 11
5 Exercise 13
6 Reference Book 13
2. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 2
CURVE TRACING
INTRODUCTION: The knowledge of curve tracing is to avoid the labour of
plotting a large number of points. It is helpful in finding the length of curve, area,
volume and surface area. The limits of integration can be easily found on tracing
the curve roughly.
1.1 IMPORTANT DEFINITIONS:
(I) Singular Points: This is an unusual point on a curve.
(II) Multiple points: A point through which a curve passes more than one
time.
(III) A double Point: If a curve passes two times through a point, then this
point is called a double point.
(a) Node: A double point at which two real tangents (not coincident) can
be drawn.
(b) Cusp: A double point is called cusp if the two tangents at it are
coincident.
(IV) Point of inflexion: A point where the curve crosses the tangent is called
a point of inflexion.
(V) Conjugate point: This is an isolated point. In its neighbour there is no
real point of the curve.
At each double point of the curve y=f(x), we get,
𝐷 = (
𝜕2 𝑓
𝜕𝑥𝜕𝑦
)
2
=
𝜕2 𝑓
𝜕𝑥2
×
𝜕2 𝑓
𝜕𝑦2
3. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 3
a) If D is +ve, double point is a node or conjugate point.
b) If D is 0, double point is a cusp or conjugate point.
c) If D is –ve, double point is a conjugate point.
2.1 METHOD OF TRACING A CURVE:
This method is used in Cartesian Equation.
1. Symmetry:
(a) A curve is symmetric about x-axis if the equation remains the same by
replacing y by –y. here y should have even powers only.
For ex: 𝑦2
=4ax.
(b)It is symmetric about y-axis if it contains only even powers of x.
For ex: 𝑥2
=4ay
(c) If on interchanging x and y, the equation remains the same then the curve
is symmetric about the line y=x.
For ex: 𝑥3
+ 𝑦3
= 3𝑎𝑥𝑦
(d)A curve is symmetric in the opposite quadrants if its equation remains the
same where x and y replaced by –x and –y respectively.
For ex: 𝑦 = 𝑥3
Symmetric about x-axis Symmetric about y-axis
Symmetric about a line y=x
2. (a) Curve through origin:
The curve passes through the origin, if the equation does not contain
constant term.
4. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 4
For ex: the curve 𝑦2
=4ax passes through the origin.
(b) Tangent at the origin:
To know the nature of a multiple point it is necessary to find the tangent at
that point.
The equation of the tangent at the origin can be obtained by equating to zero,
the lowest degree term in the equation of the curve.
3. The points of intersection with the axes:
(a) By putting y=0 in the equation of the curve we get the co-ordinates of the
point of intersection with the x-axis.
For ex:
𝑥2
𝑎2
+
𝑦2
𝑏2
= 1 put y=0 we get 𝑥 = ±𝑎
Thus, (a, 0) and (-a, 0) are the co-ordinates of point of intersection.
(b)By putting x=0 in the equation of the curve, the co-ordinate of the point
of intersection with the y-axis is obtained by solving the new equation.
4. Regions in which the curve does not lie:
If the value of y is imaginary for certain value of x then the curve does not
exist for such values.
Example 1: 𝑦2
= 4𝑥
Answer: For negative value of x, if y is imaginary then there is no curve in
second and third quadrants.
Example 2: 𝑎2
𝑥2
= 𝑦3(2𝑎 − 𝑦).
Answer: (i) For y>2a, x is imaginary. There is no curve beyond y=2a.
(ii) For negative value of y, if x is imaginary then there is no curve
in 3rd
and 4th
quadrants.
5. Asymptotes are the tangents to the curve at infinity:
(a)Asymptote parallel to the x-axis is obtained by equating to zero, the
coefficient of the highest power of x.
For ex: 𝑦𝑥2
− 4𝑥2
+ 𝑥 + 2 = 0
(𝑦 − 4)𝑥2
+ 𝑥 + 2 = 0
5. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 5
The coefficient of the highest power of x 𝑖. 𝑒 𝑥2
is 𝑦 − 4 = 0
∴ 𝑦 − 4 = 0 is the asymptote parallel to the x-axis.
(b) Asymptote parallel to the y-axis is obtained by equating to zero, the
coefficient of highest power of y.
For ex: 𝑥𝑦3
− 2𝑦3
+ 𝑦2
+ 𝑥2
+ 2 = 0
(𝑥 − 2)𝑦3
+ 𝑦2
+ 𝑥2
+ 2 = 0
The coefficient of the highest power of 𝑦 𝒊. 𝒆. 𝑦3
is 𝑥 − 2.
∴ 𝑥 − 2 = 0 is the asymptote parallel to y-axis.
(c) Oblique Asymptote: 𝒚 = 𝒎𝒙 + 𝒄
(I) Find ∅ 𝑛(𝑚) by putting x=1 and y=m in highest degree (n) terms of
the equation of the curve.
(II) Solve ∅ 𝑛(𝑚) = 0 for 𝑚
(III) Find ∅ 𝑛−1(𝑚) by putting x=1 and y=m in the next highest degree
(n-1) terms of the equation of the curve.
(IV) Find 𝑐 by the formula, 𝑐 = −
∅ 𝑛−1(𝑚)
∅′
𝑛(𝑚)
, if the values of m are not
equal, then find 𝑐 by
𝑐2
2
∅′′
𝑛(𝑚) + 𝑐∅′
𝑛−1(𝑚) + ∅ 𝑛−2(𝑚) = 0
(V) Obtain the equation of asymptote by putting the values of m and c in
𝑦 = 𝑚𝑥 + 𝑐.
For ex: Find asymptote of 𝑥3
+ 𝑦3
− 3𝑥𝑦 = 0
Solution: Here, ∅3(𝑚) = 1 + 𝑚3
and ∅2(𝑚) = −3𝑚
Putting ∅3(𝑚) = 0 or 𝑚3
+ 1 = 0
(𝑚 + 1)(𝑚2
− 𝑚 + 1) = 0
𝑚 = −1, 𝑚 =
1±√1−4
2
Only real value of m is −1.
Now we find c from the equation
6. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 6
𝑐 = −
∅ 𝑛−1(𝑚)
∅′
𝑛(𝑚)
𝑐 = −
−3𝑚
3𝑚2
=
1
𝑚
=
1
−1
= −1
On putting 𝑚 = −1 and 𝑐 = −1 in 𝑦 = 𝑚𝑥 + 𝑐, the equation of
asymptote is
𝑦 = (−1)𝑥 + (−1)
𝑥 + 𝑦 + 1 = 0
6. Tangent:
Put
𝑑𝑦
𝑑𝑥
= 0 for the points where tangent is parallel to the x-axis.
For ex: 𝑥2
+ 𝑦2
− 4𝑥 + 4𝑦 − 1 = 0
2𝑥 + 2𝑦
𝑑𝑦
𝑑𝑥
− 4 + 4
𝑑𝑦
𝑑𝑥
= 0
(2𝑦 + 4)
𝑑𝑦
𝑑𝑥
= 4 − 2𝑥
𝑑𝑦
𝑑𝑥
=
4−2𝑥
2𝑦+4
Now,
𝑑𝑦
𝑑𝑥
= 0
4 − 2𝑥 = 0.
𝑥 = 2
Putting 𝑥 = 2 in (i), we get 𝑦2
+ 4𝑦 − 5 = 0
∴ 𝑦 = 1, −5
The tangents are parallel to x-axis at the points (2,1) and (2,-5).
7. Table:
Prepare a table foe certain values of x and y and draw the curve passing
through them.
7. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 7
For Ex: 𝑦2
= 4𝑥 + 4
X -1 0 1 2 3
y 0 ±2 ±2√2 ±2√3 ±4
Note : Remember POSTER. Where,
P = point of intersection
O = Origin
S = Symmetry
T = Tangent
A = Asymptote
R = Region
3.1 Trace the following curves:
Example 1: Trace the curve 𝑎𝑦2
= 𝑥2
(𝑎 − 𝑥)
Solution: we have,
𝑎𝑦2
= 𝑥2
(𝑎 − 𝑥) ________ (i)
1) Symmetry: Since the equation (i) contains only even power of y,
∴ it is symmetric about the x-axis.
It is not symmetric about y-axis since it does not contain even power
of x.
2) Origin: Since constant term is absent in (i), it passes through origin.
3) Intersection with x-axis:
Putting y=0 in (i), we get x=a.
∴ Curve cuts the x-axis at (a, 0).
4) Tangent: The equation of the tangent at origin is obtained by
equating to zero the lowest degree term of the equation (i).
𝑎𝑦2
= 𝑎𝑥2
.
8. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 8
𝑦2
= 𝑥2
𝑦 = ±𝑥
There are two tangents 𝑦 = ±𝑥 at the origin to the given curve.
Example 2: Trace 𝑦2(𝑎2
+ 𝑥2) = 𝑥2(𝑎2
− 𝑥2)
Solution: Here we have,
𝑦2(𝑎2
+ 𝑥2) = 𝑥2(𝑎2
− 𝑥2) _____________(i)
1) Origin: The equation of the given curve does not contain constant
term, therefore, the curve passes through origin.
2) Symmetric about axes: The equation contains even powers of x as
well as y, so the curve is symmetric about both the axes.
3) Point of intersection with x-axis: On putting y=0 in the equation,
we get
𝑥2(𝑎2
− 𝑥2) = 0, 𝑥 = ±𝑎, 0,0
4) Tangent at the origin: Equation of the tangent is obtained by
equating to zero the lowest degree term.
𝑎2
𝑦2
− 𝑎2
𝑥2
= 0 ⇒ 𝑦 = ±𝑥
There are two tangents 𝑦 = 𝑥 and 𝑦 = −𝑥 at the origin.
5) Node: Origin is the node, since, there are two real and different
tangents at the origin.
6) Region of absence of the curve: For values of 𝑥 > 𝑎 and 𝑥 < −𝑎,
𝑦2
becomes negative, hence, the entire curve remains between 𝑥 =
−𝑎 𝑎𝑛𝑑 𝑥 = 𝑎.
9. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 9
Example 3: Trace the curve 𝑦2(2𝑎 − 𝑥) = 𝑥3
(cissoid)
Solution: We have, 𝑦2
=
𝑥3
(2𝑎−𝑥)
_____________ (i)
1) Origin: Equation does not contain any constant term. Therefore, it passes
through origin.
2) Symmetric about x-axis: Equation contains only even powers of y,
therefore, it is symmetric about x-axis.
3) Tangent at the origin: Equation of the tangent is obtained by equating to
zero the lowest degree terms in the equation (i).
2𝑎𝑦2
− 𝑥𝑦2
= 𝑥3
Equation of tangent:
2𝑎𝑦2
= 0 ⟹ 𝑦2
= 0, 𝑦 = 0 is the double point.
4) Cusp: As two tangents are coincident, therefore, origin is a cusp.
5) Asymptote parallel to y-axis: Equation of asymptote is obtained by
equating the coefficient of highest degree of y to zero.
2𝑎𝑦2
− 𝑥𝑦2
= 𝑥3
⟹ (2𝑎 − 𝑥)𝑦2
= 𝑥3
Equation of asymptote is 2𝑎 − 𝑥 = 0 ⟹ 𝑥 = 2𝑎.
6) Region of absence of curve: 𝑦2
becomes negative on putting
𝑥 > 2𝑎 𝑜𝑟 𝑥 < 0, therefore, the curve does not exist for
𝑥 < 0 𝑎𝑛𝑑 𝑥 > 2𝑎 .
10. Unit-1 CURVE TRACING
RAI UNIVERSITY, AHMEDABAD 10
Example 4: Trace the curve 𝑎2
𝑥2
= 𝑦2
(2𝑎 − 𝑦)
Solution: we have, 𝑎2
𝑥2
= 𝑦2
(2𝑎 − 𝑦) _________(i)
1) Symmetry:
(a) The curve is symmetric about y-axis. Since all the powers of x are even.
(b) Not symmetric about x-axis. Since all the powers of y are not even.
2) Origin: The curve passes through the origin since the equation does not
contain any constant term.
3) Region of absence of the curve: If y is greater than 2a the right hand side
becomes negative but left hand side becomes positive hence, the curve does
not exist when y=2.
4) Tangent at the origin: On putting the lowest degree term to zero 𝑎2
𝑥2
=
𝑦2
2𝑎 ⟹ 𝑎𝑥2
= 2𝑦2
⟹ 𝑦 = ±√
𝑎
2
𝑥
5) Intercept on y-axis: On putting x=0 in the equation, we get
⟹ 0 = 𝑦2
(2𝑎 − 𝑦)
⟹ 2𝑎 − 𝑦 = 0
⟹ 𝑦 = 2𝑎