HIGHER MATHEMATICS

Digital Assignment -I
Course Name: Calculus for Engineers
Course Code: MAT1711
Slot : B1/TB1
NAME: ROHITH.A
REG NO: 17MIS0298
THE MEAN VALUE THM IN DAY TO DAY LIFE:
Since Rolle's theorem asserts the existence of a point where the derivative vanishes, I
assume your students already know basic notions like continuity and differentiability.

One way to illustrate the theorem in terms of a practical example is to look at the
calendar listing the precise time for sunset each day. One notices that around the precise
date in the summer when sunset is the latest, the precise hour changes very little from
day to day near the precise date.
Mean value threom in engineering side:
Brayden’s method works because of the mean value theorem. But Taylor's theorem is just
a neat inductive application of the MVT. So maybe this is a good reason for teaching the
MVT to physicists and engineers - it provides justification (and perhaps some intuition) for
one of their favourite mathematical tools.
This show that mean value threom fundamentals for Taylor threom and braydden methods
Mean value threom contribution in
physics side:
The application of a common mean-value theorem for the description of kinetics of
heterogeneous catalytic reactions over inhomogeneous surfaces is discussed. Being
mathematically correct for integration of multiple integrals, the mean value theorem can
provide a correct value of reaction rate for a set of parameters, but physical reasons of its
application are doubtful, as it obviously contradicts the kinetic regularities. As an example,
the kinetics of a two-step sequence on an inhomogeneous surface is discussed.

Mean value threom in real in life:
Transcript of A REAL-LIFE APPLICATION OF THEMEAN VALUE THEOREM. The Mean
Value Theorem states that if y= f(x) is continuous on [a, b] and differentiable on (a, b), then
there is a "c" (at least one point) in (a, b) where f'(c)= (f(b)- f(a)) / (b-a). ... I used The Mean
Value Theorem to test the accuracy of my speedometer

increasing and decreasing function in
medical field:
We define the GVF field v(x) as the equilibrium solution to the
following vector ... Since these weighting functions are
dependent on the gradient of the edge map ... and h (·) should
be monotonically non-increasing and non-decreasing
functions ...
Research conditions in physics are differing from those ones
in medicine, see the ... Function. (Two-Dimensional.
Representation). The increasing evenness' of 'mixed' ... of a
science discipline that is subdivided into several
specialized fields ... monotonically increasing or decreasing function and both
proverbs
Increasing function, a decreasing function s ... can only be
evaluated experimentally, by imaging a flood field phantom. ...
error (decreasing function of A) and of the statistical error
(increasing function of A).

higher order derivatives in network:
he aims of those tools is to build surrogate models. We present a parameterization method
based on higher order derivatives computation obtained by automatic differentiation. The
most popular parameterization methods are neural networks, design of experiments, and
resolution of simplified models.
+……
Application of Higher Order Derivatives to Helicopter Model Control
2.1 Elevation dynamics
Let us consider the forces in the vertical plane acting on the
vertical helicopter body, whose
dynamics are given by the following nonlinear equation:
APPLICATIONS OF MAXIMA AND MINIMA:
To understand when a function is said to attain a
maximum value and a minimum value in its domain.
Objective 2: To understand the terms local maximum
value and local minimum values of a function. ...
Objective 5: To understand the behaviour of f '(x) at
local maxima and local minima.
The quantity A that we are trying to maximize is called
the objective function. The conditions that come after "subject

to" are called constraints. Thus, we are trying to maximize some
objective function subject to one or more constraints:
HISTORY OF MAXIMA AND MINIMA:
Since origin of life, all people knew, talked, applied the concept of maxima and minima in their
daily lives without even knowing about the concept of maxima and minima. In the earlier phase
of time the kings used to estimate the maximum and minimum army of the opposite side, doctors
used to record minimum and maximum symptom of any disease, cooks used to estimate the
maximum and minimum quantity of food or people before any function, the businessmen used to
estimate maximum and minimum profit or loss in any transaction. Even today also the women in
the house prepare the food according to maximum or minimum consumption by everyone.
Sir Issacs Newton, a great scientist, invented the concept of functions and hence con
The Second Derivative Test for Functions
of Two Variables
How can we determine if the critical points found above are relative maxima or minima? We
apply a second derivative test for functions of two variables.
Let (x, y) be a critical point and define
We have the following cases:
If D>0 and (,).) <0, then f (x, y) has a relative maximum at (,).).
If D>0 and (,).)>0, then f (x, y) has a relative minimum at (,).).
If D<0, then f (x, y) has a saddle point at (,).
If D=0, the second derivative test is inconclusive.
Maxima and Minima in a Bounded Region
Suppose that our goal is to find the global maximum and minimum of our model function above
in the square -2<=x<=2 and -2<=y<=2? There are three types of points that can potentially be
global maxima or minima:
Relative extrema in the interior of the square.
Relative extrema on the boundary of the square.
Corner Points.

We have already done step 1. There are extrema at (1, 0) and (-1, 0). The boundary of square
consists of 4 parts. Side 1 is y=-2 and -2<=x<=2. On this side, we have
The original function of 2 variables is now a function of x only. We set g'(x)=0 to determine
relative extrema on Side 1. It can be shown that x=1 and x=-1 are the relative extrema. Since y=-
2, the relative extrema on Side 1 are at (1, -2) and (-1, -2).
On Side 2 (x=-2 and -2<=y<=2)
We set h'(y)=0 to determine the relative extrema. It can be shown that y=0 is the only critical
point, corresponding to (-2,0).
We play the same game to determine the relative extrema on the other 2 sides. It can be shown
that they are (2,0), (1,2), and (-1,2).
Finally, we must include the 4 corners (-2, -2), (-2,2), (2, -2), and (2,2). In summary, the
candidates for global maximum and minimum are (-1,0), (1,0), (1, -2), (-1, -2), (-2,0), (2,0), (1,2),
(-1,2), (-2, -2), (-2,2), (2, -2), and (2,2). We evaluate f (x, y) at each of these points to determine
the global max and min in the square. The global maximum occurs (-2,0) and (1,0). This can be
seen in the figure above. The global minimum occurs at 4 points: (-1,2), (-1, -2), (2,2), and (2, -2).
One of the great powers of calculus is in the determination of the maximum or minimum value of
a function. Take f(x) to be a function of x. Then the value of x for which the derivative of f(x) with
respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the
function f(x).
The derivative of a function can be geometrically interpreted as the slope of the curve of the
mathematical function y(t) plotted as a function of t. The derivative is positive when a function is
increasing toward a maximum, zero (horizontal) at the maximum, and negative just after the
maximum. The second derivative is the rate of change of the derivative, and it is negative for the
process described above since the first derivative (slope) is always getting smaller. The second
derivative is always negative for a "hump" in the function, corresponding to a maximum.
A critical point (x, y) off is a point where both the partial derivatives of the functions vanish. A local
maximum, or a local minimum, is a critical point. In one variable, local maxima and minima are
the only `nondegenerate' critical points. In two or more variables, other possibilities appear. For
instance, one has the saddle point, like the critical point of at (0; 0). In some directions, this looks
like a maximum, in other directions this looks like a minimum. We try to classify critical points by
looking at the second derivatives.
APPLICATIONS OF MAXIMA AND MINIMA
IN DAILY LIFE:
There are numerous practical applications in which it is desired to find the maximum or minimum
value of a quantity. Such applications exist in economics, business, and engineering. Many can
be solved using the methods of differential calculus described above. For example, in any
manufacturing business it is usually possible to express profit as a function of the number of units
sold. Finding a maximum for this function represents a straightforward way of maximizing profits.
In other cases, the shape of a container may be determined by minimizing the amount of
material required to manufacture it. The design of piping systems is often based on minimizing

pressure drop which in turn minimizes required pump sizes and reduces cost. The shapes of
steel beams are based on maximizing strength.
Finding maxima or minima also has important applications in linear algebra and game theory. For
example, linear programming consists of maximizing (or minimizing) a quantity while requiring
that certain constraints be imposed on other quantities. The quantity to be maximized (or
minimized), as well as each of the constraints, is represented by an equation or inequality. The
resulting system of equations or inequalities, usually linear, often contains hundreds or
thousands of variables. The idea is to find the maximum value of a variable that represents a
solution to the whole system. A practical example might be minimizing the cost of producing an
automobile given certain known constraints on the cost of each part, and the time spent by each
labourer, all of which may be interdependent. Regardless of the application, though, the key step
in any maxima or minima problem is expressing the problem in mathematical terms.
Everything in this world is based on the concept of maxima and minima, every time we always
calculate the maximum and minimum of every data. Now-a-days results are also based on the
concepts of grades which is again based on the concept of maxima and minima.
APPLICATIONS OF THE DEFINITE INTERGALS AND
INDEFINITE INTREGALS:
Displacement from Velocity, and Velocity from Acceleration
High velocity train [Image source]
A very useful application of calculus is displacement, velocity and
acceleration.
Recall (from Derivative as an Instantaneous Rate of Change) that we can
find an expression for velocity by differentiating the expression for
displacement:
display style{v}=frack{{{d}{s}}} {{{left. {d}{t} right.}}} v=dads
Similarly, we can find the expression for the acceleration by differentiating
the expression for velocity, and this is equivalent to finding the second
derivative of the displacement:

display style{a}=frack{{{d}{v}}} {{left. {d}{t} right.}}
=frack{{{d}^{2}{s}}} {{{left. {d}{t} right.} ^ {2}}} a=did
=dt2d2s
It follows (since integration is the opposite process to differentiation) that to
obtain the displacement, display style{s}s of an object at time display
style{t}t (given the expression for velocity, display style{v}v) we would
use:
display style{s}=into{v} {left. {d}{t} right.} s=∫v dot
Similarly, the velocity of an object at time display style{t}t with
acceleration display style{a}a, is given BY:
INTEGRATION:
1. Applications of the Indefinite Integral shows how to find displacement
(from velocity) and velocity (from acceleration) using the indefinite
integral. There are also some electronics applications.
In primary school, we learnt how to find areas of shapes with straight
sides (e.g. area of a triangle or rectangle). But how do you find areas
when the sides are curved? e.g.
2. Area under a Curve and
3. Area in between the two curves. Answer is by Integration.
4. Volume of Solid of Revolution explains how to use integration to find
the volume of an object with curved sides, e.g. wine barrels.
5. Centroid of an Area means the canter of mass. We see how to use
integration to find the centroid of an area with curved sides.
6. Moments of Inertia explain how to find the resistance of a rotating
body. We use integration when the shape has curved sides.
7. Work by a Variable Force shows how to find the work done on an
object when the force is not constant.
8. Electric Charges have a force between them that varies depending on
the amount of charge and the distance between the charges. We use
integration to calculate the work done when charges are separated.
9. Average Value of a curve can be calculated using integration.
APPLICATION OF LAPALACE THROEM IN SIGNAL SYSTEM:

Unilateral Laplace Transform[edit]
The Laplace Transform of a function f(t), defined for all real numbers t ≥ 0, is the
function F(s)defined by:
The parameter s is the complex number:
with a real part σ and an imaginary part ω.
Bilateral Laplace Transform[edit]
The Bilateral Laplace Transform is defined as follows:
Comparing this definition to the one of the Fourier Transform, one sees that the latter is a special
case of the Laplace Transform for .
In the field of electrical engineering, the Bilateral Laplace Transform is simply referred as the
Laplace Transform.
Inverse Laplace Transform[edit]
The Inverse Laplace Transform allows to find the original time function on which a
Laplace Transform has been made.:
The properties of Laplace transform are:
Linearity Property
If x(t)⟷L. TX(s)x(t)⟷L. TX(s)
& y(t)⟷L. TY(s)y(t)⟷L. TY(s)
Then linearity property states that
as(t)+by(t)⟷Lata(s)+by(s)as(t)+by(t)⟷Lata(s)+by(s)
Time Shifting Property

Then time shifting property states that
x(t−t0) ⟷L. Te−st0X(s)x(t−t0) ⟷L. Te−st0X(s)
Frequency Shifting Property
Then frequency shifting property states that
es0t.x(t)⟷L. TX(s−s0) es0t.x(t)⟷L. TX(s−s0)
Time Reversal Property
Then time reversal property states that
x(−t) ⟷L. TX(−s) x(−t) ⟷L. TX(−s)
Time Scaling Property
Then time scaling property states that
x(at)⟷L. T1|a|X(as)x(at)⟷L. T1|a|X(as)
Differentiation and Integration Properties

Then differentiation property states that
dx(t)tilts’(s)dx(t)tilts’(s)
don(t)dtn⟷L.T(s)next(s)don(t)dtn⟷L.T(s)next(s)
The integration property states that
∫x(t)dt⟷L.T1sX(s)∫x(t)dt⟷L.T1sX(s)
∭...∫x(t)dt⟷L.T1snX(s)∭...∫x(t)dt⟷L.T1snX(s)
Multiplication and Convolution Properties
and y(t)⟷L. TY(s)y(t)⟷L. TY(s)
Then multiplication property states that
x(t). y(t)⟷L. T12πaxe(s)∗Y(s)x(t). y(t)⟷L. T12πaxe(s)∗Y(s)
The convolution property states that
x(t)∗y(t)⟷L. TX(s). Y(s)
Applications of unit step function:
The Unit Step Function (Heaviside Function) In engineering applications, we frequently
encounter functions whose values change abruptly at specified values of time t. One
common example is when a voltage is switched on or off in an electrical circuit at a specified
value of time t.
Apply the Unit Step Function to Circuit Analysis - dummies
The unit step function can describe sudden changes in current or voltage in a circuit.
The unit step function looks like, well, a step. Practical step functions occur daily, like
each time you turn mobile devices, stereos, and lights on and off.

PARATICAL APPLYCATION OF UNIT STEP FUNCTION: The unit step function can
describe sudden changes in current or voltage in a circuit. The unit step function looks like,
well, a step. Practical step functions occur daily, like each time you turn mobile devices,
stereos, and lights on and off.
CREATE A TIME-SHIFTED, WEIGHTED
STEP FUNCTION
The circuit approximation of the step function shown earlier
assumes you can quickly change from off to on at time t = 0 when
the switch is thrown.
Although the unit step function appears not to do much, it’s a
versatile signal that can build other waveforms. In a graph, you can
make the step shrink or stretch. You can multiply the step
function u(t) by a constant amplitude Viki to produce the following
waveform:
History of top Five Mathematician in India and
explain their research work.

Mahayana (800BC) Mahayana discovered the Pythagoras Theorem
around 1000 years before Pythagoras was even born. In this
book, Bandhan claustra (800 BC), he wrote, “A rope stretched along
the length of the diagonal produces an area which the vertical and
horizontal sides make together”. This is nothing, but a different way of
looking at Pythagoras theorem. Apart from this, the book contained
geometric solutions of a linear equation in a single unknown.
Aryabhata (476–550 AD)
Aryabhata is undoubtedly the most celebrated Indian mathematicians.
His most significant contributions to mathematics include approximation
of the value of pi up to five decimal places, and he also discussed the
concept of sine. Aryabhata was the one who calculated the area of the
triangle as perpendicular multiplied by the half side. He was the one to
calculate that the time that Earth takes to complete one rotation is 365
days. In algebra, he summed series of squares and cubes and solved
equations of the type as -by = c.
Brahmagupta (598-670 AD)
Brahmagupta is the man who gave the world the concept of negative
numbers and zero. He also proposed rules for solving simultaneous and
quadratic equations. He calculated the area of a cyclic quadrilateral with
semi-perimeter (s). Brahmagupta is the founder of “Numerical Analysis”,

a branch of higher mathematics. He was the one to identify that x²- y² =
(x, y) (x-y).
Bhaskar (600-680AD)
Bhaskar expanded on the work of Aryabhata, and found an
approximation of the sine function. Bhaskar laid the foundation of
differential calculus, and gave an example of the differential coefficient
and discussed the idea of what we know as Rolle’s Theorem today. He
told the world that sum of any number and infinity is infinity, and any
number divided by zero is infinity. He was the one to introduce the cyclic
method of solving algebraic equations. The “inverse cyclic”
method that we know today stems from this.
Mahavira (800-870 A.D)
Mahavira was a Jain mathematician, who derived the volume of frustum
by an infinite procedure. He worked with logarithms in base 2, base 3
and base 4. He authored the book Granita Sara Sagrada in 850 ADS,
which included teachings of Brahmagupta, but also contained
simplifications and some additional information. It includes chapters on
arithmetical operations, mixed operations, operations involving functions,
operations relating to calculation of areas and others.
Abrahamitical (505-587AD)
Abrahamitical was a mathematician, astrologer, and astronomer. His
prominent mathematical work includes the discovery of the following
trigonometric formulas:
Abrahamitical also improved the accuracy of Aryabhata’s sine tables.
Abrahamitical also defined the algebraic properties of zero and negative
numbers. He was among the first mathematicians to discover a version
of Pascal’s triangle as we know it today.

Bhaskar II (1114-1185AD)
Bhaskar II was a prominent mathematician and astronomer, who proved
that any number divided by zero is infinity. He also found that a positive
number has two square roots. Bhaskar II was the one to discover the
differential coefficient and derivative. He gave the formula
Bhaskar authored six books on mathematics.
Srinivasan Aiyana Ramanujan (1887-1920)
Ramanujan is probably the best-known mathematicians of modern India.
Some of his most credible contributions to the world of mathematics are
the Hardy-Ramanujan-Littlewood circle method, elliptic functions, work
on the algebra of inequalities, partial sums and products of
hypergeometric series, Roger-Ramanujan’s identities in the partition of
numbers and continued fractions. 1729 is known as the Ramanujan
number.
MATHMATICES IN 2017:
How are A Level Mathematics and Further Maths changing?
 Linear assessment: All assessments for A Level and
Further Maths are linear, with 100% by examination, which
means that all the exams are sat at the end of the course.
 Statistics and mechanics are compulsory: AS and A
Level Maths have 100% prescribed content, containing both
pure and applied mathematics, which means that there are
no options available to choose. All AS and A Level Maths
students will now be assessed on both statistics and
mechanics.

 Large data sets: A Level student should be familiar with
using large data sets to support their learning and
assessment of statistics.
 Use of technology and calculators: It is assumed that
students will have access to appropriate technology with the
use of scientific or graphical calculators available for all
exams.
 Mathematical understanding: There is increased focus
on problem-solving, mathematical argument, reasoning and
modelling.
 Choices at Further Maths: At AS Further Maths (30%)
and A Level Further Maths (50%), there are choices and
options for the topics you teach. This means that you can
choose topics that meet the needs and interests of your
students.
Qualifications from 2017
Individual documents, along with a selection of free resources,
are available from our new qualification webpages:
 AS/A Level GCE - Mathematics A - H230, H240 (from
2017)
 AS/A Level GCE - Mathematics B (MEI) - H630, H640
(from 2017)
 AS/A Level GCE - Further Mathematics A - H235, H245
(from 2017)
 AS/A Level GCE - Further Mathematics B (MEI) - H635,
H645 (from 2017)
 A level Content Advisory Board (ALCAB) report
on Mathematics and Further Mathematics July 2014
 AS and A Level Mathematics and Further
Mathematics: Consultation on Conditions and
Guidance, Equal, December 2015

 Equal’s A Level Mathematics Working Group Report on
problem-solving, modelling and large data sets, December
2015.
Explain your role in Mathematics and future in
Mathematics.
Real teachers in real schools face real challenges implementing the
numerous standards and recommendations for mathematics teachers
today. Many teachers reading the National Council of Teachers of
Mathematics' (NCTM's) Curriculum and Evaluation Standards for School
Mathematics (1989) and Professional Standards for Teaching Mathematics
(1991) get excited about the possibilities for new kinds of instruction just as
they are also bogged down by the overwhelming expectations about how
they should provide that instruction. Within the complex description these
two volumes provide about what teachers should do and how they should
act, ten basic metaphors seem to emerge about new teacher roles. Some
of these roles may feel comfortable to some of you, and other roles may
remind you just how hard it is to live up to you own ideals as a teacher.
The teachers as recruiter: Mathematics teachers have traditionally done a
good job of encouraging students to pursue mathematics-related fields.
Encouraging students to become mathematics teachers, however, has
often been something we are reluctant to do, especially for our favourite
students (or those we are related to). We sometimes communicate that
teaching isn't as worthy a profession as other more lucrative options. If
teaching isn't now a career worthy of our future adults, it is our
responsibility as professional educators to transform it into something that
is. The responsibility of the teacher as recruiter is to communicate not only
to students but to the broader community how important and rewarding our
profession can be. The ideal of teacher as recruiter is reflected in an
experience shared with me by Kathleen, a teacher who recently received a
Presidential Award. As she stepped down from the platform after her state-

wide award ceremony, she felt a tap on her shoulder and turned to see her
former high school mathematics teacher. Kathleen shared with her former
teacher that she was the inspiration for Kathleen choosing a career in
teaching. As they were hugging, Kathleen felt a tap on her other shoulder.
She turned to see a former student who told Kathleen that Kathleen was
her inspiration for becoming a teacher. We can all truly hope that the torch
will continue to be passed from generation to generation so that some of
our finest minds can continue to prepare new generations of students who
can think and learn mathematically.
BY: ROHITH A
REG NO: 17MIS0298
DIGITAL ASSIGNMENT -1

HIGHER MATHEMATICS

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