1. Partial derivatives describe how a function changes with respect to one variable while holding other variables constant. The partial derivative of Z with respect to x is denoted as ∂Z/∂x or fχ.
2. Optimization problems in calculus involve finding the maximum or minimum values of functions, which can be used to determine the best way to do something.
3. A function has a global/absolute maximum at c if it is greater than or equal to the function values at all other points, and a global/absolute minimum if it is less than or equal to all other points.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
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The Chasm at Depth Four, and Tensor Rank : Old results, new insightscseiitgn
Agrawal and Vinay [FOCS 2008] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Koiran [TCS 2012] and subsequently by Tavenas [MFCS 2013]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them.In an apriori surprising result, Raz [STOC 2010] showed that for any $n$ and $d$, such that $\omega(1) \leq d \leq O(logn/loglogn)$, constructing explicit tensors $T: [n] \rightarrow F$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field F. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any d such that $\omega(1) \leq d \leq n^{o(1)}$. Joint work with Mrinal Kumar, Ramprasad Saptharishi and V Vinay.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Aлеф Мед Консалт осуществляет координацию и профессиональные консультации в области лечения и медицинского туризма в Израиле. Компания работает без участия посредников с десятками врачей — специалистов в различных областях и профессорами с мировыми именами, а так же с медицинскими центрами по всей стране, с целью предоставления возможности пациентам со всего мира воспользоваться высококачественными и профессиональными услугами израильской медицины.
Ул.ха-Барзель 24, Рамат ха-Хаяль, Тель-Авив, 69710, Израиль. Тел: +972545530269 www.alefmedconsult.com
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
VISIT US @
www.anuragtyagiclasses.com
The Chasm at Depth Four, and Tensor Rank : Old results, new insightscseiitgn
Agrawal and Vinay [FOCS 2008] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Koiran [TCS 2012] and subsequently by Tavenas [MFCS 2013]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them.In an apriori surprising result, Raz [STOC 2010] showed that for any $n$ and $d$, such that $\omega(1) \leq d \leq O(logn/loglogn)$, constructing explicit tensors $T: [n] \rightarrow F$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field F. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any d such that $\omega(1) \leq d \leq n^{o(1)}$. Joint work with Mrinal Kumar, Ramprasad Saptharishi and V Vinay.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Aлеф Мед Консалт осуществляет координацию и профессиональные консультации в области лечения и медицинского туризма в Израиле. Компания работает без участия посредников с десятками врачей — специалистов в различных областях и профессорами с мировыми именами, а так же с медицинскими центрами по всей стране, с целью предоставления возможности пациентам со всего мира воспользоваться высококачественными и профессиональными услугами израильской медицины.
Ул.ха-Барзель 24, Рамат ха-Хаяль, Тель-Авив, 69710, Израиль. Тел: +972545530269 www.alefmedconsult.com
Application of partial derivatives with two variablesSagar Patel
Application of Partial Derivatives with Two Variables
Maxima And Minima Values.
Maximum And Minimum Values.
Tangent and Normal.
Error And Approximation.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
2. Let Z= f(x,y) the derivative of Z with respect
to x is, if it is, when x alone varies & y remains
constant is called partial derivative of Z w.r.t
x.
It is denoted by ¶Z/¶x or fᵪ And fᵧ for y.
3. Some of the most important
applications of differential calculus
are optimization problems.
In these, we are required to find the optimal (best)
way of doing something.
These problems can be reduced to
finding the maximum or minimum values
of a function.
4. A function f has an absolute maximum
(or global maximum) at c if f(c) ≥ f(x) for
all x in D, where D is the domain of f.
The number f(c) is called the maximum value
of f on D.
5. Similarly, f has an absolute minimum at c
if f(c) ≤ f(x) for all x in D and the number f(c)
is called the minimum value of f on D.
The maximum and minimum values of f
are called the extreme values of f.
6. If we consider only values of x near b—for
instance, if we restrict our attention to the
interval (a, c)—then f(b) is the largest of those
values of f(x).
It is called a local
maximum value of f.
7. Likewise, f(c) is called a local minimum value
of f because f(c) ≤ f(x) for x near c—for
instance, in the interval (b, d).
The function f also has
a local minimum at e.
8. In general, we have the following definition.
A function f has a local maximum (or relative
maximum) at c if f(c) ≥ f(x) when x is near c.
This means that f(c) ≥ f(x) for all x in some
open interval containing c.
Similarly, f has a local minimum at c if f(c) ≤ f(x)
when x is near c.
9. Equation of theTangent plane and Normal
line can be made with the help of partial
derivation.
Equation ofTangent Plane to any surface at P
is given by,
(X – x)¶f/¶x + (Y – y)¶f/¶y = 0
Equation of Normal Line is given by,
(X – x)/¶f/¶x = (Y – y)/¶f/¶y
10. Extreme value is useful for
1. What is the shape of a can that minimizes manufacturing
costs?
2. What is the Maximum Area orVolume which can be
obtained for particular measurements of height, length and
width?
Determination of ExtremeValue
Consider the function u= f(x , y). Obtain the
first and second order derivatives such as p=
fᵪ , q= fᵧ, r= fᵪᵪ, s= fᵪᵧ, t= fᵧᵧ.
11. Take p=0 and q=0 and solve. Simultaneously
obtain the Stationary Points.
(xₒ , yₒ),(x₁ , y₁),…. Be simultaneously points.
Consider the stationary points (xₒ , yₒ) and
obtain the value of r, s, t.
a. If rt-s²>0 then the extreme value exists.
I. If r<0, then value is Maximum.
II. If r>0, then value is Minimum.
12. b. If rt-s²<0, then the extreme value does not
exist.
c. If rt-s²=0, we cannot state about extreme
value & further investigation is required.
Follow the Same procedure for the other
stationary point.
Saddle Point
If rt-s²=0, then the point (xₒ , yₒ) is called a
Saddle point.
13. Z = f(x , y) be a continuous function of x and y
where fᵪ & fᵧ be the errors occurring in the
measurement of the value of x & y.Then the
corresponding error ¶Z occurs in the
estimation of the value of Z.
i.e. Z+¶Z = f(x+¶x , y+¶y)
Therefore, ¶Z = f(x+¶x , y+¶y) – f(x , y).
14. Expanding by usingTaylor’s Series and
neglecting the higher order terms of ¶x & ¶y,
we get,
¶Z = ¶x.¶f/¶x + ¶y.¶f/¶y
¶x is known as Absolute Error in x.
¶x/x is known as Relative Error in x
¶x/x*100 is known as Percentage Error in x.
15. 1. In measurement of radius of base and height
of a rigid circular cone are incorrect by -1%
and 2%. Calculate Error in theVolume.
Solution,
Let r be the radius and h be the height of the
circular cone andV be the volume of the
cone.
V = π/3*r^2*h
16. Thus,
¶V = ¶r.¶V/¶r + ¶h.¶V/¶h
Now,
¶r/r*100 = -1 ¶h/h*100 = 2
Again,
¶V = π/3(2rh)(r/100) + π/3(r*r)2h/100
= 0
So,
The Error in the measurement in theVolume is
Zero.