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Lesson Posted 3 days ago CBSE/Class 12/Science/Mathematics CBSE/Class 12/Science/Mathematics/Unit III: Calculus IIT JEE/IIT JEE MAINS/Mathematics +1 Exam Coaching/CET Coaching/Mathematics less

Maxima and Minima

Om Prakash Mishra

I am teaching Mathematics and Physics from past more than 12 years in coaching as well as home tuition...

To find maximum and minimum value of a given function. Differentiate the function first and then find the values of x at which the function becomes zero these are the possible points of Maximum and minimum values of the function then apply any of the following test (1) 1st derivative test: If at any... read more

To find maximum and minimum value of a given function.

Differentiate the function first and then find the values of x at which the function becomes zero these are the possible points of Maximum and minimum values of the function then apply any of the following test

(1) 1st derivative test: If at any of the above values of x dy/dX changes its sign from +ve to -ve then that is a point of local Maxima.

And if dy/dX changes sign from -ve to +ve that is a point of local minima.

(2) 2nd derivative test: Find the 2nd derivative and put the above values of x obtained if it comes -ve that is a point of local Maxima and if it comes +ve that's a point of local minima.

Note: there can be more than 1 point of local Maxima and Minima

Global Maximum and Minimum: put all the values of x obtained in the function and check which one is largest, that value correspond to Global Maxima and that vale is global maximum. Similarly check for Global minima.

Note:sometimes 2nd derivative test fails i.e it becomes zero that that point is called point of inflection. Then check with first derivative test.

 

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Asked 4 days ago IIT JEE/IIT - JEE Mains/Physics/Laws of Motion

A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction... read more

A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body.

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Asked 4 days ago IIT JEE/IIT - JEE Mains/Physics/Laws of Motion

The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the... read more

The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child. What is the average retarding force on the vehicle? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg.

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Lesson Posted on 11 Sep IIT JEE/IIT - JEE Mains/Mathematics/Vector Algebra

internal and external divison of vectors

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

If A and B are two points with position vectors a and b respectively and C is a point which divides AB internally in the ratio m: n, then the position vector of C is given by OC = (mb + na)/ (m + n). This is termed as internal division.which is shown by below figure. If A and B are two points with position... read more

If A and B are two points with position vectors a and b respectively and C is a point which divides AB internally in the ratio m: n, then the position vector of C is given by OC = (mb + na)/ (m + n). This is termed as internal division.which is shown by below figure.

If A and B are two points with position vectors a and b respectively and C is a point which divides AB externally in the ratio m : n, then the position vector of C is given by OC = (mb - na)/ (m-n)

External Division

 

 

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Asked on 01 Sep IIT JEE/IIT - JEE Advanced/Mathematics/Architectural Aptitude Test

For writing NATA, from when I should start preparing?

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Lesson Posted on 04 Aug IIT JEE/IIT - JEE Advanced/Physics/Mechanics

The Curious Case of Point of Inflection

Abhilash K S

Faculty for Engineering Mathematics in a private tuition center. Faculty for CAT/GMAT/GRE/SAT. Faculty...

In kinematics, graph plays an important part in understanding various motions. Commonly taught graphs for beginners are distance Vs time and displacement Vs time graph of rectilinear motion, in particular when an object is thrown vertically upwards. The displacement Vs time graph in such a case... read more

In kinematics, graph plays an important part in understanding various motions.

Commonly taught graphs for beginners are distance Vs time and displacement Vs time graph of rectilinear motion, in particular when an object is thrown vertically upwards. 

The displacement Vs time graph in such a case is, a parabola mouth opening downwards. The derivative of this graph gives the value of velocity which gradually decreases to zero at its highest point then increases with a negative sign. But the double derivative which gives the value of acceleration remains constant throughout which is the value of g, i.e. acceleration due to gravity.

The shape of distance Vs time graph is a bit different. It remains same as that of displacement Vs time till first half then the curvature of the graph changes. Mathematically, such a point is called the point of inflexion where the curvature of graph changes. The important point to note here is that at the point of inflexion the double derivative is zero. 

If we compare this graph with the previous one, the value of double derivative remains the same at all points except at the point of infection. It is so because the double derivative of distance Vs time graph gives "tangential acceleration". At the highest point, the velocity of the body is zero so is the value of tangential acceleration. A point which otherwise missed, if graphs are not studied properly.

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Lesson Posted on 06 Aug IIT JEE/IIT - JEE Mains/Mathematics/Statistics and Probability

Correlation, Regression, Coefficient of Correlation and Determination

Raj Kumar

I am Six Sigma Black belt trained from American Society of Quality I am 2011 pass out in B.tech from...

Correlation A measure of linear relationship between two variables independent and dependent it is used when and Y both are continuous Corelation is measure with r (Coefficient of corelation) r values ranges from 1) r=1 perfect positive relationship 2) r=0 no relationship 3) r=-1 negative... read more

Correlation

A measure of linear relationship between two variables independent and dependent 

it is used when  and Y both are continuous

 

Corelation is measure with r (Coefficient of corelation)

r values ranges from 

1) r=1 perfect positive relationship 

2) r=0 no relationship 

3) r=-1 negative relationship 

 

Regression:

Provides the basis for predicting the value of a variable from the values of one or more other variables 

used with continuous Y and Continuous Xs or continuous Y and categorical Ys

It is measured with r2 which is known as the coefficient of determination 

rtells us about the proportion of variation of Y explained by prediction equation

 

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Lesson Posted on 03 Aug IIT JEE/IIT - JEE Advanced/Physics/Modern Physics CBSE/Class 11/Science/Physics

Energy - Mass Relationship

Vidyashakti Academy

Vidyashakti Academy a registered institution powered by Ex-IITian, empowers the students by teaching...

Do you know that whenever energy is added to the system, the system gains mass? Is there any example of conversion of energy to mass and vice - versa? Yes. A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within... read more

Do you know that whenever energy is added to the system, the system gains mass?

Is there any example of conversion of energy to mass and vice - versa? Yes.

A spring's mass  increases whenever it is put into compression or tension.  Its added  mass arises from the added potential energy stored within it, which is  bound in the stretched chemical (electron) bonds linking the atoms  within the spring.

Actually neither mass gets converted to energy nor energy gets converted to mass. Both are equivalent to each other. If you have some energy E, then you also have mass E/c^2. Similarly, if you have mass m, you have energy mc^2. So, when a high energy gamma photon converts to an electron and a positron in the presence of a nucleus, some books wrongly say that energy has got converted to matter. The right thing is: Total energy of the gamma photon, E= h nu has mass h nu/c^2 and is equal to 2 m c^2/sqrt(1-v^2/c^2) where m is the rest mass of the electron, and v is the speed of electron (positron) in the rest frame of the nucleus.

 

Let's look at the reverse example.

An atom is made up of number of protons, neutrons and electrons. Each element has a mass associated with it. So logically the mass of an atom should be equal to the sum total masses of all three elements multiplied by the number of such elements present in the atom. Surprisingly, it is observed that the mass of an atom is less than the sum total masses of all elements. Where does the mass go? This mass is converted in to Energy called Binding Energy of the atom and this energy keeps the protons bound together in a nucleus which otherwise should have dismantled in to the environment due to strong repulsive forces between protons. Thus mass is converted in to the energy.

 

There are many more such examples in the wonderful nature and physics does the job of discovering and explaining such phenomena. In short Physics is Life and Life is Physics.

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Asked on 17 Jul IIT JEE

Today in these days quality faculty for iit jee are mostly given qualitative basic concept.

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Lesson Posted on 12 Jul CBSE/Class 11/Science/Physics IIT JEE/IIT - JEE Advanced/Physics/Modern Physics IIT JEE/IIT - JEE Mains/Physics/Physics and Measurement

Importance of Units and Conversions

Vidyashakti Academy

Vidyashakti Academy a registered institution powered by Ex-IITian, empowers the students by teaching...

This is the foremost important physics topic. Students very often than not forget Conversion of Units of physical quantities to one system while solving numerical questions. This leads to wrong answers and failure in competitive exams. Such errors or failures also cause disaster in real life making the... read more

This is the foremost important physics topic. Students very often than not forget Conversion of Units of physical quantities to one system while solving numerical questions. This leads to wrong answers and failure in competitive exams. Such errors or failures also cause disaster in real life making the society pay a very heavy price.

 

Physics is fundamentally an empirical science. Measurements, units & uncertainties are at the heart of it. All laws of nature are derived empirically and expressed by mathematical equations. How otherwise scientists can express relationships between measurable physical quantities? It is equally vital to express the measurement in right unit; else invite disaster.

 

The Mars Climate Orbiter costing about US$ 125 million came to a catastrophic conclusion because of confusion in UNITS. One engineering team used Metric Units while another team used English Units. As a result in September 1999 the spacecraft entered Martian atmosphere instead of reaching a stable orbit.

 

Equally important is Uncertainties in measurements e.g. pair of pants of size 34 that varies just 3 percent changes a full inch in waist size. It could end up a 35 and hang on the hips or a 33 making you wonder how you gained all that weight!

 

Measurements, Units and Uncertainties are crucially important and when properly implemented can create wonders in Life (and also exams); else a disaster. Students should pay utmost importance to Units, Conversion of Units and Uncertainties.

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