Learning

If you’re currently studying the key equations which are used in the study of implicit differentiation, you’ve come to the right place. Simply continue reading a guide to understanding implicit differentiation as well as some of the most commonly used terms that you’ll come across in your study of implicit differentiation. An example of which is the term second differentiation.

What does the term implicit differentiation refer to?

The term implicit differentiation refers to a chain of rules which are used to work with derivatives. If you plan on studying or brushing up on your calculus skills, you’ll find that the bulk of exercises which you’ll work through feature y functions which are written as functions of x.

What is the d 2y dx 2 implicit differentiation?

Now that you’re aware of what implicit differentiation refers to, it’s time to learn about one of its key terms. Firstly, it’s important to understand what the equation above refers to. The equation d 2y dx 2 refers to what is referred to in mathematics as the second differentiation. If you’ve never heard of second differentiation, simply continue reading to find out more valuable information.

What exactly is the second differentiation and how is it used in mathematics?

The second differentiation is a method which you can use to figure out the nature of multiple of stationary points. Key examples of stationary points which you can figure out the nature of using implicit differentiation include maximum points, minimum points and points of inflection.

What are maximum points?

If you’re unsure about what the term maximum points refers to a circumstance when the gradient almost reaches its maximum point. In this scenario, just before the maximum point the gradient is positive and which is reaches its maximum it reaches zero. After the maximum point, the same gradient will be classed as being negative.

The term maximum point refers to the middle position where the gradient exactly reaches its maximum value.

What are minimum points?

Now that you fully understand what the term maximum points refers to, it’s time to learn about their counterparts which are called minimum points. Minimum points are positive and occur when a gradient has not met its maximum and has not reset to zero.

Remember that any gradient which reaches the maximum point or which surpasses the maximum point and become negative, is not an example of a minimum point.

What is an inflection point?

In order to better understand the process of using the second differentiation equation when dealing with implicit differentiation it’s also a wise idea to understand the term inflection point. What is an inflection point? An inflection point is any point which you can locate on a graph where a tangent successfully crosses a curve.

What does the term point of reflection refer to in mathematics?

The term point of reflection refers to a case where the derivative has reached a local maximum.

Why you should use a d 2y dx 2 calculator:

The quickest way to accurately calculate second differentiation problems is to use a calculator.

Hopefully you now have a clear understanding of implicit differentiation and its use in calculus.

If you’re interested in learning about trigonometric functions as well as more complex formulas such as the antiderivative of csc, simply continue reading in order to discover a handy, fuss free guide to understanding trigonometric functions.

What does the mathematical term csc stand for?

The term csc refers to the word cosecant. The csc or the cosecant is a trigonometry function. In a simple, right angle triangle, the csc of an angle = the total length of the hypotenuse, divided by the total length of the opposite side.

You may be interested in learn that out of the six main trigonometric functions that exist, the cosecant or csc is the least frequently used of the six functions.

What is the antiderivative of csc?

The antiderivative of csc refers to a commonly used mathematical formula which is used to solve a wide variety of basic and complex trigonometric functions. Where sin x is a trigonometric function, cos x will be your derivative. The formula which you will use when dealing with the antiderivative of csc is a cos nx dx = (a/n) sin nx + c. In this formula the term a refers to a non zero constant and the term c refers to the integral constant.

What are the other five trigonometric functions that exist:

1. The secant

In a regular right angled triangle the secant of a given angle is simply the total length of your hyptoneuse, divided by the length of your right angle triangle’s adjacent side. In mathematical formulas the secant is usually just referred to as the SEC.

2. The cotangent

The cotangent is referred to as the COT for short. The cotangent in a simple right angle triangle refers to the ratio of the side, which is not the hypotenuse and is adjacent to the side opposite your right angle triangle’s angle.

3. Cosine

Cosine is one of the three most used trigonometric functions and often shortened to the abbreviation COS. Cosine can easily be used to find the length of all of the lines and angles in a right angled triangle. Which makes Cosine a highly useful function to remember.

4. Sine

Sine is yet another trigonometric function and is used to find a specified angle. To use the sine method simply use the ratio of the length of one side inside your right angled triangle, that is located opposite to the side of your triangle which is the longest, in order to use sine. Sin is usually written in formulas simply as SIN, to make things a little easier.

5. Tangent

The last function which needs to be discussed is the tangent. The term tangent refers to a tangent line to a plane curve which at a specific point which the straight line touches the curve. As you may have guessed Tangent is normally expressed by the three letters TAN.

So if you ever need a refresher on the five functions of trigonometry it’s well worth referring back to this simple article to discover all the information which you’ll need!

If you’re looking for advice on how to accurately measure the diameter of a ball such as a basketball, a bowling ball or a tennis ball, simply continue to discover a fuss free, easy to understand guide on measuring the diameter of a ball, of any size. As a bonus, you’ll also discover how to work out the volume of a ball.

How to measure diameter of a ball:

In order to calculate the diameter of a ball use a tape measure to measure the circumference of your ball. To measure the circumference of your ball wrap your tape measure around your ball once.

Once you’ve written down the number of your ball’s circumference divide the number which you have by pi. If it’s been a few years since you’ve graduated from school, pi = 3.14. Once you’ve divided your circumference by 3.14, you’ll successfully have calculated the diameter of your ball.

As an example, if you measure a yoga ball and its circumference measures 67.71 inches, the diameter of the yoga ball which you’re measuring will be 21.65 inches.

How to measure the diameter of a circle:

If you’re not looking to measure the diameter of a three dimensional ball and are looking to calculate the diameter of a two dimensional circle, you’ll need to use an alternate formulaic equation to find the number that you’re looking for.

To accurately find out the diameter of a circle, first figure out the radius of your circle. The radius of a circle is the distance between the center point of your circle and the outside of your circle. Once you’ve found the radius of your circle, multiply the number which you have by 2 and then multiple the number which you get with pi which is 3.14.

If you’re curious as to why it’s a better idea to calculate the diameter of a circle by first finding the radius of your circle, instead of your circle’s circumference, the answer is that you should find it a lot easier to accurate measure your circle’s radius with a ruler than to try and measure the circumference of your circle.

The answer is that you should find it a lot easier to accurate measure your circle’s radius with a ruler than to try and measure the circumference of your circle. This method is used in making Pavers And Turf also

How to measure the volume of a sphere:

If you’re also interested in calculating the volume of a three dimensional sphere such as a ball, you can also use the following equation to figure out the volume of your ball. V = r. In this equation, V = volume and r = the radius of your ball or sphere.

Firstly you’ll need to find out the radius of your sphere. Next, cube your radius and multiply your cubed radius by 4/3. Now your equation should read V = . Lastly multiple your equation by pi which is 3.14 and you should get your answer and the volume of the sphere which you’re measuring.

So if you ever need to accurately measure the diameter of a sphere, such as a ball or a circle. Or you need to work out the volume of a sphere, it’s well worth using the simple equations listed above to find out the answers to your questions.

Whether you’re already slightly familiar with imaginary numbers and their purpose in mathematics or you’re a little confused or overwhelmed by the term, simply continue reading in order to discover a comprehensive guide to understanding and using imaginary numbers. Which are sometimes to referred to as complex numbers.

A guide to understanding imaginary numbers:

A simple definition of the term imaginary numbers: An imaginary number refers to a number which gives a negative answer when it is squared. So if you assumed that the term imaginary numbers would refer to a complicated type of number, that would be hard to wrap your head around, think again.

While imaginary numbers are sometimes referred to as complex numbers, they aren’t as complex as you may think. If you’re interested in further understanding imaginary numbers, they are a form of a complex number that can not be classed as a real number and which have simply been multiplied by an imaginary unit, which is always referred to simply as i.

Why are imaginary numbers useful?

Imaginary numbers are used in a wide variety of scenarios. As examples imaginary numbers are often used in advanced calculus and to solve advanced algebraic equations. Imaginary numbers are normally used to stand in for calculations that have to be worked out at a later stage of your equation solving.

In these scenarios you’ll often need to find further information out by performing other calculations, so that you can find the intended value of your imaginary number, which will always be expressed by the letter i.

Why is the unit when you are dealing with imaginary numbers important: When you’re dealing with the theoretical concept of imaginary numbers, the term unit is used to describe first term and is equivalent to how the numeral one, is the first number which exists.

In regards to imaginary units the formula for a single unit is squared root, minus one. However, in equations the term unit is more commonly referred to simply as the letter i. If you’re curious about why the letter i is used to denote the unit, the answer is that i stands for imaginary.

If a number is not an imaginary number, what could it be?

Numbers which aren’t imaginary numbers are defined as real numbers. However there is another subset of numbers which are classed as real numbers, which are referred to as complex numbers and which still feature imaginary numbers in their equations. One unique number is zero which is both an i number and a real number.

What is an example of an imaginary figure?

A simple example, which you can take a look at is 5i. The square root of which equates to -25.

In conclusion hopefully you found the information and answers that you were seeking and now fully understand what imaginary numerals are as well as what complex and real numbers are and how they all differ from each other. If you’d like to learn more information about imaginary numbers, make sure to refer to an accurate imaginary numbers chart.

To discover a fuss free guide to understanding integrals and using calculations which feature improper integrals, simply continue reading. As a bonus, you’ll also get to understand the major differences between improper integrals and their opposite, which are simply referred to as proper integrals.

What is an improper integral?

Whether you’re confused about what an improper integral is or you’re never come across the term improper integral, it may be helpful to be provided with a clear definition of an improper integral.

An improper integral, features a denominator which as a larger degree, then the degree of its numerator. Furthermore, an improper integral is a type of definite integral which has two infinite limits or which features an integrand which reaches infinity at one or more points in a defined range of integration.

It’s also worth keeping in mind that you can’t use a Riemann integral with improper integrals, like you can with other integrals.

What is an integrand?

If you’re unsure of what the term integrand refers to, in mathematics integrand refers to a function, which you will then integrate.

How you can use improper integrals in your calculations and equations:

One of the most common equations which features improper integrals, seeks to convert improper integrals into mixed numbers.

How to perform calculations which feature improper integrals:

While it’s certainly possible to perform calculations which involve the use of improper integrals manually, it’s far simpler to use an improper integral calculator to get quick answers and solutions to your problems.

As an example, you’ll easily be able to turn any improper integral into mixed numbers, in less than a minute, which is incredibly convenient and well worth experimenting with. Especially if you forsee yourself performing such calculations on a regular basis as if you use a calculator to turn improper integrals into mixed numbers, you’ll save yourself a lot of time and energy in the long run!

What is the opposite of an improper integral?

The opposite of an improper integral is simply referred to as a proper integral.

What is a proper integral?

A proper integral features a denominator degree which is less than the degree of its numerator. Remember to keep in mind that in mathematics the term denominator refers to the bottom figure in a fraction, while the term numerator is used to refer to the top number in a fraction.

How does a proper integral differ from an improper integral:

If you’re a bit perplexed by how a proper integral differs from an improper integral, while a proper integral has a denominator degree which is less than the degree of its numerator an improper integral has a denominator degree which is larger than the degree of its numerator.

Hopefully you now have a clear understanding of the use of the both improper and proper integrals and know how to go about using a calculator to solve any equations which involve improper integrals. If you ever forget the difference between improper integrals and proper integrals simply refer back to this handy page.

To discover everything you need to know about the element Barium, simply continue reading to discover a handy guide to Barium.

Everything you need to know about Barium:

Classification:
Is Barium a metal nonmetal or metaloid? Barium is classed as an Earth metal.

Atomic number:

Barium has the atomic number 56.

Symbol:

Barium’s atomic symbol is BA.

Appearance:

Barium is silver in color and is a soft, pliable metal. Barium has a tendency to tarnish quickly when it is exposed to air and also is known to react to water. When Barium is ultra pure it sometimes has a slightly golden tinge.

Typical uses:

Barium is sometimes used in drilling fluids, that businesses use that operate in the mining and oil industries. Less commonly Barium is also sometimes used as an ingredient in paint.

Barium is also used in the health care industry. A type of non toxic Barium, Barium Sulfate is occasionally prescribed to patients who complain of serious digestive issues.

Once ingested Barium Sulfate is processed by the patients’ stomach and intestines and shows up on x-rays and allows doctors to analyse a patients stomach and intestines as Barium Sulfate is a heavy element and shows up well in x-rays. You may be surprised to read that Barium has been used in x-rays since 1908.

Is Barium toxic?

Yes, Barium is toxic and all Barium compounds are classed as being toxic. Which is one reason why Barium is not used as commonly as other metals. However, Barium Sulfate which was mentioned above is not toxic and can be safely consumed.Barium Ion is often used in pesticide products as its so toxic.

How toxic is Barium to humans? Barium is extremely poisonous and even one gram of Barium can be lethal if its ingested by humans. Individuals who accidentally consume Barium are instructed to drink water in order to flush the barium which they have ingested out of their system.

Does Barium have a known biological role?

No, Barium doesn’t have a clear, known biological purpose. However, Barium Sulfate can be found in one species of algae.

Is Barium ever found with other compounds?

Yes, in fact Barium is only found with other compounds. As examples Barium combined with Barite is called Barium Sulfate and Barium which is combined with Whiterite is known as Barium Carbonate.

How is Barium created?

There are two known ways to produce Barium. The first way to produce Barium involves performing electrolysis on Molten Barium Chloride, while the second method of producing Barium entails heating Barium Oxide with Aluminium Powder.

When was Barium first discovered and who is attributed with discovering Barium?

Barium was first discovered by the scientist Humphrey Davy in 1808. When Davy managed to isolate Barium by utilizing electrolysis. Another scientist Carl Scheele identified Barium in 1774 but Scheele is not credited with discovering Barium as he was unable to succesfully seperate and isolate Barium.

How did Barium get its name?

Barium stems from the Greek word barys which roughly translates to the English word heavy.

Hopefully you now have a clear understanding of the properties, uses and dangers of the solid metal element Barium.

If you’re confused about whether negative numbers and decimal figures can be rational numbers, continue reading to demystify both types of numbers!

What is the key difference between rational numbers and their irrational counterparts?

Before you’re able to figure out whether a given number is a rational number or an irrational number, it’s important to be able to tell the key difference between both types of numbers.

Rational numbers are figures which can be easily written down as a whole fraction. Whereas their opposite, irrational numbers can’t be written down as a simple whole fraction.

You may be surprised to read that both decimal figures and negative figures can be classed as irrational numbers as well as rational numbers. Remember that if a negative number or a decimal number can be written down as a whole fraction, you’re dealing with a regular figure.

Are negative numbers rational?

As mentioned briefly above negative numbers can be rational or irrational, to figure out whether your negative number is rational, go ahead and try and turn your negative number into a whole fraction.

Another tip which you may want to use in order to tell if the number that you’re looking at is rational or irrational, check whether your negative number’s positive equivalent is a rational number. As if a negative number’s positive counterpart is negative, there’s a high likelihood that the number that the negative number that you’re trying to place is also an irrational number.

Is negative 1.5 a rational number?

If you want to have a go at classifying a negative number as a rational number or an irrational number, you may want to start off with an easy example, that is simple to sort such as -1.5. If you’re stuck, just make sure to try and write down -1.5 as a whole fraction. If you can successfully write down -1.5 as a whole fraction which has a whole numerator and a whole denominator, then -1.5 is a rational number. However, if you can’t turn -1.5 into a whole fraction, then -1.5 is an irrational number, not a rational number.

Examples of negative irrational numbers:
If you find it helpful to look at a list of examples of negative irrational numbers, in order to more easily be able to classify negative numbers, the following numbers are all negative irrational numbers, -2, -13 and -8.
Examples of decimal numbers that are irrational:

Just as negative numbers can be irrational, decimal numbers can be irrational too. One prime example that is great to keep in mind is the number pi. Pi is an infinite number, which doesn’t have reoccurring digits and which therefore is an irrational number. As there is no logical way that you can turn an infinite, endless number which has no end into a neat whole fraction.

Just be careful when you’re classing numbers not to automatically assume that decimals are always irrational as decimals which recurring numbers are almost always rational numbers.

If you’re ever confused about negative numbers or decimal numbers and whether they are rational or not, simply refer back to this handy article for answers.

If you’re interested in studying statistics and are interested in learning about the definitions of key statistical terms such as least to greatest, mean, mode, modal and median, simply continue reading to discover a guide to commonly used statistical terms. That will help you solve simple and complex statistical equations with ease.

Least to greatest definition:

In statistics data is always always arranged from least to greatest, which will always allow you to easily find the mean, median and mode of data.

The process of rearranging data from least to greatest is referred to simply as ordering. So if you’re ever asked to order your data while completing a statistical exercise, simply order your data from least to greatest.

What does the term mean refer to?

In the study of statistics, the term mean refers to the answer which you’ll get if you add all the items in your list and then divide your answer by the number of items in your list.

What does the term mode refer to?

The term mode refers to answer which you’ll get if you look at the items in your list and choose the value which appears in the highest frequency. Another word for mode is the modal value, so don’t feel thrown off if you see the term modal referred to in a statistics question.

If it’s not easy to find an answer by simply looking at your list of numbers, you can make things a lot easier for yourself by creating a simply tally chart.

When you create a tally chart, make sure that you have a unique column for each different value which is represented in your list. Then simply add a mark in each column each time that a specific value occurs in your list, to find your mode.

What does the term median refer to?

In statistics the term median refers to the middle number, in a series of numbers that has been sorted from least to greatest. If there are two numbers in the middle of your list, simply find the average of your two numbers, which can be done by adding both of your numbers together and then dividing your total by two.

What does the term range refer to in statistics?

It may be useful to remember that your list of numbers which range from least to greatest is always referred to as your range. Making sure to recall statistical terms such as range, mode, median, mean is crucial to be able to understanding statistical problems which you’re given.

As half the difficulty of being able to complete complex problems is being able to clearly understand what each problem is asking you to do. As if you understand each key term, you’ll know what functions and operations to use to solve your problems.

In conclusion:

Hopefully understanding the satistics terms which are listed above which include least to greatest, mean, median, modal, range and mode, will help you better understand the statistics equations which you’ll come across in your statistics studies!

The Lewis base list is commonly referred to in the study of chemistry and contains substances which donate an electron pair which is then shared by the base and the covalent bond. To learn more about some of the bases, which are contained in the Lewis base list, simply continue reading to discover everything you need to know about the Lewis base list.

Everything you need to know about the Lewis bases:

1. Most types of oxygen and nitrogen can act as Lewis bases
If a type of oxygen or nitrogen feature organic compounds, the oxygen or nitrogen type in question can be used as a Lewis base as it will feature a pair of electrons.
2. Atoms, ions and molecules can all act as Lewis bases
You may be surprised to read that atoms, ions and molecules can all be used as Lewis bases.
3. For an atom, ion or molecule to be used as a Lewis base, there also needs to be an atom which will accept a Lewis bases’ pair of electrons
If an atom, ion or molecule isn’t able to donate its pair of electrons to another atom, which will accept them, the atom, ion or molecule in question won’t become a Lewis base.
4. You can find out whether or not an atom, ion or molecule is by drawing the Lewis structure
If you’re curious about how you can accurately discover whether a particular atom, ion or molecule is a Lewis base, you can draw the Lewis structure in order to see whether you can find a pair of electrons which can be used to form a bond to another atom. If you can identify a pair of electrons which can be donated to another atom, the atom, ion or molecule which you’re researching can in fact act as a Lewis bond.
As an example, ammonia may be able to be used in a Lewis base if it’s able to donate a pair of electrons to the element H+, in order to form a bond with the element H+.
5. In organic chemistry compounds with pi electrons are usually Lewis bases
If you study organic chemistry, you’ll discover that the vast majority of compounds which boast pi electrons can be classified as Lewis bases.
6. Some compounds can be used either as a Lewis base or a Lewis acid
To further complicate things, some compounds, including atoms, ions and molecules actually have the potential to turn into either a Lewis base or a Lewis acid. As an example, most forms of alcohol act as acids when they donate H+ but can also act as a Lewis base when their oxygen atom decides to accept H+.
So if you’re interested in learning further information about Lewis bases, it’s well worth researching further types of compounds which can be turned into Lewis bases and Lewis acids.
So if you were confused about how compounds such as ions, molecules and atoms can be used as Lewis bases, hopefully you now have a clear understanding of Lewis bases.

In order to discover how you can use the radial acceleration formula in order to perform complex calculations about items which move in a circular motion, simply continue reading to discover a guide to radial acceleration.

What does the term radial acceleration refer to?

Whether you’ve heard the term radial acceleration before and were confused about its definition or you’ve just come across the term radial acceleration, you may be curious about the definition of radial acceleration.

Simply put, radial acceleration refers to the speed of any moving object, which has been altered, sometimes by the moving object rapidly changing direction. Radial acceleration typically measures acceleration of a moving object, which moves in a circular motion or pattern.

What are the units of measurement that are used to measure radial acceleration?

The units of measurement, which you’ll need to use in order to be able to measure radial acceleration include radians per second and meters per second squared.

What is a simple definition for the term centripetal acceleration?

The term centripetal acceleration is also referred to sometimes as radial acceleration and describes a moving object which continues to move in a circle and which has an acceleration which is directed to the middle of its circle.

What are the units of measurement that are used to measure centripetal acceleration?

The units of measurement that you would use in order to accurately measure centripetal acceleration include meters per second squared. Which happens to be the most used unit of measure to measure centripetal acceleration.

How to calculate the radial acceleration:

In order to calculate the radial acceleration, you’ll need to use the radial acceleration formula, to get an accurate answer. The radial acceleration formula involves dividing the velocity by the given radius.

Why does radial acceleration occur:

If you were wondering why radial acceleration occurs, the simple answer is that radial acceleration occurs when there is a change in the direction of the velocity.

What types of situations would you use a radial acceleration formula in?

Any time that you seek to measure the acceleration of an item which is moving in a circular pattern such as a merry go round, you can’t go wrong using the formula velocity divided by the given radius.

Another key example of a moving object that continuously moves in a circular pattern is planet Earth, which moves in a radial pattern around the sun and which can also be described as being directed by the centrifugal force, which in this case is the sun.

The Earth moving around the sun as an example of radial acceleration:

If you would like to measure how fast the Earth moves around the sun, you could use the acceleration formula which you’ve read about above, to find out. Assuming of course that you’d have all the numbers which you’d need to complete your calculation.

If you’re curious, the radius of planet Earth’s continuous orbit is 149.6 million kms and the Earth’s current rate of velocity is approximately 29800 meters per second. So the answer which you should get to your maths problem is 0.00593 meters per second squared.

Hopefully, you now have all the information which you need in order to be able to solve complex radial acceleration problems on your own.