Finding Limits In Calculus – Follow These Steps

Finding limits in calculus isn’t easy. We’re here to help! Using these simple methods, we’ll be able to find the limit quickly no matter the situation.

If you’ve been asked to find a limit by your teacher, there are a range of different methods that you can use. It’s much simpler than it sounds and at the end of this guide, we have a nifty strategy that you can follow so that you’ll always know which method to use and when.

Finding limits isn’t easy, and a lot of people struggle with it. If this is you, don’t worry, by the end of this guide, we’ll have you finding limits in a few minutes at most. Read carefully and try to follow along, but don’t be afraid to start from the beginning to perfect your technique and to memorize each of the strategies properly.

Why Would You Want to Find Limits?

Limits are incredibly essential, and without them, we would be unable to do more advanced forms of calculus. A limit is the limit of a function f(x) as x approach c but never reaches it. Remember, x can approach c from either side. Picture a graph; it can come from either side of the axis.

Limits allow us to find out how a function will behave even if it doesn’t exist at a specific value of x. The result is that finding limits will allow you to derive the angle of a slope at a given point, even if you don’t have a specific value of x for every point along the line. Without knowing how to find limits, we would have little information about the gradient between points.

If we take the function f(x) = x – 1 / x – 1 and then imagine that x could be any number.

We know that if x = 1 the function would look like this:

f(1) = 1 – 1 / 1 – 1 which would equal 0 / 0.

Pen

Image by Edmund Fung from Pixabay

The result is that when x = 1, the function itself is undefined because the fraction 0 / 0 is undefined. On a graph, this would look like a straight line across, parallel to the x-axis, but there would be a gap where x = 1 because it’s just not defined. But what if we wanted to know what the function was when x = 1?

Well, we can’t do it. But what we can do is to get as close as possible to x = 1 so that we can know approximately what the value of the function is at that point. This idea is necessarily a limit. It’s the idea of being able to get as close as possible to an undefined point so that we can approximate it with a high level of accuracy.

Finding the Limit by Plugging in X

The first technique that we’ll look at is plugging x into the function to see the limit. In an ideal world, this would work all of the time. Therefore, we always start with this technique because it’s the simplest and allows us to get more information about what to do next. The idea is that you make x equal to the number it ’s approaching.

So, if we are trying to find the limit as we approach 2, we make x = 2 and then run the function.

When you do this, you’ll get one of three results:

  • f(a) = b / 0 where b is not zero.

  • f(a) = b where b is a real number.

  • f(a) = 0 / 0.

In the first circumstance, you’ve probably found an asymptote. An asymptote is when a line continually approaches a given value, but it will never reach it at any finite point.

In the second situation, you have probably found the correct limit through the substitution method.

Finally, in most complicated questions you will end up with a situation where the function is undefined, and therefore you’ll need to try other techniques. If this is the case, we will need to rearrange the function so that we can consider the limit in an identical but differently arranged form using one of the following three techniques.


 

Factoring Method

Factoring is a great method to try and is often one of the easiest to learn because it relies on skills that you’ve already practiced. If you’ve already tried to plug in a number have ended up with 0 / 0, you need to start factoring.

Often you’ll see that either the numerator or the denominator is more ‘friendly’ to factoring. Usually, x with the highest power is the best place to start. Let’s consider the following equation:

x^2 – 6x + 8 / x – 4 where x is approaching 4.

In this example, the numerator is the only place for you to factor. It’s also obvious because of the x^2 which can factor. In this case, we can factor to:

(x – 4)(x – 2) / (x – 4)

As you can see, we can then cancel the two matching x – 4 on both the top and the bottom. Pretty simple, right? It won’t also be this easy, but if you continue to factor you can often find places to simplify the expression.

This simplification leaves us with:

f(x) = x – 2 where x is approaching 4.

If we try to substitute 4 into the equation now, you’ll find the f(x) = 2. See, by factoring you’ve shown that the equivalent function has a specific value and that value is 2 when x is approaching 4.

If you were to create a graph of this function, you would still see a gap where x = 4 because the original equation is still undefined. However, you’ll know that when approaching 4, the function equals 2.

After factoring, you might find that there is no way for you to cancel and simplify. In this case, you should try another method to ensure that there is no limit of the function at the specific value of x.

Rationalizing the Numerator

The third technique requires you to rationalize the numerator so that you can try substitution again. You’ll know if you should rationalize the numerator because you’ll see a square root on the top and a polynomial expression on the bottom. Let’s look at the following example:

f(x) = sqr(x-4) – 3 / x – 13 as the function approaches 13.

We know that substitution fails when you get 0 in the denominator, and therefore substitution would fail in this example. Factoring would also fail because there is no polynomial to factor in this example.

However, if you were to multiply the numerator and denominator by the conjugate of the top (numerator), then you’ll be able to cancel and find the limit.

The conjugate of the numerator is: sqr(x – 4) + 3 and therefore we can multiply through to get:

(sqr(x – 4) – 3)(sqr(x – 4) + 3) / (x – 13)(sqr(x – 4) + 3)

We can then FOIL the numerator to get the following:

(x – 4) + 3sqr(x – 4) – 3sqr(x – 4) – 9

When simplified the above expression will become x – 13 because the middle terms cancel and then you can combine like terms.

If we go back to the full equation you can now see that we have:

(x – 13)  / (x – 13)(sqr(x – 4) + 3)

The terms cancel, and we have:

1 / (x – 13)(sqr(x – 4) + 3)

From there, we can plug in 13 into the function because we have all of the unknowns on one side of the fraction. The result is that the limit is ⅙.

Trig Identities

So far we’ve only looked at situations which don’t include any trigonometry. These require unique methods like factoring and conjugates to ensure that you can simplify and be able to easily plug in a number for x to find the limit. We want to do the same with this equation, but it contains trigonometry which complicates things a little.

For you to solve these equations, it’s vital that you know all of the trig functions so that you can rewrite equations and more effectively address them.

The most common are as follows:

Cos (x) = 1 / Sin (x)

Sec (x) = 1 / Cos (x)

Cot (x) = 1 / Tan (x)

It’s highly likely that you’ll also need to know the double angle identities in order to simplify more complex functions.

Sin (2a) = 2Sin(a)Cos(a)

Cos (2a) = Cos^2(a) – Sin^2(a)

Tan (2a) = 2Tan(a) / 1 – Tan^2(a)

This equations might seem confusing, but they are actually very simple. They are each used for different purposes, but when finding limits we only need to know them for rewriting equations.

Let’s look at the following example:

Sin (x) / Sin (2x) when x is approaching 0

We can use the double angle identities formula to simplify to:

Sin (x) / 2Sin(x)Cos(x)

From there, the Sin(x) can cancel and we are left with:

1 / 2Cos(x)

If we plug in 0 as x, we will get ½ because cos(0) = 1 and therefore you have 1 / 2*1 which is ½.

The Strategy to Finding Limits in Calculus

Now that we’ve covered all of the tactics that you can use to find limits let’s discuss which you should use and when. There is a straightforward rule. You should always do a direct substitution first.

If you get f(a) = b / 0 then you have an asymptote.

If you get f(a) = b then you have a limit.

If you get f(a) = 0 / 0 then you should try factoring, rationalizing the numerator or trig identities depending on which seems most likely to work.

 
–>

Understanding Linear Functions in Calculus

Introduction Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs. A simple function can be as follows: [{1, 2}, {2, 4}, {3, … Continue reading "Understanding Linear Functions in Calculus"

Introduction

Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs.

A simple function can be as follows:

[{1, 2}, {2, 4}, {3, 6}, {4, 18}, {5, 10}]

The above function has five pairs where the first members are 1, 2, 3, 4 and 5.

Functions usually have alphabetical letter as their names. So if we term this function ‘f’, which is the most common letter used for functions, then it will be properly written as:

f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10

Here are two definitions to keep in mind:

The entire set of first numbers in the function is called a domain and the first members are called arguments. In this particular example, the domain has 5 numbers and the numbers 1, 2, 3, 4 and 5 are the arguments of the function.

The whole set of second numbers in the function is called the range and the second members are called the values. Going back to the above function, the range also has 5 numbers and the numbers 2, 4, 6, 8 and 10 are the values of the function.

As mentioned before, the standard naming of a function is f. Thus we can explain this function in a sentence as follows:

The value of the function (f) at argument 1 is 2, its value at argument 2 is 4, its value at argument 3 is 6, its value at argument 4 is 8 and its value at argument 5 is 10.

Therefore a function can also be defined as a set of assigned values (the second numbers) to arguments (the first numbers)

This can be expanded further to say that the condition is that the first member of every pair is different; therefore each argument of the domain of function ‘f’ gets an exclusive value in its range.

The linear function and its importance to calculus

The linear function is the basic and essential function, on which calculus is based upon. This is a function that has a straight line running through the domain of its graphs.

Such a line can be determined by two points that lie on it. Look at the function [a, f{a}], [b, f{b}]. You can pick an “a” and “b” in the domain and determine this line defined by the two values f{a} and f{b}.

Let’s look at the formula for such a function.

It is possible to determine the linear function for the two values mentioned above by using the following formula.

f{x} = [f{a} x – b/ a – b] + [f{b} x – a/ b – a ]

Effectively this means that the first term is 0 when x is equal to b, and it becomes f{a} when x equals a. The second term is 0 when x is equal to a and it becomes f{b} when x equals b.

Another important aspect of a linear function is its slope.

This is defined as the ratio of the change of function f between x = a and x = b the change in x between the two arguments. The y-intercept is the point at which the line passes the y-axis.

The intercept of the line on the y axis is also an essential part of the linear function.

As we have seen, a linear function can be defined one that has a graph with a straight line, and can be described by its slope and y-intercept.

Special linear functions are often useful and they all have an important and unique property – they all have linear functions whose y-intercepts go through the point 0. Their graphs pass through the origin of the x and y axes. They are aptly called homogenous linear functions, and they all share the same property which is:

Their value at any permutation of two arguments is equal to the same permutations of their values at those arguments.

This can be explained by the following formula:

F{ak + bc} = af{k} + bf{c}

The above property is called the “property of linearity”.

NOTE: not all linear functions have this property of linearity. The property implies that once you know the value of a linear function and any two distinct arguments, then you can find the value at any other point or pair of arguments. This is not always true.

Practical applications of the linear functions

There are several real life applications of calculus linear functions. Remember that this is the most basic function on which other functions are based upon. The function is applied in various fields, such as meteorology, pharmaceuticals, engineering, and a lot more.

Whenever you have to create a graph in a straight line, no matter what the slope or y-intercept is, you are applying this basic principle.

NOTE: One should not confuse linear functions in calculus to linear equations in algebra. They have different properties even if sometimes their graphs can be identical. You can find a graph for a linear equation of algebra having the same slope and y-intercept as a graph for linear function of calculus, but they do not

Conclusion

Starting off by understanding this basic formula of calculus will make it very easy for you to move on and understand the deeper functions or integration and differentiation. Calculus should not be a behemoth to be feared but a friend to be understood. Try out some basic exercises on the linear functions in calculus and you will get a better grip on the topic.

The video may take a few seconds to load.Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.

How To Factor Cubic Polynomials In Calculus

An important part of learning how to factor cubic polynomials in calculus and other forms of math is learning how to simplify.

Polynomials are algebraic expressions that have a higher degree than the standard x + 3 or y – 2. A polynomial refers to anything with a degree, or highest exponent, above 2, but usually means at least 4. Some of the polynomials you’ll see most often are cubic polynomials or expressions with a cube as their highest variable.

Often in higher math such as calculus, you’ll be asked to solve these expressions, However, if you’ve been working with higher math for any time, you know that the more complex that a math problem is, the more difficult its variables and solutions will be. One way, and often a mandatory step, in solving cubic polynomials is to factor the expression.

We’ll go over some ways to factor cubic polynomials, but first, let’s review what these are.

 

What is a Polynomial?

 

A polynomial is any expression that has multiple terms in it. Terms are variables or numbers such as x2, 3x3, and 5. In algebraic expressions, you don’t know what the variables represent so you can’t add them. You can only add or subtract terms that have the same degree and the same variable. For example, you could subtract 5x from 8x, but not from 8y.

Therefore, an important part of learning how to factor cubic polynomials in calculus and other forms of math is learning how to simplify.

 

Simplifying Cubic Polynomials

 

The first thing to remember is that not all polynomials can be factored. Many if not most of them, can, but there are a few that have no roots at which x is equal to zero.  However, most polynomials can be simplified into a single expression multiplied by a quadratic expression. For example, you might see (x2 – 2x +4)(x + 3).

It’s even possible that the quadratic equation can factor further, but we’ll get to that later. The first step to factoring a cubic polynomial in calculus is to use the factor theorem.

The factor theorem holds that if a polynomial p(x) is divided by ax – b and you have a remainder of 0 when it’s expressed as p(b/a), then ax – b is a factor. It’s a roundabout way of saying that if an expression divides evenly into a polynomial, then it follows that the expression is a factor.

One way to factor is to set the expression to equal 0, and then substitute various values of x until the equation is satisfied. Once you do that, you can determine that one of the factors is (x – whatever the number is. If it’s a negative, the expression would instead be x + the number. Subtracting a negative is the same as addition. Bear in mind that this is for only one factor.

The degree of an expression directly indicates how many factors it has. An expression leading with x2 has two factors.

Factoring with a Constant

 

If your polynomial contains a constant – that is, not a variable – you can sometimes factor it using that number. If the constant has no factors other than itself and one, this makes your job a little harder, but not impossible. The defining factor as to whether this is a solution is whether setting the expression equal to 0 results in a true statement.

To start, rewrite the expression as an equation that equals 0. Then, look at your constant. Start by taking the first factor of the constant, which is always going to be 1. For example, if you have the expression x3 – 3×2 – 10x + 24 = 0, you can assume that the factors are +-1, 2, 3, 4, 6, and 12.

Check both positive and negative results, because negative numbers require opposite signs and any of them could be the solution. Place each term in the equation one by one until you get as many true statements as possible:

  •  

    (1)3 – 3(1)2 – 10(1) + 24 =/= 0

  •  

    (-1)3 – 3(-1)2 – 10(-1) + 24 =/= 0

  •  

    (2)3 – 3(2)2 – 10(2) + 24 = 0

  •  

    (-2)3 – 3(-2)2 – 10(-2) + 24 =/= 0

  •  

    (3)3 – 3(3)2 – 10(3) + 24 =/= 0

  •  

    (-3)3 – 3(-3)2 – 10(-3) + 24 = 0

  •  

    (4)3 – 3(4)2 – 10(4) + 24 = 0

  •  

    (-4)3 – 3(-4)2 – 10(-4) + 24 =/= 0

  •  

    (6)3 – 3(6)2 – 10(6) + 24 =/= 0

  •  

    (-6)3 – 3(-6)2 – 10(-6) + 24 =/= 0

  •  

    (12)3 – 3(12)2 – 10(12) + 24 =/= 0

  •  

    (-12)3 – 3(-12)2 – 10(-12) + 24 =/= 0

You might be wondering about checking negative factors for a positive constant. In a cubic polynomial, this is impossible because you have three possible factors. Only three positives can result in a positive result.

 

Uses of Polynomials

 

When learning how to factor cubic polynomials, it helps to think of real-life applications. One of the most pervasive in modern life is the way our electricity functions. Alternating current, which powers our homes, constantly fluctuates in voltage and current. The function used for this is described as a sine wave, which is expressed as a polynomial.

Any type of curved function, such as the curve of a roller coaster, is expressed as a polynomial. Most often, you’ll have computers to help graph the difficult equations, but knowing the basics can help you understand the concepts.

 

Final Thoughts

The methods we’ve listed are just a few ways you can solve cubic polynomials. Note that not every expression can be factored. For example, if you have a polynomial with no solutions when you attempt to solve for zero, you can conclude that it never touches the x-axis. The graph of a cubic polynomial, however, may have three possible solutions, or two places where it curves.

 

Area Under Curve In Calculus: How To Find It

The area under curve in calculus can be determined through the use of a specific formula that uses both definite and indefinite integrals. Learn it here!

How To Find Area Under Curve In Calculus

Figuring out how to find the area a under curve in a graph can be a simple process once you understand the formula and the surrounding pieces of information given in the problem. Finding the area is part of integration mathematics, and by using the appropriate formula, we can calculate not just the area, but any given quantity.

A typical graph has an x-axis and a y-axis, and when you add a curve to this structure, you’ll immediately see where the area under the curve lies. By finding the points along the curve, we can input them into the formula for finding the area under a curve, and solve for the final answer.

What Is Calculus?

Calculus is one kind of mathematics and deals with finding different properties of integrals of functions and other derivatives. To see these values, specific mathematical methods are used that are based on the sums of differences which are infinitesimal.

There are two primary types of calculus, which are integral calculus and differential calculus. Overall, calculus can be summed up as a system of calculating and reasoning different values as they undergo constant change.

These two kinds of calculus are connected to each other by a theory called the fundamental theorem of calculus. Both kinds of calculus use notions such as infinite sequences, convergence, and infinite series as it applies to limits that are well defined.

The calculus that is taught currently is credited to the 17th-century mathematicians Isaac Newton and Gottfried Leibniz. Calculus today is now widely used in other disciplines such as economics, engineering, and science.

Calculus is now part of the modern curriculum taught in most schools, and it can serve as a gateway to more complicated mathematical learning, and other studies regarding limits, functions, and additional mathematical analysis.

It’s possible that you may have also heard calculus referred to by other names such as infinitesimal calculus, or “the calculus of infinitesimals. Calculus has also been employed as a name for different kinds of mathematical notation such as Ricci calculus, lambda calculus, propositional calculus, process calculus, and calculus of variations.

Calculus dates back to ancient and even medieval times where it was used to develop different ideas surrounding calculations such as area, volume, and early ideas about limits. Calculus has been used to discover different values such as the area of a circle, and the volume of a sphere.

Definite Integrals And Finding The Area Under A Curve

What Is The Arclength Formula In Calculus

If you want to know about arc length formula then this guide is for you. In this section we will discuss how to find the arc length of a parametric curve.

Finding the distance from one point to another is critical for a variety of reasons. On a straight surface, this is incredibly easy to achieve. But what do you do with a curve? Well, that’s what mathematicians tried to figure out for many centuries. Fortunately, this is now a solved problem, and it has allowed us to advance our knowledge of calculus significantly.

From biblical times, the leading figures in science have wrestled with the issue. Some, like Archimedes, even came up with reliable solutions that could give good estimations, albeit not equal to the correct answer. Building upon his and others new theorems, later mathematicians have been able to make improvements and additions.

But it wasn’t until relatively recently, at least in terms of mathematics, that we figured out precisely how to calculate the length. Even as late as Newton, it still wasn’t completely fleshed out. Fortunately for you, we now have a simple and easy to understand formula to relate to and use to calculate distances of any length.

In this guide, we’ll look at what the formula is. We’ll also discuss why it’s important to know and how Fermat and other mathematicians throughout history derived it. By obtaining it, you’ll be able to understand the equation more fully and therefore, should be able to apply it with a higher degree of success and comprehension.

Why Do We Care About the Arclength?

If you want to do any advanced calculus, it’s critical that you can identify the length of a curve. In many cases, curves are infinitely long, and therefore your goal might be to find the distance between two values of x or y such that you can solve or implement the number into further equations.

In a practical sense, this formula could be for engineering. Imagine creating a bridge with a given height and span. You’d need to know precisely how long the metal should be before it’s going to bend into the specific curvature that you need to support the span and keep it upright.

Without this formula, it would be incredibly hard to figure this distance out. By using it, companies can save time and wasted materials, allowing them to calculate to within inches the length of piping or metal that they need. Like much of mathematics, there are often efficient uses that you may not think of if you don’t look into it.

Although most of us will never do this type of engineering, this formula was groundbreaking at the time of discovery. Without it, we would be unable to advance and do further calculations. It solved a roadblock that has allowed us to discover more advanced functions and theorems that the leading scientists are still using today.

Students often ask what the point of the formula is. You could ask the same about the Pythagorean theorem. But as you’ll see when we derive the arclength below, without Pythagoras is would be impossible for us to have discovered how to calculate the length of a given curve or arc.

Truthfully, what most people misunderstand about mathematics is that it’s a story. Teachers and students alike make the mistake of jumping forwards and backward and then complaining that it’s difficult and they can’t understand it. It’s necessary to step to step, in a logical order that the leading mathematicians took so that you can follow every part of the story.

For example, if you don’t understand graphing, you cannot understand calculus. Without algebra, you won’t grasp graphing and so on. Therefore, if you’re struggling with arcs, go back to graphing and keep going back until you get to a point where you’re comfortable. Then, work forward and don’t skip a step. Fix the foundations so that you have a better understanding.

What Is The Arclength Formula?

The arclength formula allows us to figure out the distance between two points on a curve that isn’t straight. We can also do the same for a circle or different shape. But the primary use of this equation is to solve for distances between irregular and complex curves. Here’s a quick history of the formula:

  • It’s been worked on for centuries

  • Early mathematicians like Archimedes had crude alternatives
  • Fermat and another mathematician simultaneously were working on the problem
  • Femat found fame for his discovery

  • Newton and others made improvements later

Deriving the Arclength Formula

Imagine that we have an arc on a graph, whereby the gradient is changing at a constant rate to create a smooth curve. How would we go about finding the length of the arc? This problem is what led to the derivation of the arc length formula which allows us to calculate it.

Pierre de Fermat, arguably the greatest mathematician in history, was the one who discovered and popularized this method. His work built upon that of Archimedes and other mathematicians before Fermat’s time. His idea was that by taking a tangent, you could create a formula for the line and then use small segments to estimate the length of the curve.

If we take two points, x0 and x1, on the curve, we can create a right-angled triangle to connect them. The longest side of the triangle will not perfectly follow the curve, but it’s close enough to get a reasonably accurate estimation for the length of the curve between the two points. Where S1 is the distance on the curve and x and y are points on the curve respectively.

S1 = √ (x1 − x0)2 + (y1 − y0)2

In this example, we use the mathematical symbol  Δ (delta) which means the difference between two values, in this case, x0 and x1 and y0 and y1.

S1 = √ (Δx1)^2 + (Δy1)^2

Imagine that instead of just using one large triangle, we used a lot of small ones. The result would be that we could get the longest side on each triangle closer to the correct path of the arc. Therefore, we will be able to get a much closer estimation of the true length of the curve on the graph.

S2 = √(Δx2)^2 + (Δy2)^2

S3 = √(Δx3)^2 + (Δy3)^2

Sn = √(Δxn)^2 + (Δyn)^2

Rather than having a bunch of equations, we can simplify them all into a single formula. By doing so, we can look at the curve in its entirety by segmenting it into an infinite number of infinitely small segments.

Arclength Formula

S ≈  Σ √(Δxi)^2 + (Δyi)^2

      i=1

This formula above looks complicated, but it’s incredibly simple. All it’s saying is that the length S is roughly equal to the sum of all of the longest sides of the triangles, where we use n number of triangles. The problem is that it will take us years to add up all of those numbers! Hmm, there has to be a better way.

Fortunately, there is, and this is what Fermat discovered and why he’s remembered as such a talented mathematician. This calculation seems simple now because we can find it online, but this discovery was groundbreaking and change the direction of the universe for good. Without his genius, we wouldn’t be able to do much of the engineering we do today.

The solution is to make all of the Δxi in the formula be the same, so we can remove them from inside of the square root and create an integral.

To achieve this, we need to divide and multiply Δyi by Δxi, leaving us with:

S ≈  Σ √(Δxi)^2 + (Δxi)^2(Δyi/Δxi)^2

      i=1

Next, we must factor out the (Δxi)^2:

S ≈  Σ √(Δxi)^2 + (1 + (Δyi/Δxi)^2)

      i=1

Take (Δxi)^2 out from inside of the square root:

S ≈  Σ √(1 + (Δyi/Δxi)^2) * Δxi

      i=1

We now that as n approaches infinity, we get closer to an infinite number of slices over a given distance. Therefore, the segment that each cover gets smaller, approaching an infinitely small size. As a result, we can get a more accurate estimation of the length of the arc, to the point of it becoming equal rather than approximate.

S =  lim Σ √(1 + (Δyi/Δxi)^2) *  Δxi

N→∞    i=1

Now, we have an integral. Typically we replace Δx with dx and likewise with dy. The reason for this is to symbolize that we are approaching a zero width and therefore can assume equality.

S =  lim Σ √(1 + (dy/dx)^2) *  dx

N→∞    i=1

We know that dy/dx is the derivative of the function f(x) and therefore you can substitute that into the formula if you wish. However, for most people, it’s easier to remember the above the arclength formula so that you can more easily solve the equation and use it for calculations.

In a few short steps we’ve been able to derive the formula so that rather than needing to add up a bunch of slices for an approximate answer, we can calculate a precise number. The only thing you need to do is to solve the differential and integral for the curve or arc which will allow you to find the distance and also other figures like space under the curve.

It’s important to remember that the integral will also work concerning y. This fact is useful if we already know that x = f(y) and therefore can solve.

Featured Image by Gerd Altmann from Pixabay  

The video may take a few seconds to load.

Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.



How To Find The Square Root Of X Graph

How to find the square root of x graph is easier for some students than many other pre-calculus subjects because visual representation is involved.

Image source: Pixabay.com

You’ll encounter the square root of x graph in your calculus and geometry classes. Graph-related work is easy to understand even if you don’t have much-advanced math experience. Draw graphs to accompany equations to help you find answers without a struggle.  

Learn all you can about square roots before using graphs to determine solutions to assigned problems. A square root is a number multiplied by itself. Square roots are used to take measurements of spaces and buildings. There are also many other scientific and real-world uses for square roots.  

What are Square Roots?

Square roots are indicated by putting a two (commonly called squared, or multiplying the number by 2) above the number you’re multiplying.

If you calculate a number’s square root, you need to find the number that was squared in the first place. For example, three is the square root of nine. Three squared equals nine and the square root of nine equals three. All squares and square roots are displayed in pairs.

There are many different square roots, including decimal squares, perfect squares and square roots for negative numbers.

Square roots of most numbers aren’t whole and easy to determine. You probably know the square roots of four and nine by heart, but what about five or seven? The square root of seven is 2.64575131106!

You will use a calculator or table to determine most square roots. However, some teachers may want you to figure it out long-hand – or at least know the manual process for determining odd square roots.

You can figure out square roots by hand by:

  • Adding a zero to the front of numbers with an odd amount of digits before the decimal place
  • Adding a zero to the end if there is an odd number of digits behind the decimal place
  • Making sure you have an even number of digits to work with

Once you’ve adjusted the numbers, draw an area diagram and use it to determine the square root. You probably won’t need to use this method often, but it will help you understand the method behind determining square roots. Your teacher may recommend other manual methods.

Square roots are used by factory workers, engineers, and scientists. Square roots are used to calculate volume and area. When you are looking for a new house or apartment, square roots help you determine the real size of a room. A 400 square foot apartment equals 20 feet by 20 feet, for example.  

Carpenters use square roots to determine building and room measurements.

Graphs

You can draw graphs on your computer or Smartphone. Go old-school and use graph paper or a calculator to solve problems if you prefer. A visual representation of the equation always helps you to calculate the answer more efficiently.

Define the variable in your equation and locate the point on your graph for the function. Buy a graphing calculator if you need to solve lots of square root graphing problems. A graphing calculator shows plotted graphs and has tangent, sine and cosine capabilities.  You can buy these calculators online or at Staples or Office Depot.

Example #1


Here’s an example of how to find the square root of x graph that involves a table. Determine the domain for this equation, determine the table for the values of the f function, and graph it. Then figure out the range.

The equation is:  f(x) equals the square root of (x squared negative 9). Find the function’s domain with x squared negative 9 is greater than or equal to zero. The domain is shown by negative 00negative 3 U (3 plus 00).

Now find values of x in the f domain and make a table of values. Consider that f(x) equals f (negative x). Check the graph symmetry and the y-axis to determine the answer.

F’s range is indicated by (0 plus 00).

Example #2


Determine the graph for the radicand (the number under the radical symbol) and devise a table of values of function f, and graph f and determine its range.

F(x) equals the square root of x squared plus 4x plus 6. You complete the square by rewriting the numbers under the square root as follows:

X squared plus 4x plus six equals (x plus 2) squared plus 2

The expression underneath the square root is positive. Therefore, f’s domain consists of all real numbers. The graph of (x plus 2) squared plus 2, or a parabola. The graph of f has the same axis of symmetry. X equals two the vertical line, is above the graph. The table of graphs can be written numerically as:

X equals the square root of (x plus two squared plus 2). On the graph, negative 2 equals 1.4, zero equals 2.4, two equals 4.2, and four equals 6.2

The f range is shown by the interval the square root of 2 plus 00.

Example #3


Here’s something a little different. Show the graph of y equals the square root of x minus one plus two as an extension of the parent graph y equals the square root of x.

Draw a graph representing y equals the square root of x. Then move the graph a unit to the right, and you’ll get y equals the square root of x minus one. Move the graph of y equals the square root of one up two units, and you’ll get y equals the square root of x minus one plus two.

The visual representation of the graph makes it easy to see the answers. Now you can figure that x is greater than or equal to one is the domain of the function y equals the square root of x minus one plus two.

The graph also shows the range of the function y equals the square root of x minus one plus two is greater than or equal to two.  

How to Study

Most people have a hard time studying for calculus, geometry or algebra exams. Most problems can be attributed to math anxiety, a common classroom ailment. Math is only hard if you think it is – like any other subject, your attitude will make or break your study habits.

Whether you are studying how to find the square root of x graph or something else, you’ll encounter obstacles if you don’t break down the process into manageable steps.

All-nighters won’t garner the best results in any school subject, and math is the hardest subject of all for last-minute study. Plan your study schedule even if you don’t have a test coming up –  daily study is the only way to familiarize yourself with calculus formulas and definitions.

Make sure you get enough sleep and eat right, especially the day before an exam. If you’re bleary-eyed and running on caffeine and sugar, you won’t be able to concentrate on the formulas and terms you need to solve problems.

You build on one theory and set of formulas in algebra, geometry or calculus before you can move on to another, more complicated aspect of advanced math. You won’t get answers on the first try unless you’re a math genius or you’re working on simple problems. Be prepared for some trial and error.

You have several options besides solo study and reviewing class notes to help you study. Need extra help? Consider the following options:

  • Math tutor

  • Study groups

  • Online math sites

  • Extra textbook problems

There are many video tutorials on YouTube and math websites to help you if written explanations are confusing. Check websites for colleges and universities, as many math departments include PDFs of handouts from some of their math classes.

Don’t ignore concepts or problems you don’t understand and move onto another lesson. You need to fully grasp one concept before moving on to the next chapter or function. Practice on as many problems as you can, and pinpoint where you made the error in your thinking.

Always ask your teacher, tutor or classmates to help you find real-life applications for math problems. When you are aware of how to use calculus or other math formulas in the real world, you’ll be less apt to have math anxiety or complain that calculus is useless and won’t help you when you graduate.

Calculus is used in manufacturing, science, engineering, astronomy, aviation, and other industries. Try to find “common ground” with your math assignments. Look for how you can use the formulas in your daily life, or a hobby or your chosen field.

Keep in mind that most math and calculus problems can be solved in several different ways. There might be a second or third way if one method is hard for you to understand. If your book or teacher only suggests one way to solve a problem, look for other methods online or ask a friend about alternate methods.

Control your math anxiety by having a set schedule for homework. Always have at least one other trustworthy person to help you review assignments and help you study for tests. Keep a positive attitude about math classes; don’t dread them.

Be open to new study methods and learn about practical ways to use calculus and other advanced math in your life outside the classroom.  

Featured Image by Gerd Altmann from Pixabay

The video may take a few seconds to load.

Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.



Trapezoid Calc: Find P

Learning to solve trapezoid calc: find P problems is far easier if you learn first to do it algebraically instead of relying on a graph.

A trapezoid, also called a trapezium in some countries, is a geometric shape with one pair of  parallel sides. It looks like a complex shape when you’re trying to figure out things like area, perimeter, and angles, but it is actually quite simple when you break it down into its basic part, then use a few mathematical formulas to derive other quantities.

Perimeter, as you may recall from geometry, is nothing but the sum of the length of all the sides of a shape.

There are some instances in geometry, algebra, calculus, and practical applications that will require you to find the perimeter of a trapezoid when you may not necessarily have all the quantities. When doing a trapezoid calc: find P operation, you can add up the known sides and figure out the rest.

Properties of Trapezoids

The only constant property of trapezoids is the existence of a single pair of parallel sides. Beyond that, a trapezoid can take any configuration. The two parallel sides can have any difference in length, or they can be the same length. In other words, a rectangle or a square is technically a trapezoid, just a special type of one.

Finding the Perimeter of a Simple Trapezoid

To find the perimeter of a trapezoid (or any shape except an ellipse or a circle) you just have to add the lengths of all the sides. However, if you don’t know the side lengths, you might have a problem. By using properties of geometry, trigonometric identities, and basic logic, you can infer or find the values of the unknown sides of a trapezoid.

For example, if you know the angle formed by the base and one of the sides, and if that angle is the same on the other side, you can then infer that the opposite side must be the same length because it has to cover the same distance to reach the parallel top of the trapezoid. However, you may not be given this information outright and you’ll have to figure it out on your own.

The best way to think of a trapezoid is as a rectangle with two right triangles on the sides. Then, by using the properties of right triangles, you can figure out the length of the unknown side.

We’ll start by working with a trapezoid ABCD, where each letter represents a vertex where two line segments intersect. We’ll assume, that line segment AB and line segment CD are parallel. AB has a length of 5, and CD has a length of 11. We also know that the trapezoid has a height of 4, if you were to take a perpendicular line connecting AB to CD.

Finally, angles ACD and BCD are given to be the same.

Because we have the top base has a length of 5, you can think of the bottom base as including a rectangle with a length of 5. Because we know that the angle is the same for both sides, you know that the two sides have to be the same length. Therefore, you now have to imagine the trapezoid broken down into other shapes.

The height is 4, and the total length of the bottom base CD is 11. Because AC and BD are the same lengths, it makes things easier. Now, on each side of a length of 5 on CD, you have 6 unaccounted for, so you can think of a right triangle with 3 on one leg and 4 on the other because you already know that height.

The next step in trapezoid calc: find p is to work out the length of the right triangle’s hypotenuse. The formula for the Pythagorean theorem is a2 + b2 = c2. Square the lengths of the two legs.

The square of 3 is 9, and the square of 4 is 16. Adding these up, you get 25. Now, take the square root, which is 5. In this example, we’ve used the simplest right triangle there is.

It won’t always be this simple, but it serves as an example. Because you know that the two right triangles are the same dimensions, you can infer that the hypotenuse is the same on both sides.

Finally, you can go back to your original trapezoid and substitute the two values into the original shape. Then all you have to do is add the sides as you normally would.

In this example, 5 + 5 + 5 + 11, for a total of 26. Therefore, the trapezoid in our example has a perimeter of 26.

Using Trigonometric Identities

There may be times when you have to use some knowledge of trigonometry and a scientific calculator to help you out.

Here are the three identities you’ll use most often:

  • Sine = opposite/hypotenuse

  • Cosine = adjacent/hypotenuse

  • Tangent = opposite/adjacent

The tangent is also equal to the sine divided by the cosine.

You can also use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).

This is how you’ll find any unknown angles or, with the help of some algebra, the sides if you need them.

We’ll assume for now that you’re just working with a right triangle with another known angle of 60 degrees and a hypotenuse 5 units long, and you want to find the lengths of the other sides. Remember that as per geometry, the three angles of any triangle add up to 180 degrees. Because one of them is by default 90 degrees, you have 90 degrees to account for.

You know your angles: a/sin(60) = b/sin(30) = 5/sin(90). Simplifying, you get b/0.5 = 5. Therefore, b equals 10. You can get the other sides this way as well.

Finding Perimeter of a Trapezoid on a Graph

Having the figures sketched out in geometric diagrams is one thing, but having it on a graph coordinate system can make it even easier to find the perimeter. Assuming that each increment of x and y is one unit, the only part for which you might have a problem is the hypotenuse. However, even that is easy to calculate.

All you have to do is count the units of x and y that make up the right triangle, then square the units and take the square root of the result. This gives you the diagonal of the trapezoid. To get the lengths of the sides, all you have to do is count the units down to the x-axis and across to the other parallel line.

What if You Don’t Have a Graph?

Pencils, eyeglass and graph notebook

Image by marijana1 from Pixabay

Even if you don’t have a graph and are just given points, you can do simple subtraction. For example, take the points (0, 3) and (0, 9). To determine the length of the line segment and the slope, you use the formula (y1 – y0)/(x1 – x0). Substituting the values, you get 0 and 6.

The formula above is meant for slope, but it’s also an easy way to visualize. If your x-values are the same, you have a vertical line.

Now, you need to find your diagonal line of the trapezoid. It doesn’t matter what slope it is; all you’re looking for is the length. By finding the change in y and the change in x, you get two sides of your right triangle. You should have a good idea of what to do from here: square the lengths of the two sides, add them, and take the square root of the result.

Now, unless you were fortunate enough to get a rational root – i.e., one that simplifies to a fraction or decimal, you’re not going to be able to write it as a standard integer. For example, say that you got a diagonal with a length √2. You would have to add the other sides on their own, such as 3 + 3 + 5, which gives you 11. Then, you add √2.

From here, there’s nothing further you can do. If you have something like √8, you can simplify that into 2√2, which is √2 * √4. Always reduce math expressions to their simplest forms when you can. If you’re in school, odds are you’ll get only partial credit if you don’t simplify your answers.

Learning to solve trapezoid calc: find P problems is far easier if you learn first to do it algebraically instead of relying on a graph. Not all graphs will resolve to neat points and therefore risk giving you inaccurate answers.

Final Thoughts

The perimeter of a trapezoid is one step in helping you to find its area. Because the area is a function of height and width, you need to gather as many concrete measurements as you can. If you’re working on a graph, it might help to visualize the trapezoid as lying sideways, with the vertical lines being the bases and the horizontal lines being the height.

Featured Image via Flickr

The video may take a few seconds to load.

Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.



Trapezoid Calc: Find A

Trapezoid calc: find a is easy. However, it does take some manipulation of variables and measurements when you get into more complex expressions.

A trapezoid, as you’ll recall from geometry, is a shape with two parallel sides. That’s the only definition and the only set of properties that it has. There are special trapezoids, but you might also know them as rectangles or squares. You’ll need to be able to determine the area of a trapezoid in various studies like geometry and calculus, as well as use it in practical applications.

For starts, the area of a trapezoid is helpful in calculus when getting the approximation for integrals, assuming you don’t want to go through the trouble of taking integrals directly and can just provide a rough approximation. There are several things you can do when faced with trapezoid calc: find A math problems.

What is Area?

Area is nothing but the space outlined by a two-dimensional shape, expressed in square units, such as in2 or cm2. Multiplying the length by the width provides you with this information for simple quadrilaterals, but it doesn’t work as well with trapezoids. Therefore, you have to be a little creative and be diligent about separating your trapezoid into other shapes.

Here’s what you need to find a trapezoid’s area:

  • Absolute maximum

  • Lengths of two parallel bases

  • Height

  • Dimensions of triangular sections if one of the bases is not available

Dividing Up a Trapezoid

The simplest form of an area is length times width, but this only works for quadrilaterals like squares or rectangles whose sides all make right angles. A trapezoid can be thought of as a rectangle with a triangle attached to either side, so you’ll have to find the area of each of these sections separately.

The formula for the area of a triangle is ½ bh, where b is the base and h is the height. Therefore, if you know the extra length that isn’t a part of the basic quadrilateral, as well as the height, you can easily calculate the area of the trapezoid by adding each section.

Let’s start with a simple example. Assume that you have a trapezoid ABCD with all known lengths. AB, the top line, is 5 units long, and CD the bottom line, is 8 units long.

You also know that the height is 3 units.

To start, take the basic area of the quadrilateral in the form of 5 * 4, or 20 square units. Then you’ll have to take the areas of the two triangles. Unlike with perimeter, you don’t need to know the length of the hypotenuse. All you need is that extra length and the height. Because the total is 8 and the part you’ve already calculated is 5, you have 3 units left over.

Multiply 3 by 4 to get 12, then divide by 2 to get an area of 6. Multiply by 2 to account for both sides, leaving you with 12 square units, and adding this measurement to the 20 square unit you calculated earlier leaves you with a final answer of 32 square units.

What if You’re Missing Sides?

If you’re missing a side measurement, it makes it harder to complete a trapezoid calc: find A operation, but not impossible. You’ll usually have at least one angle, and if you have an angle, you can find sides using the Law of Sines. You’ll also have at least one or two sides. It’s rare to impossible that you’ll be asked to solve a trapezoid with no measurements.

This is a simple ratio: a/sin(A) = b/sin(B) = c/sin(C)

Now because your basic part of the trapezoid is a quadrilateral, you already know that the rest of it is composed of right triangles. Let’s also assume you have a hypotenuse, side c, of 5. As such, you know that one of your angles is already accounted for. Therefore, a/sin(A) = b/sin(B) = c/sin(90). The sine of a 90-degree angle is 1.

Now, let’s assume that b, your base side, is 10 units. So, you can now put that in as well and cross- multiply: 10/sin(B) = 5. Simplifying the equation gives you 2/sin(B) = 1, or sin(B) = ½, or 30.

Plug 30 into (B) to continue with the equation to have your other value present. Also, remember from geometry that all three angles of any triangle add up to 180 degrees. Therefore, you can subtract and figure out that your missing angle is 30 degrees.

By using the Law of Sines, assuming you have a known side (it’s rare that you’ll have all sides or all angles missing), you can then take the sines of the angles you have, cross-multiply, and simplify.

Once you do that, you can take the side of the base of the triangle and the height and use those two measurements to calculate area as you normally would: ½(bh). Do this for both sides, add them, and then add the rest to the quadrilateral base. You should have this in your formula; if you don’t, it may not be possible to solve for the area of a trapezoid.

Applications

Now we can get into a slightly tricker subject: using the area of a trapezoid on a graph, such as you would do when approximating integrals. Your first step is to decide how thin you want the trapezoid to be on the graph. The more you have, the more closely you can approximate the area under the curve. For simplicity, try starting with slices with a width of 1.

This corresponds to a change of 1 for x on the graph, so this should make it easy to calculate.

Using the relevant function, such as 2x + 3, use x to find the appropriate values of y In this case, if you start at x = 1, you’ll have the vertices (0, 0), (3, 0), (0, 3), and (3, 7).

In a way, this makes it simpler to calculate because you have the points plotted out and can, therefore, find your lengths. You have an imaginary line traveling across the graph at y = 3, dividing your trapezoid into a quadrilateral and a right triangle. You should have 3 increments of x and 4 for y.

As always, you’ll find your base area first. Then find the dimensions of your triangle and find its area. Then, just add the two numbers.

How is This Useful?

If we take multiple trapezoidal segments of an integral, calculate all their areas, then add them together, we can easily approximate the area under the integral without having to go through the steps of doing derivatives and solving regarding calculus.

This sort of function has many applications in science, such as determining the total volume of liquid given a certain rate of flow over time. Of course, all this works only when the value for y is not at or below zero, at least as far as practical applications are concerned. Few, if any, practical applications use negative numbers.

Trapezoid calc: find A Another Method to Find Area

Another way that you can do trapezoid calc: find A equations is to use the formula [(b1 + b2)/2]h, where you take the average of the two bases and then multiply by the height. However, this requires you to know the bases, which isn’t always possible. If you know both of them, you can use this formula. If you only know one, you’ll use the alternate method; that is, sectioning it.

Visualizing

If you use the basic formula for the area of a trapezoid [(b1 + b2)/2]h, you’ll need to do a little visual tweaking. Functions travel along the x-axis of a graph, meaning that the trapezoids used to approximate them are sideways. In this case, you need to envision the graph as being sideways. The height of the trapezoid, in this case, lies parallel to the x-axis.

Meanwhile, the bases are vertical. Even though it looks vice-versa, don’t let it throw off your formula. You’re essentially treating the trapezoid as a rectangle with distorted sides.

Irrational Area Measurements

Some of the example problems we’ve covered have side measurements that work out to neat and clean integers. This isn’t always going to be the case. Because you’re dealing with the Pythagorean theorem to find the measurements of a right triangle, you’re likely going to have to deal with square roots that don’t simplify. An example is √6. You cannot simplify it any further.

Something like √12, however, can be reduced to √4 * √3, or 2√3. If you see a number like this, keep it as is unless your instructor specifically asks you to use a calculator and round to 3 or 4 decimal places. Because these square roots are irrational, you won’t get an exact match if you try to check your work. You might get something like 2.9998.

Final Thoughts

Before you start with your trapezoid calc: find A work, take a look at the problem and determine what data is present and what you need to find. This will help you determine which method you need to use to find the shape’s total area. Many times, you’ll be able to use the basic trapezoid formula. If you don’t have all the data, you’ll need to split the shape into pieces.

The video may take a few seconds to load.

Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.



What is Euler’s Method Formula in Calculus?

Learn about Euler’s Method formula and how it works along with some real-world examples to make it easy to grasp. Click here to find out more!

A common joke found all over the internet is a comment about teachers telling students who wouldn’t always have a calculator in their pocket. Obviously, the following comment is usually just an image of a modern cellphone. While we can all laugh at the joke behind this dialogue, the teacher is still right. A calculator on your phone can’t read your mind and write the equation for you.

That’s a broad statement, but it’s true for all forms of math from simple word problems to complex calculus of algebra problems. You need to know how to write out the equation and which equation to use. Understanding formulas like Euler’s Method are critical to solving many real-world problems and passing a college calculus class.

What Is Euler’s Method?

To understand Euler’s method, you’ll need to understand a few other math terms or formulas as well. The focus of this article is Euler’s method, so expect our definitions to be short and summarized. If you need more information on any of the topics, a quick search online will explain anything we leave out. That said, our definitions should convey all the knowledge you need to understand this method.

On occasion, you may need to solve a differential equation where you can’t use separation of variables, or you may get specific conditions to satisfy. Some of the methods you learn to conquer these types of equations simply won’t work. They should work, but the problem either ends up with an obviously incorrect answer and keep rolling over.

You can solve these types of differential equations using Euler’s method almost without failing. Why this isn’t the standard method taught in class is beyond us. That said, we’re going to help you understand Euler’s Method and how to use it. Before we get into the examples and a better explanation, let’s define some terms we’re going to use. The key terms we may use include:

  • Differential equations: This is an equation that uses a derivative to solve the equation for a function. 
  • Functions: This describes the relationship between the variables you put in and what comes out. It’s typically written out as f(x) or g(x). Each variable you put in has a relationship with the answer. That relationship is the function. Sine and tangent are trigonometry functions.
  • Derivatives: A derivative is usually used to define a slope or the change in a slope. You would use the slope formula to find the derivative of y = f(x). 
  • Separation of variables: This is typically how you solve differential equations by moving like terms to one side of an equation. The equation is generally noted by an equal sign. 
  • Tangent line: This is any line that comes in contact with a curve and mimics the curve where it runs into it. Small tangent lines are the basis of Euler’s Method. Tangent lines are usually outside of a curve if that curve is a circle.
  • Slope: This is any number that tells us the direction of a line and how steep that line may be along its path. A slope is equal to the rise divided by the run of the line.

Any other terms that require a definition will get defined in the same section. For now, those are the basic terms aside from understanding the basics of how to graph or work with fractions and variables. If you’re reading this, we assume you know how to create a graph or work with variables on some level whether it’s this advanced or on a lower tier.

Using tangent lines, Euler’s Method helps you approximate the solution to any equation, almost, if you know the initial value. If the problem changes rapidly or changes direction more than once, Euler’s Method may not work. That said, if the graph changes direction you’ll end up with multiple curves instead of one which rules out using Euler’s Method altogether. 

Euler’s Method is one of three favorite ways to solve differential equations. As we mentioned earlier, you may be able to use separation of variables, or you might find slope fields are the best method. Euler’s equation is must have a starting value or an assumed starting value in order to work. If you don’t have either of those things, refer to the other two methods we mentioned.

Using Euler’s method, we can see what goes on over a segment of our curve by intersecting or paralleling it with our tangent line. In short, Euler’s Method is used to see what goes on over a period of time or change. For instance, it can approximate the slope of a curve or define how money market funds changed over time. 

Using Euler’s Method, we can draw several tangent lines that meet a curve. Each line will match the curve in a different spot. By getting the approximate solution or equation where each line meets the curve, we can begin to put together a picture of what is happening along our curve. In short, Euler’s Method is just a lot of tangent lines strung together to help us guess at what the cure is doing as it travels.

Cons Of Using Euler’s Method

Since Euler’s Method only gives us approximate values, there may be room for error in the final result. If the curve is sharp and changes rapidly at any point, the solution we find at these sharp turns using Euler’s Method may lack accuracy. However, this is precisely where Euler’s excels if you need to crudely calculate why something sped up like rates of deaths due to disease or sales over a specified period.

While many people refer to Euler’s Method as a formula, and you can write a pseudo formula for it, it’s not a formula; it’s a method. It produces a solution without variables which may be considered an approximate value of the current problem. It also causes some issues with math teachers if they want you to use a specific formula or method.

When Would I Use Euler’s Method Outside Of Class?

Differential equations end up being a big part of our lives whether directly or indirectly. Mathematicians sometimes work with biologists to develop programs for monitoring diseases or population problems. Other mathematicians may work in banking or economics while some find their home in writing about things like Euler’s Method and exploring the different ways to solve differential equations.

Differential equations help scientists monitor everything from the Moon’s orbit to the rate at which a glacier may melt. We could keep giving examples, but we believe you understand how vital understanding this part of math is to your daily life and possibly your future career. Many professions beyond being a mathematician rely on approximations and solving differential equations.

You can check the Bureau of Labor Statistics for specific information on many job titles including those that use math a lot. Many careers at NASA require a firm grasp of applied mathematics and calculus. It’s how they figure out how to fly by a planet without hitting it which is probably essential. Pharmaceutical science uses calculus and is required for many jobs at places like Bayer as well.

Aerospace engineers spend their time creating and designing satellites, space vessels, space stations, and other human-made objects that need to survive in space. They rely on math heavily to do their jobs and ensure the safety of the equipment the build and the astronauts that us it.

Earthquake Safety Engineers rely on all forms of math to design better buildings and materials that withstand earthquakes. They also use math to build models to test their designs in a computer simulation. The same modeling steps may also be used to determine the damage existing structures might suffer during an earthquake

Aside from the complicated math used in the professions, we mentioned above, Euler’s method has many practical applications and may help determine simpler things like the rate of flow for running water. Similar methods and functions got used to help figure out how to shut off an oil leak that was 1,800 feet below the ocean’s waters. 

It’s impressive how math affects our lives every minute of the day. Many people don’t realize its importance, but we’re hoping we’ve conveyed the message well. That said, most common functions and formulas come preprogrammed into computers and calculators used in science-based fields. However, you still need to understand why and when to use them.

Some Final Notes

Euler’s Method is undoubtedly one of the most exciting formulas we’ve come across. Approximations usually find their home in less precise math problems. However, Euler’s method gets used across the spectrum of physics and various disciplines that use calculus. We don’t always need to know changes at every point along a curve, or we may have a starting value, and that’s when Euler’s Method works best.

The video may take a few seconds to load.

Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.



Slope Formula In Calculus Math: Our Comprehensive Guide

What is Slope Formula in calculus? To calculate the slope of a line you need only two points from that line, (x1, y1) and (x2, y2). Learn more here!

​The history of calculus and its many problems is almost as fascinating as its many modern applications and challenges. Gottfried Wilhelm Leibniz and Sir Isaac Newton discovered calculus in the 17th century. However, they did so independently then accused each other of stealing the other’s work. Their mathematical feud continued until the end of their respective lives.

Some historians argue that calculus was commonplace much earlier than the 17th century. Some claim it got used as early as 2000 BC. It’s true that some parts of calculus had been around for centuries before Newton’s and Leibniz’s discoveries, but they get credited with bringing all the functions, derivatives, integrals, and other terms into a group and calling it infinitesimal calculus. 

For clarity, calculus is here to help us solve problems, so we use the term problems as a way to describe an equation since it’s a problem until we answer it. The challenge in solving many problems is determining which formula applies and how to write the equation. Even a small error in a formula can wildly change the outcome when you’re dealing with math.

Photo of a board with formula in it

​Image via Pixabay

It sounds simple enough, right? However, if you knew a house cost X dollars in 1980 and it’s value increased until it was sold again in 1990 and you need to show at what rate the value of the house increased over time, how do you write that statement in mathematical terms? It sounds mind-numbingly hard, but the solution is simple if we use the right formula.

The calculus way to finding a solution to our problem is by creating a graph and finding the slope of the line that connects the original value of the house to its value at the time it got sold. The slope of the line indicates the rate at which the house increased in value. We’re assuming that the rate of increase created a straight line for this problem.






























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































Defining the Slope Formula in Calculus

The slope of a line is always the same anywhere along the route. If we get technical, the slope is merely a ratio that defines the changes in a line as it goes from its starting point to its end. It’s easier to understand if you picture a graph and imagine the line starting at the bottom left corner of a graph and growing until it reaches any point along the right wall of the chart.

The changes in the line’s rise as it travels to its ending point on the right wall is the line’s slope. Let’s look at a real-world scenario like determining the slope of a roof. Picture a graph with a bottom row of evenly spaced numbers from zero to nine. The right wall of our graph has the same numbers on it except they go up vertically beginning with zero.

Our roof begins at the bottom left over the number zero and travels up until it ends at the number seven on the right wall of the graph. We know the graph is nine points across the bottom and the roof intersects the right wall at the number eight, we can write a simple formula to explain the slope which is: the slope equals eight divided by nine or 0.888 repeating. 

If the numbers on our graph indicate feet, we now know the roof rises about 0.88 feet per foot of roof length. This application of the slope formula has a lot of applications in the real world. For instance, an accessible wheelchair ramp at a public facility in some states must maintain a specific ratio. The ramp must be one foot long for every inch of rising. If the ramp is four inches tall, it must be four feet long. 

Our problem is simple to keep the math simple for the sake of explaining the slope formula. The math gets more complicated based on the type of slope. There are four types of slopes to contend with including:

  • Zero slope: the line is perfectly horizontal
  • Positive slope: this is when a line increases in height
  • Negative slope: this is a positive slope in reverse
  • Undefined slope: primarily used to define the slope of a vertical line because you can’t divide by zero





























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































  • Real World Examples of the Importance of the Slope Formula

    We use math every day in hundreds of ways. Without math, we couldn’t drive cars or browse the internet. If you look closely, you’ll find some form of math in everything you do from cooking to exercising. Aside from built-in math that we all take advantage of, math is essential to many careers. You just can’t exist and not use math directly or indirectly.

    For most of us, we use math indirectly almost constantly. For instance, we know it takes 20 minutes to drive from home to work when traffic is slow. However, some days traffic is heavier than others so we use variables and math to determine when we should leave to ensure we make it to our job on time. Imagine the horror of baking without using math to measure ingredients.

    You may use math in a more direct sense if you work in medicine, construction, or marketing. If you do work in these fields, you know that the slope formula may be one of the most critical math formulas that we take for granted. It has different names depending on the occupation, but it’s all the same when you reduce it to pure math.

    The most apparent jobs that require a solid understanding of the slope formula are architects and construction workers. They use the slope formula or a variation of it when designing or building roofs, stairs, ramps, and just about every part of a building above ground. A typical house is little more than a box with a triangle nested on top of it. The slope formula is used to build the triangle on top. 

    The slope of the roof determines the number of materials needed to build it along with how to measure and cut each piece of material. It is much easier to construct a triangle out of wood and nails if you know how long each piece of wood needs to be to make the slope correct. In some cases, getting the slope wrong could cause the roof to collapse or perform poorly in high winds.

    Photo of different kinds of formula

    ​Image via Pixabay

    The slope formula makes everything around us, in respect to construction, safer. A slope is used to make roads safer and easier to travel across. It makes stairs and accessibility ramps safer and easier to use as well. Imagine building stairs without using the slope formula. Some might stretch out into the street while others may be too steep to climb. The same principle applies to roads and bridges.

    Social scientists and marketing professionals use the slope formula as well. Remember our example of a graph and determining the rise of a roof based on its run? The same principle gets used to determine things like how well a product is increasing concerning sales or how a population is performing based on specific economic indicators. 

    The slope of a line on a marketing graph tells a company how their product is performing. A negative slope indicates poor performance while a positive slope means the product is doing well. Determining the slope of the line on the marketing graph lets them know exactly how well the product is doing over time and may even indicate when the product’s sales are at their best.

    People that work in healthcare use the slope formula in a more critical sense. For example, epidemiologists may use a graph and the slope formula to determine how rapidly a disease is progressing over time. Doctors may use it to determine how long it might take for a treatment to cure a patient. The slope of a line may tell a doctor to increase or decrease medications as well.

    It’s critically important to healthcare to know the slope of lines on a chart if we expect doctors to find cures for terminal disease and many chronic ailments that don’t have treatments. It’s all about knowing how high a line rises over a specific period. That’s all math and 100 percent slope formula. 

    The list of uses for the slope formula is vast. If your career is impacted by it, you recognize its importance and understand why we think it’s one of the most crucial calculus topics. Beyond the examples we touched on above, the slope formula has many other uses including:

    • Determining speed over time
    • Determining distance over time
    • Calculating stock prices over time
    • Determining weight loss over time
    • Calculating win versus loss rates for sports teams
    • Calculating pay rate increases over time






























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































    Some Final Notes

    Nothing we see or do today would be possible without math. Okay, a few things might be possible, but roads and stairs would be unsafe while some buildings might fall randomly. You wouldn’t be able to read this article without math because the internet and your computer could not exist without it. Appreciate the power of it and how the slope formula impacts your life at every angle.

    The video may take a few seconds to load.

    Having trouble Viewing Video content? Some browsers do not support this version – Try a different browser.