What are vertical asymptotes?

 

Vertical asymptotes are important boundary lines for a function, because, if you can find them, they're a line that the graph cannot cross, which can really help you sketch a more accurate picture of the curve.

Vertical asymptotes are usually found in rational and logarithmic functions, but they can be found in other functions, too. And, not every rational function or logarithmic function has a vertical asymptote.

But if there is a place along the curve where the graph is undefined, like maybe because the denominator of a fraction in the function is equal to 0, or because the argument inside a logarithm is negative, then you'll have a vertical asymptote at that point.

While the curve will get infinitely close to a vertical asymptote on one or both sides of it, it will never ever cross the asymptote.


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What are critical points?

 

Critical points are one of the best things we can do with derivatives, because critical points are the foundation of the optimization process. Optimization is all about finding the maxima and minima of a function, which are the points where the function reaches its largest and smallest values. And if we know which values maximize function and which ones minimize it, that has all kinds of real-world applications.

In order to find critical points, which are the points where the function might have extrema, we just take the derivative of the original function, set that derivative equal to 0, and then solve that equation for x. That gives us the critical points of the function. Then, we can test those critical points to determine whether they represent maxima, minima, or neither.
 


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What is linear approximation?

 

0:00 // What is linear approximation?
0:44 // When do you use linear approximation?
1:28 // Estimating square roots using linear approximation
5:23 // Estimating trig functions using linear approximation
6:37 // How to find the error in a linear approximation
7:48 // Summary

Linear approximation, or linearization, is a method we can use to approximate the value of a function at a particular point. The reason liner approximation is useful is because it can be difficult to find the value of a function at a particular point.

Square roots are a great example of this. We know the value of sqrt(9); it’s 3. That’s easy to figure out. But we don’t know the value of sqrt(9.2). We can guess that it’s a little bit more than 3, since we know that sqrt(9) is 3, and 9.2 is a little bit more than 9, but other than that, we don’t know how to find a better estimate of sqrt(9.2). That’s where linear approximation comes in to help us.

Since we’re dealing with square roots, if we imagine the graph of the function sqrt(x), we know one point on that function is (9,3). If we find the tangent line to the function sqrt(x) through the point (9,3), then we can see that, since the tangent line is really close to the graph of the function around the area of (9,3), that the value of the function and the value of the tangent line will be pretty close to each other at x=9.2.

So to get an estimate for sqrt(9.2), we’ll use linear approximation to find the equation of the tangent line through (9,3), and then plug x=9.2 into the equation of the tangent line, and the result will be the value of the tangent line at x=9.2, and very close to the value of the function at x=9.2.

That’s why linear approximation is so helpful to us, because it’s a quick, simple method that let’s us estimate a value that would otherwise be very difficult to find.

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What is average rate of change?

 


We already know that the slope of a function at a particular point is given by the derivative of that function, evaluated at that point. So we can easily find the slope, or the rate of change, in one particular location, and so we could call this the instantaneous rate of change, because it’s the rate of change at that particular instant.

If instead we want to find the rate of change over a larger interval, then we’d need to use the average rate of change formula. After all, we’re looking for the average rate at which the function changes over time, or over this particular interval. To do that, all we need is an equation for the function, and the endpoints of the interval we’re interested in. We can plug those things into the average rate of change formula, and we’ll have the average rate of change over the interval.

So in the same way that the derivative at a point, which we can also call instantaneous rate of change, is equal to the slope of the tangent line at that point, the average rate of change over an interval is equal to the slope of the secant line that connects the endpoints of the interval.

0:27 // Formula for the average rate of change
0:49 // Average rate of change is equal to the slope of the line
1:17 // When average rate of change is negative, positive, and zero
2:30 // Average rate of change vs. instantaneous rate of change
3:12 // Average rate of change from a table


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What is implicit differentiation?

 

Most often in calculus, you deal with explicitly defined functions, which are functions that are solved for y in terms of x. In that case, finding the derivative is usually really simple, because you just call the left-side of the equation y', and then you differentiate the right side with respect to x.

But when you come across an implicitly defined function, finding the derivative isn't always that easy. Implicit functions are functions where the x and y variables are all mixed up together and can't be easily separated. That's when implicit differentiation comes in handy.

Implicit differentiation lets us take the derivative of the function without separating variables, because we're able to differentiate each variable in place, without doing any rearranging.

When we use implicit differentiation, we differentiate both x and y variables as if they were independent variables, but whenever we differentiate y, we multiply by dy/dx. That's because y, as a dependent variable, is actually a function of x. Therefore, when we take its derivative, it's as if we're taking the derivative of a composite function, and we therefore have to apply chain rule. When we do apply chain rule, we multiply by the derivative of y, which is dy/dx.


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What is L'Hospital's rule?

 

0:45 // What does L'Hospital's rule do?
2:36 // How many times can you apply L'Hospital's rule?
3:26 // An example using L'Hospital's rule
5:13 // When you can and can't use L'Hospital's rule
7:35 // What to do when L'Hospital's rule fails
7:51 // Two examples that avoid L'Hospital's rule
10:11 // Another example using L'Hospital's rule

L'Hospital's rule is so named because i was discovered (maybe) by the mathematician L'Hospital in the 1600's. While L'Hospital is credited with the rule, he worked closely with Johann Bernoulli, who claimed after L'Hospital's death that L'Hospital had paid him off in order to take credit for the rule.

But no matter who discovered it, L'Hospital's rule is a really helpful tool that you can use to solve limit problems. You want to use L'Hospital's rule when you evaluate a limit and the result is an indeterminate form, like 0/0 or infinity/infinity. For a limit that gives an indeterminate form, L'Hospital's rule says that you can replace the numerator and denominator of the original function with their respective derivatives. In other words, replace the numerator with its derivative, and replace the denominator with its derivative. Then use substitution to try to evaluate the limit again. Oftentimes, L'Hospital's rule will have simplified the function to the point where substitution no longer gives an indeterminate form, but instead a real-number answer.

And L'Hospital's rule can actually be applied multiple times. So even if you use it once, and then try substitution and you still get an indeterminate form, you can just try applying L'Hospital's rule again and again until you eventually do get a real-number answer. The only drawback is that there are a few instances in which L'Hospital's rule can't be used.

 

 

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When is a curve differentiable?

 

0:00 // What is the definition of differentiability?
0:29 // Is a curve differentiable where it’s discontinuous?
1:31 // Differentiability implies continuity
2:12 // Continuity doesn’t necessarily imply differentiability
4:06 // Differentiability at a particular point or on a particular interval
4:50 // Open and closed intervals for differentiability
5:37 // Summary

When we talk about differentiability, it’s important to know that a function can be differentiable in general, differentiable over a particular interval, or differentiable at a specific point. In order for the function to be differentiable in general, it has to be differentiable at every single point in its domain. If there’s just a single point where the function isn’t differentiable, then we can’t call the entire curve differentiable. In that case, we could only say that the function is differentiable on intervals or at points that don’t include the points of non-differentiability.

So how do we determine if a function is differentiable at any particular point? Well, a function is only differentiable if it’s continuous. So if there’s a discontinuity at a point, the function by definition isn’t differentiable at that point. This applies to point discontinuities, jump discontinuities, and infinite/asymptotic discontinuities.

But there are also points where the function will be continuous, but still not differentiable. Remember, differentiability at a point means the derivative can be found there. If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there.

So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined. That means we can’t find the derivative, which means the function is not differentiable there. In the same way, we can’t find the derivative of a function at a corner or cusp in the graph, because the slope isn’t defined there, since the slope to the left of the point is different than the slope to the right of the point. Therefore, a function isn’t differentiable at a corner, either.

Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Barring those problems, a function will be differentiable everywhere in its domain.

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What is Stokes theorem?

 

Where Green's theorem is a two-dimensional theorem that relates a line integral to the region it surrounds, Stokes theorem is a three-dimensional version relating a line integral to the surface it surrounds. For that reason, Green's theorem is actually a special case of Stokes Theorem.

Again, Stokes theorem is a relationship between a line integral and a surface integral. Before you use Stokes theorem, you need to make sure that you're dealing with a surface S that's an oriented smooth surface, and you need to make sure that the curve C that bounds S is a simple, closed smooth boundary curve with positive orientation.

Since Stokes theorem can be evaluated both ways, we'll look at two examples. In one example, we'll be given information about the line integral and we'll need to evaluate the surface integral. In the other example, we'll be given information about the surface integral and we'll need to evaluate the line integral.

0:00 // About Stokes theorem
2:41 // Example 1
10:46 // Example 2


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What is the tangent line?

 

Tangent lines are absolutely critical to calculus; you can’t get through Calc 1 without them! In this video, we’re talking all about the tangent line: what it is, how to find it, and where to look for vertical and horizontal tangent lines.

The tangent line is useful because it allows us to find the slope of a curved function at a particular point on the curve. We learned a long, long time ago in a math class far, far away that we could find the slope of a line, but we’ve never learned how to find the slope of a curved function. Since the slope of a curved function is always changing, the best we can do is find the slope of the curved function at one particular point on the function. And to do this, we actually don’t look at the function at all. Instead, we look at the tangent line to the curve that passes through the particular point we’re interested in, and we find the slope of the line instead.

To find the slope of the curve, all we have to do is take the derivative of the curve (because the derivative represents the slope), and then find the line with the correct slope that passes through the point of tangency. That’ll give us the tangent line, and the tangent line will have the same slope as the slope of the curve at the point of tangency.

We’ll also look at where to find vertical tangent lines, and where to find horizontal tangent lines, since that’s something you’ll be asked to do often. Horizontal tangent lines exist where the derivative of the function is equal to 0, and vertical tangent lines exist where the derivative of the function is undefined.

0:24 // The definition of the tangent line
1:16 // How to find the equation of the tangent line
3:10 // Where the tangent line is horizontal and vertical


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How do you multiply matrices?

 

In this video we’re talking about everything you need to know about matrix multiplication. We’ll start simple and look at what it means to multiply a matrix by a scalar, and then move on to multiplying matrices together using the dot product, and a special trick for remembering how to do matrix multiplication.

We’ll cover rules of algebra for matrices, including commutative, associative, and distributive properties, and show that each of these properties applies to matrix multiplication, except for the commutative property.

We’ll talk about matrix dimensions and how to describe a matrix by its rows and columns. We’ll address the fact that not all matrix multiplication is defined, and that you in fact need the number of columns in the first matrix to match the number of rows in the second matrix in order for the multiplication to work at all. You can also use matrix dimensions to figure out the dimensions of the resulting matrix after the multiplication.

Finally, we’ll look at identity matrices and zero matrices, which act like the special numbers 1 and 0 when it comes to matrix multiplication.

0:17 // Multiplying a matrix by a scalar

0:37 // Applying algebra to matrices

0:44 // Matrix multiplication is not commutative

1:21 // Associative property and distributive property with matrices

2:00 // Matrix dimensions

2:21 // Which matrices can be multiplied together?

2:55 // How to figure out whether matrices can be multiplied together. How to figure out the dimensions of the resulting matrix.

3:40 // How to multiply matrices using the dot product

4:26 // How to remember what to put where

5:16 // Identity matrices

5:51 // Multiplying by the identity matrix is like multiplying by 1

6:31 // Multiplicative inverses

6:56 // Zero matrices

7:23 // Summary


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What are differential equations?

 

Differential equations are usually classified into two general categories: partial differential equations, which are also called partial derivatives, and ordinary differential equations.

Partial derivatives can be first-order, second-order, or higher order, and they’re what you get when you have a multivariable equation, but take the derivative of that equation with respect to just one variable.

Ordinary differential equations are equations that include a derivative, and that derivative can be of any order (first-order, second-order, or higher-order). Ordinary differential equations are more commonly classified into other specific types of differential equations, like linear differential equations, separable differential equations, and exact differential equations.

0:18 // What are partial differential equations (partial derivatives)?

1:31 // What are ordinary differential equations?

2:53 // What are linear differential equations?

3:22 // What are separable differential equations?

4:26 // What are exact differential equations?

4:52 // Summary


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Why can't logarithms be negative?

 

The only numbers you can plug into a logarithm are positive numbers not equal to 1. Negative numbers, and the number 0, aren’t acceptable arguments to plug into a logarithm, but why?

The reason has more to do with the base of the logarithm than with the argument of the logarithm. To understand why, we have to understand that logarithms are actually exponents. The base of a logarithm is also the base of a power function.

When you have a power function with base 0, the result of that power function is always going to be 0. In other words, there’s no exponent you can put on 0 that won’t give you back a value of 0. Or, put a different way, 0 raised to anything is always still 0. In the same way, 1 raised to anything is always still 1.

If you raise a negative number to a positive number that’s not an integer, but instead a fraction or a decimal, you might end up with a negative number underneath a square root. And as you know, unless we’re getting into imaginary numbers, we can’t deal with a negative number underneath a square root.

So 0, 1, and every negative number presents a potential problem as the base of a power function. And if those numbers can’t reliably be the base of a power function, then they also can’t reliably be the base of a logarithm.

For that reason, we only allow positive numbers other than 1 as the base of the logarithm. Then what we know is that, if the base of our power function is positive, it doesn’t matter what exponent we put on that base (it could be a positive number, a negative number, of 0), that power function is going to come out as a positive number.

So in summary, because the base can only be a positive number, that means the argument of the logarithm can only be a positive number. Which means that in order to protect our bases, we have to only allow positive arguments inside the logarithm.

0:00 // The argument can’t be negative

0:19 // Parts of the logarithm

0:30 // The argument of the logarithm can’t be negative because of how the base of the logarithm is defined

0:47 // The logarithm is a power function

1:36 // What kind of numbers can the base of the logarithm actually be?

3:11 // How does the base of the logarithm effect the argument of the logarithm?

4:32 // Summary and conclusion


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What are imaginary numbers?

 

Imaginary numbers are any numbers that include the imaginary number i. A mix of imaginary and real numbers gives you what’s called a complex number.

The primary reason we use imaginary numbers is to give us a way to find the root (radical) of a negative number. There’s no way to use real numbers to find the root of a negative number, which prevents us from solving even simple polynomial equations.

But with imaginary numbers, we define the square root of -1 as the imaginary number i, and that allows us to easily find the roots of imaginary numbers.

0:21 Are imaginary numbers real numbers?

0:45 Examples of real and imaginary numbers

1:22 The definition of the imaginary number i

1:39 Why do we need imaginary numbers?

1:53 Why is the imaginary number helpful?

2:52 What happens when you square the imaginary number?


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What are similar triangles?

 

You’ve heard about similar triangles, but do you know what technically makes two triangles similar?

Informally, we can say that two triangles are similar if their associated angles are congruent. In other words, their angle measures have to be the same. However, the triangles don’t necessarily have to be the same size, in order for them to still be similar.

The formal definition of similar triangles tells us that two triangles are similar when the associated angles are congruent, and when the associated side lengths are proportional. If both of those conditions are present, then the triangles are similar. If either or both of those conditions are missing, then the triangles are not similar.

0:17 Informal definition of similar triangles // In order for two triangles to be similar, their angles have to the same, but the triangles themselves can still be different sizes.

0:32 Examples of similar triangles 

0:45 Formal definition of similar triangles // In order for two triangles to be similar, their matching angles must be congruent, and the ratio of their matching sides are the same (their sides must be proportional).

2:02 Summary // If two triangles have the same angles and the same side lengths, then they’re congruent (they’re identical). If two triangles have the same angles but different side lengths, then they’re similar. It’s helpful to remember that, just like all squares are rectangles, all congruent triangles are similar triangles.


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What are partial sums?

 

Ever wondered what a partial sum is? The simple answer is that a partial sum is actually just the sum of part of a sequence. You can find a partial sum for both finite sequences and infinite sequences.

When we talk about the sum of a finite sequence in general, we’re talking about the sum of the entire sequence. Because the sequence is finite, we can find the sum of the entire thing by adding all the terms together. When we talk about the sum of an infinite sequence, we’re talking about the sum of the entire sequence. But because an infinite sequence goes on forever, we have to use special tricks to find the sum of the whole thing. So we use a special name for the sum of an infinite sequence; we call it a “series”.

A partial sum, on the other hand, is just the sum of part of a sequence. In other words, we just take the sum of a few terms of the sequence to find the partial sum. For that reason, we can find the partial sum of either a finite sequence or an infinite sequence. Oftentimes, you’ll be asked to find the partial sum of something like the first five terms of the sequence. In which case, you simply add the first five terms together, and you’ve got the partial sum of the first five terms of the sequence.

00:18 // What is a sequence? What’s the difference between a finite sequence and an infinite sequence? How do you find the sum of a finite sequence or an infinite sequence?

01:12 // What are partial sums? What kind of sequence do you need in order to find a partial sum?

01:41 // How do you express a partial sum in summation notation? How do you expand the summation notation to show the partial sum?


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Why do we rationalize the denominator?

 

Our math teachers always tell us to "rationalize the denominator", but most of the time they don't tell us why. I'll leave it up to you to decide whether or not you think the reasons for rationalizing are good ones, but here are some of the reasons why we do it.

0:13 What we mean when we say "rationalize the denominator" // We're basically just saying "get the root out of the denominator". To do this, we have to multiply both the numerator and denominator by the root that's in the denominator. That way, the roots will cancel. Pro tip: If you have more than just a single root in your denominator, try conjugate method to eliminate the root from the denominator.

1:28 Why should be bother rationalizing the denominator at all? // We'll go over a few reasons why it might be a good idea:

1:34 It's easier for teachers to grade work when everyone's giving their answers in the same format.

1:51 Historically, we think about rationalized fractions as being reduced to lower terms, compared with non-rationalized fractions. And we always want to have our fractions reduced to the lowest possible terms. The reason rationalized fractions are in lower terms is because, before we had calculators, it was easier to do the long division for a rationalized fraction, than it was to do the long division for a non-rationalized fraction.

2:35 Agreeing to rationalize our fractions means we can always recognize like-terms when we have them, which might help us further simplify our answers.

3:00 Summary


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How do you convert a Riemann sum to a definite integral?

 

0:25 // Definition of the Riemann sum
0:50 // What you need in order to use a Riemann sum to find area
1:09 // The difference between a Riemann sum and an integral
1:34 // Converting a Riemann sum into a definite integral
2:09 // Converting an example Riemann sum into a definite integral
2:35 // Summarizing how to use the Riemann sum versus the definite integral

We know that Riemann sums estimate area, and we know that integrals find exact area. But how do we convert a Riemann sum into a definite integral?

The simple answer is that we just use an infinite number of rectangles to find the area, instead of a finite number of rectangles, like we normally would when we're using a Riemann sum. If we want to express the fact that we're going to use an infinite number of rectangles, we need to add a limit out in front of the integral as n (the number of rectangles) approaches infinity, and then the Riemann sum has become a definite integral.

As long as we know the interval over which we're trying to find area, we can change the limit and summation notation into integral notation, with the limits of integration reflecting the interval we're interested in. We change the function f(x_i) into a function f(x), and we change the delta x into dx. And that's actually all it takes to change the Riemann sum into a definite integral.


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How does integration by parts work?

 

In this video we talk about why the integration by parts formula actually works. What we conclude is that there are two reasons.

0:36 Where does integration by parts come from? // First, the integration by parts formula is a result of the product rule formula for derivatives. In a lot of ways, this makes sense. After all, the product rule formula is what lets us find the derivative of the product of two functions. So, if we want to find the integral of the product of two functions, why wouldn’t we try integrating the product rule? In fact, that’s exactly how we get to the integration by parts formula. We start with the product rule, and we integrate both sides. Through some fancy rearranging, we end up with the integration by parts formula. And so we then have the product rule formula for taking the derivative of a product, and we have the related integration by parts formula for taking the integral of a product.

1:09 Why does the integration by parts formula work? // Second, the integration by parts formula works because it takes an integrand that we CAN’T integrate, and turns it into an integrand that we CAN integrate. And that’s the same as any other method of integration, like substitution, partial fractions, or trig substitution, to name a few. They all reformat the integrand so that it’s easier for us to integrate. If we look at the integration by parts formula, we can see that it changes the integral of u times dv to the integral of v times du. So if the integral of v du is easier for us to integrate than the integral of u dv, then the integration by parts formula is going to be a huge help to us!


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When does trapezoidal rule overestimate?

 

In this video we talk about when trapezoidal rule overestimates the area under the curve, when it underestimates the area under the curve, and when it finds exact area.

In general, when a curve is concave down, trapezoidal rule will underestimate the area, because when you connect the left and right sides of the trapezoid to the curve, and then connect those two points to form the top of the trapezoid, you’ll be left with a small space above the trapezoid. The small space is outside of the trapezoid, but still under the curve, which means that it’ll get missed in the trapezoidal rule estimate, even though it’s part of the area under the curve. Which means that trapezoidal rule will consistently underestimate the area under the curve when the curve is concave down.

The opposite is true when a curve is concave up. In that case, each trapezoid will include a small amount of area that’s above the curve. Since that area is above the curve, but inside the trapezoid, it’ll get included in the trapezoidal rule estimate, even though it shouldn’t be because it’s not part of the area under the curve. Which means that trapezoidal rule will consistently overestimate the area under the curve when the curve is concave up.

So if the trapezoidal rule underestimates area when the curve is concave down, and overestimates area when the curve is concave up, then it makes sense that trapezoidal rule would find exact area when the curve is a straight line, or when the function is a linear function.


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How do you find discontinuities?

 

In this video we talk about how to find discontinuities in a function.

0:02 How do you find the discontinuities when you have a picture of the graph of the function? // You need to look for any point where there’s any kind of hole, break, jump, asymptote, or endpoint in the graph. These will all be discontinuities. If you’re tracing the graph from left to right, and you have to pick up your pencil to continue tracing, then there’s a discontinuity at that point. Endpoints are technically discontinuous because there’s only a one-sided limit on one side, which means there’s no general limit, which means the endpoint is technically discontinuous. In order for the function to be discontinuous at an asymptote, the curve needs to exist on both sides of the asymptote. If the curve only exists on one side of the asymptote, then there’s no discontinuity at that point.

1:32 How do you find the discontinuities of a rational function? // Rational functions are fractions with polynomials in the numerator and denominator. Any value that makes the denominator of the fraction 0 is going to produce a discontinuity. If the zero value can be canceled out by factoring, then that value is a point discontinuity, which is also called a removable discontinuity. If the zero value can’t be canceled out by factoring, then that value is an infinite discontinuity, which is also called an essential discontinuity. You can also think of this as just a vertical asymptote for the function.

2:52 How do you find the discontinuities of a piecewise defined function? // You can have discontinuities within each “piece” of the piecewise defined function, just like you could for any other kind of function. But piecewise functions can also be discontinuous at the “break point”, which is the point where one piece stops defining the function, and the other one starts. If the two pieces don’t meet at the same value at the “break point”, then there will be a jump discontinuity at that point. For that reason, jump discontinuities are really common for piecewise functions.

3:17 Summary


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