Everyone thinks calculus is super hard! And it is, kind of. But with this series, you'll get a superb foundation of conceptual understanding regarding differentiation and integration, as well as lots of practice with all the types of problems you will see on your tests. Let's do this!

An alternating series is one in which the terms alternate sign, so positive, then negative, then positive, etc. How can we generate a series like this, and how can we find out whether it is convergent or divergent? Find out now!

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OVERSEAS

Alternating Series, Types of Convergence, and the Ratio Test

We now have a pretty good grasp of what integration is, and how to do it. But what about when we see an integral without any limits of integration listed? This is extremely common, and these are called indefinite integrals. These won't be any harder to evaluate, because we just take the antiderivative as usual, and then leave it at that! There are a few details to go over, and then we can practice with some tougher examples.

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Evaluating Indefinite Integrals

When we learned about definite integrals, we saw that we can evaluate the antiderivative over the limits of integration to get a number, the area under the curve over that interval. But what if that interval is infinitely large? Rather surprisingly, an infinitely large interval can actually yield a finite area! Let's see how this works by checking out a variety of different improper integrals and attempting to evaluate them.

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Evaluating Improper Integrals

We are just about done with calculus! Before we go, let's talk about one more topic that brings together differentiation and integration. It's called the mean value theorem. There is one version that utilizes differentiation, and another version that uses integrals. Let's learn both, and see how they can both be used to compute the average value of a function over some interval.

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The Mean Value Theorem For Integrals: Average Value of a Function

Now that we have the basics down regarding integration, it's time to start looking at trickier functions, and eventually more complex integrands. First, we will see trig functions inside integrands all the time. How do we integrate these? Well if we remember the derivatives of the trig functions, it won't be too hard!

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Evaluating Integrals With Trigonometric Functions

Our bag of tricks for integration is complete! Now it's time to put all our knowledge to use and evaluating integrals without any context. What strategy should we use in each case? Watch me work through them or pause the clip on each example and give them a try! Remember, we know how to use the substitution rule, integration by parts, and integration by trigonometric substitution. We may have to do other clever manipulations too. You can do it!

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Advanced Strategy for Integration in Calculus

Ok, we've wrapped up differential calculus, so it's time to tackle integral calculus! It's definitely the trickier of the two, but don't worry, it's nothing you can't handle! First things first, what is integration? Well it's that thing we mentioned way back in the introduction clip, when we talked about adding up the infinite tiny rectangles. Just watch and you'll see!

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What is Integration? Finding the Area Under a Curve

Differentiation and integration are the two main operations in calculus. We just discussed one way to interpret differentiation by finding the equation for a tangent line. Let's look at one other way to interpret differentiation, by discussing the rate of change in the position of an object, or its velocity. As it turns out these two approaches are really the same thing, so let's see how this works!

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Understanding Differentiation Part 2: Rates of Change

Sometimes it can be tricky to calculate the precise sum of a series, but we can still use some nifty tricks to determine whether it is convergent or divergent. These are the integral test, where we approximate a sum by evaluating a corresponding improper integral, and the comparison test, where we compare a series to a simplified version with known behavior. They're not too hard, check it out!

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Estimating Sums Using the Integral Test and Comparison Test

We learned a little bit about sequences and series earlier in the mathematics course, but now its time to work with these some more, now that we understand calculus! First up, what does it mean for a sequence or series to be convergent or divergent, and how can we tell which one it is? Let's find out!

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Convergence and Divergence: The Return of Sequences and Series

We are pretty good at taking derivatives now, but we usually take derivatives of functions that are in terms of a single variable. What if we have x's and y's in there? We need a new technique! It's called implicit differentiation, and it goes like this.

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Implicit Differentiation

We now know one method for finding the volume of a solid of revolution. But there are tricky examples where the normal method won't work, like when both the inner and outer radius of a washer are being determined by the same function. Luckily there is another method we can use! It involves cylindrical shells. What's that, you ask? Watch this!

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Calculating Volume by Cylindrical Shells

So we know a lot about differentiation, and the basics about what integration is, so what do these two operations have to do with one another? Everything! And the discovery of their relationship is what launched modern calculus, back in the time of Newton and pals. Check it out!

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The Fundamental Theorem of Calculus: Redefining Integration

We've gone through a few different types of series, so let's learn another type, power series. What are these, and how can we tell if they will converge or diverge? The ratio test is back!

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Power Series

We've learned how to use calculus to find the area under a curve, but areas have only two dimensions. Can we work with three dimensions too? Yes we can! We can find the volume of things called solids of revolution, again by integration, it's just slightly more involved. Let's learn this neat trick!

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Calculating the Volume of a Solid of Revolution by Integration

Now that we can take the derivative of polynomial functions, as well as products and quotients thereof, it's time to start looking at special functions, like trigonometric functions. How can we take the derivative of these if we can't use the power rule? Good thing we understand the concept of differentiation so well! This will help make sense of the whole task. Check it out!

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Derivatives of Trigonometric Functions

Now we know how to take derivatives of polynomials, trig functions, as well as simple products and quotients thereof. But things get trickier than this! We may want to take the derivative of a composite function, where some function is operating on some other function. How can we do this? With the chain rule! It's easier than you think, I promise.

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Derivatives of Composite Functions: The Chain Rule

Let's wrap up our survey of calculus! We have one more type of series to learn, Taylor series, and special case of those called Maclaurin series. This utilizes differentiation, and you'll see some familiar polynomials in this one!

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Taylor and Maclaurin Series

Now that we know where the power rule came from, let's practice using it to take derivatives of polynomials! Furthermore, when we have products and quotients of polynomials, we can take the derivative of these as well, but we need special rules. Let's learn all the rules and practice them now!

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Derivatives of Polynomial Functions: Power Rule, Product Rule, and Quotient Rule

Now that we know that integration simply requires evaluating an antiderivative, we don't have to look at rectangles anymore! But integration can also be a very difficult technique. Let's start simple by learning some properties of integrals, and getting some practice with evaluating simple definite integrals.

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Properties of Integrals and Evaluating Definite Integrals

We know how to graph functions, and we know how to take derivatives, so let's graph some derivatives! Many students find that this hurts their brain, but it's just about practice! Remember that the value of a function and the rate of change of that function at a particular point are completely unrelated! A function may be positive but have a negative rate of change, or vice versa. Let's learn how to graph derivatives intuitively, and then some applications of this in physics!

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Graphing Functions and Their Derivatives

We learned about limits earlier in this series. We know what they represent, and we know how to evaluate them. Then we found that we don't need them that much, because we have better methods for differentiating functions than all that business with tangent lines and limits. But limits still have applications, and we can use them to find out the value of a function at a certain point when we can't figure this out from the function itself. How? With L'Hospital's rule!

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Understanding Limits and L'Hospital's Rule

Now that we understand differentiation, it's time to learn about all the amazing things we can do with it! First up is related rates. Sometimes the rates at which two parameters change are related to one another by some equation. With our newfound understanding of implicit differentiation, it's not too hard to find this precise relationship so that we can do important scientific calculations. Check it out!

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Related Rates in Calculus

The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!

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Understanding Differentiation Part 1: The Slope of a Tangent Line

What else is differentiation good for? Well if we are looking at the graph of a function, differentiation makes it super easy to find where any local maxima and minima occur. This act in itself has many applications, but before we learn those, let's just learn how to find the maxima and minima!

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Finding Local Maxima and Minima by Differentiation

What good is calculus anyway, what does it have to do with the real world?! Well, a lot, actually. Optimization is a perfect example! If you want to figure out how to maximize your profits or minimize your costs, or if you want to maximize an area or minimize a distance, you are finding the maxima and minima of a function, and that's doing calculus! All we need to know is how to take derivatives and set them equal to zero, so let's go make some money!

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Optimization Problems in Calculus

With the basics of integration down, it's now time to learn about more complicated integration techniques! We need special techniques because integration is not as straightforward an algorithm as differentiation. The first technique we will add to our bag of tricks is the substitution rule. Check it out!

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Integration Using The Substitution Rule

We've got two techniques in our bag of tricks, the substitution rule and integration by parts, so it's time to learn the third and final, and that's integration by trigonometric substitution. This is probably the toughest one, and it only works in very specific circumstances, but when those circumstances arise, it works like a charm! So let's make sure we know how to execute this bad boy.

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Integration By Trigonometric Substitution

In introducing the concept of differentiation, we investigated the behavior of some parameter in the limit of something else approaching zero or infinity. This concept of limits is how calculus got started. As the field developed, new techniques arose such that we don't have to find limits very often anymore, but we still use them sometimes, so let's make sure we know what they are, how they work, and how to evaluate them!

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Limits and Limit Laws in Calculus

After discussing differentiation at great length, it is time to connect this concept with the act of taking the derivative of a function. In actuality these mean the same thing, but using the power rule to take the derivative of a function is actually much simpler than all that business with limits and tangent lines! But it is important to know what the power rule is and where it comes from, so let's derive it together. This way, when you take the derivatives of functions, you'll know exactly what is really happening.

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What is a Derivative? Deriving the Power Rule

By now we are very familiar with the concept of evaluating definite integrals to find the area under a curve. But this always gives us the area between a curve and the x-axis. Is there some other way to apply integration to find some different area? Yes there is! We can also find the area between any two curves on the coordinate plane over some interval, simply by subtracting two integrals. It's easier than you think! Let's find the area between two curves.

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Finding the Area Between Two Curves by Integration

With the substitution rule, we've begun building our bag of tricks for integration. Now let's learn another one that is extremely useful, and that's integration by parts. Let's learn how this rule was derived and then get some practice using it! You're gonna love it.

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Integration By Parts

When we take the derivative of a function, we get another function. So what's to stop of us from taking the derivative of that function? Nothing! If we take the derivative of a derivative, we get the second derivative. And it doesn't stop there. This sounds abstract, but it has tremendous application in classical physics, specifically kinematics.

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Higher Derivatives and Their Applications

Let's learn how to differentiate just a few more special functions, those being logarithmic functions and exponential functions. These are a little funky, but there are some simple rules we can memorize!

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Derivatives of Logarithmic and Exponential Functions

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Calculus #8

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