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Calculus: Help and Review13 chapters | 148 lessons

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Lesson Transcript

Instructor:
*Shaun Ault*

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

This lesson is an introduction to differential calculus, the branch of mathematics that is concerned with rates of change. If you ever wanted to know how things change over time, then this is the place to start!

What is **differential calculus**? Let's take a car trip and find out!

Suppose we take a trip from New York, NY to Boston, MA. That's roughly 200 miles, and (depending on the traffic), it will take about four hours. Now, we all know that distance equals rate multiplied by time, or *d* = *rt*. In this example, we have distance and time, and we interpret **velocity** (or speed) as a rate of change. So we could figure out our **average velocity** during the trip by dividing distance over time. That's:

(200 miles) / (4 hours) = 50 miles/hour

However, if you've ever driven in Boston or New York, you know that when you're in the city, you're driving much slower than 50 mph! On the other hand, when you get to the highways, you expect to be going more than 50 mph. The calculation we did is just an average, and it's answering the question: if my velocity stayed the same throughout the entire trip, what would it be?

But your car knows better. It has a speedometer that keeps track of the speed (velocity) at any given instant. When you look at the speedometer and it reads 61 mph, that is telling you the **instantaneous velocity** at the particular instant of time you decided to look at it. How does it know? Your car is doing something like differential calculus to figure it out!

**Differential calculus** is the study of **rates of change** of functions, using the tools of limits and derivatives.

Now I know some of these words may be unfamiliar at this point in your journey, but we'll take some time to explain them in this lesson. Remember, this is only an introduction to the important topic of differential calculus. A full treatment takes a semester or more to explain!

Okay, let's dig into these ideas further. There are two related concepts to discuss, average rate of change and instantaneous rate of change.

Average rates start with the idea of a *y* = *f*(*x*). You input *x*, and get back output *y*. Let's assume you know this from algebra or pre-calculus. In calculus, a **rate of change** is a measurement of how the *y*-values of a function change with respect to changes in the *x*-values. Sound confusing? Well, let's just think of **rate** in the same way as velocity. It's a fraction, with the change in *y* on top divided by the change in *x* on the bottom.

Suppose we start at *x1* and end at *x2*. Then the change in *x* is the difference, *x2* - *x1*, which is similar with the *y*-values as we see here. The triangle notation is a Greek capital *delta*, which means 'change in' a quantity.

This is really nothing new. It's the same slope formula you've seen way back in algebra class! It's just rise over run. And it measures what we call the **average rate of change** of *f* on the interval from *x1* to *x2*. The only difference now is that we have to use the given function *f* to find the *y*-values, like this:

Ok, but what if we want to know the rate of change at a particular instant, or our instantaneous rate? The average rate of change formula requires two different points, *x1* and *x2*, and it doesn't work so well when *x1* = *x2* because then *x2* - *x1* = 0 on the bottom of the fraction. And, we're never allowed to divide by 0, right?

This is where limits come in. You see, even though we're not allowed to let *x2* = *x1* (since that would produce a division by 0, we might just allow *x2* to get closer and closer to *x1*.

However, the details get tricky here. Let's say we fix *x1* = *x* (for some arbitrary value *x*), and let *x2* = *x* + (a little bit). Then if the little bit is small enough, the calculation for average velocity should be a fairly accurate estimate of instantaneous velocity at *x*. Now, instead of saying 'a little bit', let's use a variable, such as *h*. That is, *x2* = *x* + *h*. Then the rate of change formula looks like this:

Finally, we should assume that *h* will eventually approach 0 (though in practice, this only can occur after some algebra has been used to simplify the expression ). We write this using limit notation, and this becomes the definition of **instantaneous rate of change**, or the derivative of *f*, with the limit definition of the derivative looking like this:

Now that we've gotten our feet wet a little bit, let's talk about more terminology. The instantaneous rate of change of a function is called the function's derivative, as we mentioned just a minute ago. In later lessons, you'll develop tools and formulas to help find derivatives. For example, if *f*(*x*) = *x^2*, then something called the **power rule** implies that the derivative is *f*(*x*) = 2. And the process of finding a derivative is called **differentiation**.

There are tons of different techniques used to differentiate functions. There are also tons of different applications for differential calculus! As such, differential calculus may be thought of as the branch of mathematics in which you learn how and why to differentiate functions.

Okay, let's quickly review. **Differential calculus** is the branch of mathematics concerned with rates of change. The idea starts with a formula for average rate of change, which is essentially a slope calculation. Then, using limits, a formula for the instantaneous rate of change can be developed, which is called the derivative of a function.

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Calculus: Help and Review13 chapters | 148 lessons

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