Differential and Integral Calculus

Differential and Integral Calculus
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Instructors wishing to consider this option should consult the mathematics undergraduate office for more information. The goal of Math31AB is to provide a solid introduction to differential and integral calculus in one variable. The course is aimed at students in engineering, the physical sciences, mathematics, and economics. It is also recommended for students in the other social sciences and the life sciences who want a more thorough foundation in one-variable calculus than that provided by Math 3. Students in 31AB are expected to have a strong background in precalculus mathematics, including polynomial functions, trigonometric functions, and exponential and logarithm functions.

In order to enroll in 31A, students must either take and pass the Mathematics Diagnostic Test at the specified minimum performance level, or take and pass Math 1 at UCLA with a grade of C- or better. Most students entering the sequence at UCLA have taken a calculus course in high school and enter directly into Math 31B, for which there is no enforced prerequisite. The course 31A covers the differential calculus and integration through the fundamental theorem of calculus.

The first part of course 31B is concerned with integral calculus and its applications. The rest of the course is devoted to infinite sequences and series. Single-variable calculus is traditionally treated at many universities as a three-quarter or two-semester course.

Elements of the Differential and Integral Calculus/Contents

Thus Math 31AB does not cover all of the topics included in the traditional single-variable course. The main topics that are omitted are parametric curves and polar coordinates, which are treated at the beginning of 32A. Ample tutoring support is available for students in the course, including the walk-in tutoring service of the Student Mathematics Center.

Math 31A is not offered in the Spring Quarter. Students wishing to start calculus in the Spring may take 31A through University Extension in the Spring or in the Summer. Some imitative words are more surprising than others. How to use a word that literally drives some people nuts.

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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral. has been added, containing simple applicationsof integration. In both the Differential and Integral Calculus, examples illustrat- ing applications to Mechanics and.

The awkward case of 'his or her'. Which of these things doesn't belong?

Test your vocabulary with our question quiz! Definition of integral calculus. Examples of integral calculus in a Sentence Recent Examples on the Web His work -- which included the development of differential calculus and integral calculus -- helped lay the groundwork for the computer and smartphone technology used in today's society. First Known Use of integral calculus circa , in the meaning defined above.

Learn More about integral calculus. Share integral calculus Post the Definition of integral calculus to Facebook Share the Definition of integral calculus on Twitter. Resources for integral calculus Time Traveler! We have a bounded region and we want to compute the area. To this end, notice that, if I use the precise definition of a definite integral, the area of the region 'R' is what?

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What does that mean? It means a particular limit. What limit? In other words, what we want to do is to compute this limit. Now, the thing to notice is, this thing may be a mess but it's computable.

We could use specially designed graph paper or measuring devices to count the units of area under here. We can find all sorts of ways of getting estimates, even the long hard way of the 'U sub n's and the 'L sub n's to pinpoint the area of the region 'R' to as close a degree of accuracy as we want, et cetera. The point is that, if we have never heard of a derivative, the area of the region 'R' is given precisely by this sum. And admittedly, the sum is a mess.

So let's try to do it the so-called easier way. The skeptic looks at this thing and says, who needs this?

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He assumes a background in elementary differential and integral calculus and college physics. Analyzing concavity and inflection points : Analyzing functions Second derivative test : Analyzing functions Sketching curves : Analyzing functions Connecting f, f', and f'' : Analyzing functions Solving optimization problems : Analyzing functions Analyzing implicit relations : Analyzing functions Calculator-active practice : Analyzing functions. History at your fingertips. Translated from English. Derivative of a function of function Chain Rule : dy dy du , where u h x If y f h x , then dx du dx dy 2 Exp: If y log 1 x , then find. John Crescitelli. Using a sequence of test functions, we prove that the subspace of real-valued continuously differentiable functions on a finite dimensional Euclidean space is dense in the space of Lipschitz maps equipped with the L-topology.

And the answer to that is yes. So far, so good. Remember, this is calculus revisited. For those of you who remember calculus from the first time around-- and I'll talk about this later-- it's going to turn out that the required 'G' is a logarithmic function. For those of you don't remember that, there's no harm in not remembering that.

That's what we mean by saying we can't exhibit 'G' explicitly. What do I mean by that? I say, well, that's simple. It's 'G of x'. He says, well, what's 'G of x'?

Well, you see, that implicitly tells me what 'G' is like. But in terms of concrete measurements, I don't know anything about 'G'. I can't express it in terms of well-known, familiar functions. You see, I'm hung up now.

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Namely, this is precise. But I don't know explicitly-- yikes-- I don't know, explicitly such a 'G'.

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And I put the exclamation point out here to emphasize that particular fact. That's the hang-up. The statement that says that the area is 'G of b' minus 'G of a', where 'G prime' equals 'f' hinges on the fact that you can explicitly exhibit such as 'G'. Certainly, if you can't exhibit that 'G', you can still compute this area as a limit.

Certainly, this region 'R' has an area. And by the way, there's plenty of drill on this. This is a hard concept. And as a result, you'll notice that the exercises in this section hammer home on this point, because it's a point that I'm positive that, if you're having trouble at all with integral calculus, this is certainly the most sophisticated part of what we're doing right now.

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The idea looks something like this. I'm on the interval from 1 to 2 here. The x-axis and the line, 'x' equals 'x1'. This is my region 'R'. And what I say is, look it, what do I know about this area? Remember, what I started this lecture with was the knowledge that a prime of 'x' is 'f of x'. In other words, what property does this area function have? In other words, if I now say, look it, boys, I can compute this area, if not exactly, at least to as many decimal place accuracy as I wish, so that the area function is something I can construct. Let me define 'G of x1' simply to be the area under this curve when 'x' is equal to 'x1'.

The beauty is what? That 'G prime of x1' is, by definition, 'A prime of 'x1'. And we have already seen that the derivative of 'A' is 'f', where 'f' is the top curve.

Calculus 1 Introduction, Basic Review, Limits, Continuity, Derivatives, Integration, IB, AP, & AB

In other words, the area under the curve explicitly can be computed as a function of 'x'. And by the way, let me make just a quick aside over here. I happened to throw in the word "logarithms" before.