### Derivatives (Differential Calculus)

With limits, you can rewrite. Note from Tim in the comments: the limit is coming from the right, since x was going to positive infinity. Have 5? Have ?

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Make it And so on. The rules of calculus were discovered informally by modern standards. Yet engines whirl and airplanes fly based on his unofficial results. BetterExplained helps k monthly readers with friendly, insightful math lessons more. An Intuitive Introduction To Limits. How do we make a prediction? Zoom into the neighboring points. Why do we need limits? But for most natural phenomena, it sure seems to. Unfortunately, the connection is choppy: Ack!

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Our prediction is feeling solid. Can we articulate why? Limits are a strategy for making confident predictions. What does perfectly continuous growth look like? Can we find the speed at an instant? About The Site BetterExplained helps k monthly readers with friendly, insightful math lessons more. The integrated function is sometimes called the integrand. We have the lower limit a and the upper limit b , giving the integrating interval [ a , b ]. The variable x is called dummy variable because it is not really important.

Since the Riemann integral is related to the area under the graph of f , the only important information is the shape of the graph. So if we decide to use a different variable in the same formula, the shape and therefore the integral stay the same. Thus, for instance,. Indeed, the area under the same piece of the given parabola is always the same, regardless of what letter we write next to the horizontal axis.

For a more thorough explanation of the meaning of the integral notation not mathematically correct, but very useful for understanding the concept , click here. In our definition, we put the smaller limit the left endpoint as a lower limit. Sometimes we may want to "integrate backward", from b to a.

Often we want to be able to simply write the integral without worrying about the order, so we need a more general definition.

We define. Now that we can integrate with any order of limits, the above equation becomes a general rule: We can switch the limits in the integral, provided we also add the minus sign in front. Example: We investigate the integral.

### Key Analogy: Predicting A Soccer Ball

We need to decide on some partitions that would involve smaller and smaller segments, hoping that the corresponding upper and lower sums will get closer until they agree. Unless there is a good reason to do otherwise, it is usually a good idea to try a regular partition, that is, given a natural number N , split the interval [2,4] into N equal segments.

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Thus we have the following partition check. Now we need to determine the suprema and infima for the sums, but this should be easy just by looking at a picture:. Since our partitions are very specific, one can not expect that in general they would already give us the area in the definition, we investigated all possible partitions.

## Analysis, Complex Analysis

However, here the function is very nice and it turns out that when we send N to infinity, the sums approximate the area well. We have to be a little bit more careful to do it properly by definition:. We can check that this answer is correct by direct calculation from the picture using the formula for the area of a trapezoid. The above calculation was not easy, even though we were lucky that we remembered the formula for adding first N natural numbers.

For more complicated functions, it may be impossible to determine an explicit formula for the upper and lower sums. This is the reason why we usually use other means than the definition for evaluating Riemann integrals see The Fundamental Theorem of Calculus. In our pictures we always had a positive function; the Riemann integral is then equal to the geometric area of the region between the graph of f and the x -axis.

What if we have a negative function? Since the value of f determines the height of rectangles, we get areas with negative sign. We again prove by going back to the original definition of the Laplace Transform. The last integral is just the definition of the Laplace Transform, so we have the time delay property.

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To properly apply the time delay property it is important that both the function and the step that multiplies it are both shifted by the same amount. All four of these function are shown below. Important : To apply the time delay property you must multiply a delayed version of your function by a delayed step.

## Limits of integration - Wikipedia

To prove this we start with the definition of the Laplace Transform and integrate by parts. The first term in the brackets goes to zero as long as f t doesn't grow faster than an exponential which was a condition for existence of the transform. In the next term, the exponential goes to one. The last term is simply the definition of the Laplace Transform multiplied by s. So the theorem is proved. We will use the differentiation property widely.