# Calculus for Sequences: Understand Basics of Differentiation & Integration

• terryds
In summary, the conversation discusses the use of calculus for determining the order of a polynomial in a sequence of values. By looking at the differences between terms in the sequence, one can determine the order of the polynomial. Calculus is used to find derivatives, which show the rate of change of the function, and the number of derivatives needed to eliminate all power terms indicates the order of the polynomial.
terryds
First, please take a look at http://www.purplemath.com/modules/nextnumb.htm (the second-order sequence problem)

http://www.purplemath.com/modules/nextnumb.htm :
"Since these values, the "second differences", are all the same value, then I can stop. It isn't important what the second difference is (in this case, "2"); what is important is that the second differences are the same, because this tells me that the polynomial for this sequence of values is a quadratic.(Once you've studied calculus, you'll be able to understand why this is so. For now, just trust me that this works.)"

And now, I've studied introductory (basic differentiation and integration)
But, I still have no idea what calculus for sequence is.

Please explain me or give me a link to a web which clearly explains it.

For a linear equation, y=mx +c, the first derivative ( y' = m ) is a constant.
For quadratic, y=ax2 + bx + c, the first derivative ( y'= 2ax + b ) is linear and the second derivative (y''=2a) is constant.
For cubic first deriv is quadratic, second deriv is linear, third deriv is constant,
and so on.

Derivatives are the rate of change of the function, so derivative of a constant is zero. There is no point in calculating more derivatives, because they are now all zero.
For polynomials each differentiation reduces the powers by one and loses any constants (which is why you need to add a constant in integration.)In your table of differences, the first differences reflect the first derivative of the sequence function - the size of the difference is how much the numbers are changing term by term.
The second difference reflects the second derivative, the third difference the third deriv, etc.
Once your differences are constant you know how many differentiations are needed to eliminate all power terms, so you know the order of the polynomial.

If you consider the sequence $x_n = n^k$ for integer $k$, then you see that $x_{n+1} - x_n = (n+1)^k - n^k = kn^{k-1} + \dots + 1$. Continue this and you'll find that the $k$th difference is $k!n^0 = k!$. The (k+1)th difference is zero, since the difference of a constant sequence is zero.

## 1. What is the purpose of calculus for sequences?

The purpose of calculus for sequences is to study the behavior of sequences and understand how they change over time. It helps us analyze and predict the behavior of sequences by using concepts such as differentiation and integration.

## 2. What is the difference between differentiation and integration in calculus for sequences?

Differentiation in calculus for sequences is the process of finding the rate of change of a sequence at a specific point, while integration is the process of finding the area under the curve of a sequence. In other words, differentiation focuses on the instantaneous change of a sequence, while integration looks at the cumulative change over a certain interval.

## 3. How are differentiation and integration useful in real-world applications?

Calculus for sequences, specifically differentiation and integration, has many real-world applications. For example, it is used in physics to study the motion of objects, in economics to analyze supply and demand curves, and in biology to understand the growth and development of populations.

## 4. What are some common techniques used in calculus for sequences?

Some common techniques used in calculus for sequences include the power rule, product rule, quotient rule, and chain rule for differentiation; and the fundamental theorem of calculus, substitution, and integration by parts for integration.

## 5. How can I improve my understanding of calculus for sequences?

To improve your understanding of calculus for sequences, it is important to practice regularly and work through problems to apply the concepts. You can also seek help from a tutor or join a study group to clarify any doubts and discuss different approaches to solving problems. Additionally, reviewing and understanding the underlying concepts and principles will also help in strengthening your understanding of calculus for sequences.

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