Palm City Life 2026

Calculus for Beginners

Pretty much everyone would know how to count to 10, add, subtract, multiply, and divide. And most of the US population would know how to graph lines, solve equations, and factor polynomials. However, a layman may see that the biggest monster of mathematics is Calculus. To be truthful, not a large percentage of the world would know how to do Calculus. But I have completed worse ones before.

However, if you’re curious on what Calculus is like, I might teach some baby steps. Two of the most common topics in Calculus are derivatives (differentiation) and anti-derivatives (integration). While Algebra and less would describe number patterns, Calculus measures the rate of change.

Number Patterns:

So let’s take a look at these number patterns. In all example sequences, there are ten terms. What can you notice about them?

  • Sequence A: 5,5,5,5,5,5,5,5,5,55, 5, 5, 5, 5, 5, 5, 5, 5, 5
  • Sequence B: 1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10
  • Sequence C: 2,4,6,8,10,12,14,16,18,202, 4, 6, 8, 10, 12, 14, 16, 18, 20
  • Sequence D: 1,5,9,13,17,21,25,29,33,371, 5, 9, 13, 17, 21, 25, 29, 33, 37
  • Sequence E: 1,3,6,10,15,21,28,36,45,551, 3, 6, 10, 15, 21, 28, 36, 45, 55
  • Sequence F: 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000
  • Sequence G: 2,4,8,16,32,64,128,256,512,10242, 4, 8, 16, 32, 64, 128, 256, 512, 1024
  • Sequence H: 1,2,6,24,120,720,5040,40320,362880,36288001, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800

Let’s take a walkthrough in each sequence.

  • The first one, Sequence A, has only 5’s. That means, no matter where you are in the sequence, the number will always be 55. In Algebra, we write this sequence as a(n)=5a(n)=5.
  • Sequence B is the basic counting sequence. When you learn addition, each number is one larger than the previous number. 2−1=12-1=1. 3−2=13-2=1. 4−3=14-3=1. No matter what two consecutive terms we put in a subtraction problem, the difference will always be 11. In Algebra, we write this sequence as a(n)=na(n)=n.
  • Sequence C is the same as Sequence B but only using even numbers. All consecutive odd numbers have a common difference of 22, as well as consecutive even numbers. In Algebra, we write this sequence as a(n)=2na(n)=2n.
  • Sequence D, like Sequences B and C, has a common difference. 5−1=45-1=4. 9−5=49-5=4. 13−9=413-9=4. No matter what two consecutive terms we put in a subtraction problem, the difference will always be 44. While first terms may differ, all sequences using these terms only are variants of a(n)=4n+1a(n)=4n+1.
  • Sequence E is when it starts to get spicy. As you see, there is no common difference between two consecutive terms. 3−1=23-1=2. 6−3=36-3=3. 10−6=410-6=4. However, the difference is increasing by 11 for every two consecutive terms you go up. So instead of focusing on the main sequence, focus on the sequence of changes, which is the sequence of the differences between consecutive terms. If the first sequence of changes has a common difference, but not the main sequence, this is what we call a quadratic sequence. In Algebra, we write this sequence as a(n)=12n2+12na(n)=\frac{1}{2}n^{2}+\frac{1}{2}n, or a(n)=12n(n+1)a(n)=\frac{1}{2}n(n+1).
  • Sequence F is similar, but what we have is a cubic sequence. This means the main sequence doesn’t have a common difference, as well as the first sequence of changes. But the second sequence of changes has a common difference. In Algebra, we write this sequence as a(n)=n3a(n)=n^{3}.
  • If you tried subtracting terms in Sequence G, you’re out of luck. That’s because there are no common difference between the terms. However, it looks like each term is multiplied rather than added. 4/2=24/2=2. 8/4=28/4=2. 16/8=216/8=2. No matter what two consecutive terms we put in a division problem, the quotient will always be 22. In Algebra, we write this sequence as a(n)=2na(n)=2^{n}.
  • Sequence H is another multiplicative sequence. However, the quotient between two consecutive numbers increases by 11 for each two consecutive numbers. 2/1=22/1=2. 6/2=36/2=3. 24/6=424/6=4. This is called a factorial sequence, which is the product of all consecutive natural numbers. We write this sequence as a(n)=n!a(n)=n!. You’ll see more of these in combinatorics, probability, and statistics.

As I go over this math lesson, I’m only going to focus on additive and subtractive sequences. But some sequences are out of order.

Differentiation:

As stated above, Calculus is about finding the rate of change. The base sequence is the first sequence you’ll see. The derivative sequence, on the other hand, is the sequence of changes. For starters, we’ll be dealing with these four sequences:

  • 4,4,4,4,4,4,4,4,4,44, 4, 4, 4, 4, 4, 4, 4, 4, 4
  • 20,25,30,35,40,45,50,55,60,6520, 25, 30, 35, 40, 45, 50, 55, 60, 65
  • 36,33,30,27,24,21,18,15,12,936, 33, 30, 27, 24, 21, 18, 15, 12, 9
  • 1,4,9,16,25,36,49,64,81,1001, 4, 9, 16, 25, 36, 49, 64, 81, 100

Another thing. We’re only subtracting current terms from next terms. Not next terms from current terms. So if the sequence is 1,2,31, 2, 3, we only consider 2−12-1 and 3−23-2, not 1−21-2 or 2−32-3. If it was the other way around, 3,2,13, 2, 1, we’ll consider 2−32-3 and 1−21-2, but not 3−23-2 or 2−12-1.

First Sequence:

As we know, the number will always be 44 regardless of what order the term is. The first term is 44. The tenth term is 44. The one hundredth term is 44. This also means that at all points in the sequence, two consecutive terms are 44. So what happens if you subtract two consecutive terms or two terms from any part of the sequence? The answer is 00, because any number minus itself is zero. 4−4=04-4=0, and all numbers are 44. Therefore, the sequence of changes is a series of zeroes.

Differences of Terms:
1. 4−4=04-4=0
2. 4−4=04-4=0
3. 4−4=04-4=0
4. 4−4=04-4=0
5. 4−4=04-4=0
6. 4−4=04-4=0
7. 4−4=04-4=0
8. 4−4=04-4=0
9. 4−4=04-4=0
  • Base Sequence: 4,4,4,4,4,4,4,4,4,44, 4, 4, 4, 4, 4, 4, 4, 4, 4
  • Sequence of Changes: 0,0,0,0,0,0,0,0,00, 0, 0, 0, 0, 0, 0, 0, 0

Second Sequence:

What do all ten terms have in common? They’re all multiples of 55. All consecutive multiples. That means, if we take the difference between two consecutive terms, you’re only going to get 55 as your difference. Therefore, the sequence of changes is a series of fives.

Differences of Terms:
1. 25−20=525-20=5
2. 30−25=530-25=5
3. 35−30=535-30=5
4. 40−35=540-35=5
5. 45−40=545-40=5
6. 50−45=550-45=5
7. 55−50=555-50=5
8. 60−55=560-55=5
9. 65−60=565-60=5
  • Base Sequence: 20,25,30,35,40,45,50,55,60,6520, 25, 30, 35, 40, 45, 50, 55, 60, 65
  • Sequence of Changes: 5,5,5,5,5,5,5,5,55, 5, 5, 5, 5, 5, 5, 5, 5

Third Sequence:

Just like the second sequence, the focus is on multiples, with 33 being the central focus. But the sequence isn’t an increasing sequence. It’s a decreasing sequence. That means, the next term is always smaller than the previous term, and when you subtract a bigger number from a smaller number, the result will be negative. For this reason, the sequence of changes is a series of negative threes.

Differences of Terms:
1. 33−36=−333-36=-3
2. 30−33=−330-33=-3
3. 27−30=−327-30=-3
4. 24−27=−324-27=-3
5. 21−24=−321-24=-3
6. 18−21=−318-21=-3
7. 15−18=−315-18=-3
8. 12−15=−312-15=-3
9. 9−12=−39-12=-3
  • Base Sequence: 36,33,30,27,24,21,18,15,12,936, 33, 30, 27, 24, 21, 18, 15, 12, 9
  • Sequence of Changes: −3,−3,−3,−3,−3,−3,−3,−3,−3-3, -3, -3, -3, -3, -3,- 3, -3, -3

Fourth Sequence:

Our last sequence is not just an increasing sequence. It’s increasing at a faster rate. This means that there’s no common difference in the main sequence. But since this is a quadratic sequence, the sequence of changes has a common difference. 4−1=34-1=3. 9−4=59-4=5. 16−9=716-9=7. And so on. What we got was a sequence of consecutive odd numbers.

Differences of Terms:
1. 4−1=34-1=3
2. 9−4=59-4=5
3. 16−9=716-9=7
4. 25−16=925-16=9
5. 36−25=1136-25=11
6. 49−36=1349-36=13
7. 64−49=1564-49=15
8. 81−64=1781-64=17
9. 100−81=19100-81=19
  • Base Sequence: 1,4,9,16,25,36,49,64,81,1001, 4, 9, 16, 25, 36, 49, 64, 81, 100
  • Sequence of Changes: 3,5,7,9,11,13,15,17,193, 5, 7, 9, 11, 13, 15, 17, 19

And that’s how Calculus works. If you can describe the pattern in the sequence of changes, you’ll find the rate of change.

As a bonus, here are some Calculus facts regarding observations with these sequences:

  • When there’s no change in the function, the derivative is zero.
  • When the change is constant across the function, the derivative is constant.
  • When the change varies across the derivative is an algebraic expression.

Increasing and Decreasing Sequences:

One property of differentiation is that wherever you see an increase, the derivative is positive. And wherever you see a decrease, the derivative is negative. Let’s take a look at these three sequences:

  • 1,2,3,4,3,2,1,2,3,41, 2, 3, 4, 3, 2, 1, 2, 3, 4
  • 0,1,3,6,10,15,19,22,24,250, 1, 3, 6, 10, 15, 19, 22, 24, 25
  • 100,95,85,70,55,45,40,40,45,50100, 95, 85, 70, 55, 45, 40, 40, 45, 50

They don’t look like standard sequences, but Calculus considers all patterns.

First Sequence:

From just the first four terms, it looks like the sequence is increasing by 11 after each term. But when we get to the next three terms, the sequence is decreasing by 11. And then it increases by 11 in the last three terms. The sequence of changes is a series of ones and negative ones.

Differences of Terms:
1. 2−1=12-1=1
2. 3−2=13-2=1
3. 4−3=14-3=1
4. 3−4=−13-4=-1
5. 2−3=−12-3=-1
6. 1−2=−11-2=-1
7. 2−1=12-1=1
8. 3−2=13-2=1
9. 4−3=14-3=1
  • Base Sequence: 1,2,3,4,3,2,1,2,3,41, 2, 3, 4, 3, 2, 1, 2, 3, 4
  • Sequence of Changes: 1,1,1,−1,−1,−1,1,1,11, 1, 1, -1, -1, -1, 1, 1, 1

What can we infer about the base sequence and its derivative sequence? The derivative is positive wherever the base sequence is increasing, but negative wherever the base sequence is decreasing.

Second Sequence:

Throughout the entire sequence, the sequence is increasing. But when we take the sequence of changes, notice the changes. The consecutive terms at both ends are closer than the two consecutive terms in the middle. As you begin the sequence, it not only increases, but also increases at a faster rate. And then it starts to slow down before reaching the end.

Differences of Terms:
1. 1−0=11-0=1
2. 3−1=23-1=2
3. 6−3=36-3=3
4. 10−6=410-6=4
5. 15−10=515-10=5
6. 19−15=419-15=4
7. 22−19=322-19=3
8. 24−22=224-22=2
9. 25−24=125-24=1
  • Base Sequence: 0,1,3,6,10,15,19,22,24,250, 1, 3, 6, 10, 15, 19, 22, 24, 25
  • Sequence of Changes: 1,2,3,4,5,4,3,2,11, 2, 3, 4, 5, 4, 3, 2, 1

What can we infer about the base sequence and the sequence of changes? Wherever the sequence is increasing, the derivative is increasing when the base sequence is increasing at a faster rate, but decreasing when the base sequence is increasing at a slower rate.

Third Sequence:

The sequence starts out decreasing, and then it increases. While it’s decreasing, there’s a point where it decreasing at its highest rate. It then slows down, and stops, before it turns around.

Differences of Terms:
1. 95−100=−595-100=-5
2. 85−95=−1085-95=-10
3. 70−85=−1570-85=-15
4. 55−70=−1555-70=-15
5. 45−55=−1045-55=-10
6. 40−45=−540-45=-5
7. 40−40=040-40=0
8. 45−40=545-40=5
9. 50−45=550-45=5
  • Base Sequence: 100,95,85,70,55,45,40,40,45,50100, 95, 85, 70, 55, 45, 40, 40, 45, 50
  • Sequences of Changes: −5,−10,−15,−15,−10,−5,0,5,5-5, -10, -15, -15, -10, -5, 0, 5, 5

What can we infer about the base sequence and the sequence of changes? For the first part, we can see a contrast of how the sequences of changes works when the sequence is decreasing. Unlike an increasing sequence, the derivative is decreasing when the base sequence is decreasing at a faster rate, but increasing when the base sequence is decreasing at a slower rate. But in both parts, when the base sequence is increasing or decreasing at the same rate, the derivative is constant.

Going over all three inferences, here are some Calculus facts:

  • Wherever a function is increasing:
    • The derivative is positive regardless.
    • If the function is increasing at a faster rate, then the derivative is increasing.
    • If the function is increasing at a constant rate, then the derivative is constant.
    • If the function is increasing at a slower rate, then the derivative is decreasing.
  • Wherever a function is decreasing:
    • The derivative is negative regardless.
    • If the function is decreasing at a faster rate, then the derivative is decreasing.
    • If the function is decreasing at a constant rate, then the derivative is constant.
    • If the function is decreasing at a slower rate, then the derivative is increasing.
  • Wherever a function is constant, at a maximum, or at a minimum, the derivative is zero.

Higher Order Differentiation:

In Calculus, each function doesn’t just have a derivative, but also a family of derivatives. As you continue taking the derivative of the same function, you will get a higher order of derivatives. The derivative of the base function is the first derivative. The derivative of the first derivative is the second derivative. The derivative of the second derivative is the third derivative. The derivative of the third derivative is the fourth derivative. And so on.

Some functions have a finite set of derivatives. That means, you can keep differentiating until…

  • You get to a derivative that is zero.
  • You create a cycle of repeating base functions.

Now how do they work with the sequence of changes as discussed in this entry? You find the sequence of changes to a previous sequence of changes. Our sequence is:

  • 5,40,135,320,625,1080,1715,2560,3645,50005, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000

This sequence has ten terms. The first derivative sequence will have nine terms. The second derivative will have eight terms. The third derivative sequence will have seven terms.

First Derivative:

So let’s start by subtracting the current terms from the next term. 40−5=3540-5=35, so the first term of the first derivative sequence is 3535. 135−40=95135-40=95, which makes the next term 9595. 320−135=185320-135=185, so the next term in the sequence is 185185. Going even further, the differences are 305305, 455455, 635635, 845845, 10851085, and 13551355. So our first derivative sequence is:

  • 35,95,185,305,455,635,845,1085,135535, 95, 185, 305, 455, 635, 845, 1085, 1355

Second Derivative:

When we look for the second derivative, we must look for the sequence of changes from the first derivative sequence, not the base sequence. That being said, 95−35=6095-35=60, which makes the first term of the sequence 6060. 185−95=90185-95=90, which makes 9090 as the next term. 305−185=120305-185=120, giving us 120120 as the third term. Going even further, the differences are 150150, 180180, 210210, 240240, and 270270. Our second derivative sequence is:

  • 60,90,120,150,180,210,240,27060, 90, 120, 150, 180, 210, 240, 270

Third Derivative:

We are now at the third derivative. This time, we are looking for the sequence of changes from the second derivative sequence. 90−60=3090-60=30, which makes our first term 3030. 120−90=30120-90=30, making our second term 3030 as well. 150−120=30150-120=30, giving us 3030 as our third term. The next four terms are 3030, 3030, 3030, and 3030. Our third derivative sequence is:

  • 30,30,30,30,30,30,3030, 30, 30, 30, 30, 30, 30

And that’s how higher order differentiation works. For a review, here are all of our sequences we found:

Family of Derivatives:
  • Base Sequence: 5,40,135,320,625,1080,1715,2560,3645,50005, 40, 135, 320, 625, 1080, 1715, 2560, 3645, 5000
  • 1st Derivative Sequence: 35,95,185,305,455,635,845,1085,135535, 95, 185, 305, 455, 635, 845, 1085, 1355
  • 2nd Derivative Sequence: 60,90,120,150,180,210,240,27060, 90, 120, 150, 180, 210, 240, 270
  • 3rd Derivative Sequence: 30,30,30,30,30,30,3030, 30, 30, 30, 30, 30, 30

Other Sequences:

Let’s go over two other sequences. A seven-term sequence and a nine-term sequence to be exact:

  • 3,6,12,24,48,96,1923, 6, 12, 24, 48, 96, 192
  • 2,5,8,5,2,5,8,5,22, 5, 8, 5, 2, 5, 8, 5, 2
Exponential Sequence:

In Calculus, there’s at least one base function that retains its base function each time you differentiate. This is the exponential function. What we are given is this geometric sequence of 2n2^{n} times 33. Starting with the first derivative, 6−3=36-3=3, 12−6=612-6=6, 24−12=1224-12=12, 48−24=2448-24=24, 96−48=4896-48=48, and 192−96=96192-96=96. Notice that the first six terms in the first sequence of changes is the same as the first six terms of the base sequence. We could try it again for the second sequence of changes, the third sequence of changes, and so on, but the results aren’t going to change.

  • Base Sequence: 3,6,12,24,48,96,1923, 6, 12, 24, 48, 96, 192
  • 1st Derivative Sequence: 3,6,12,24,48,963, 6, 12, 24, 48, 96
  • 2nd Derivative Sequence: 3,6,12,24,483, 6, 12, 24, 48
  • 3rd Derivative Sequence: 3,6,12,243, 6, 12, 24
Sine Sequence:

Not like the first sequence, the second sequence will at least have different terms in the first sequence of changes. 5−2=35-2=3, 8−5=38-5=3, 5−8=−35-8=-3, and 2−5=−32-5=-3. Our first sequence of changes is 3,3,−3,−3,3,3,−3,−33, 3, -3, -3, 3, 3, -3, -3. Going to the next sequence of changes, 3−3=03-3=0, −3−3=−6-3-3=-6, −3+3=0-3+3=0, and 3+3=63+3=6. Our second sequence of changes is 0,−6,0,6,0,−6,00, -6, 0, 6, 0, -6, 0. Going to the last sequence of changes, −6−0=−6-6-0=-6, 0+6=60+6=6, 6−0=66-0=6, and 0−6=−60-6=-6. Our third sequence of changes is −6,6,6,−6,−6,6-6, 6, 6, -6, -6, 6.

What have we noticed? Although the sequences are different, the patterns are the same. For the first and third sequences, we have three types of terms, which are the high term (88 and 66), the medium term (55 and 00), and the low term (22 and −6-6). It goes as high, mid, low, mid, high, mid, low, and so on. For the second and fourth sequences, we get two of the same positive term, followed by two of its negative equivalent, then it goes back and forth. The terms will continue to double for every two sequences of changes we find, but that won’t change the pattern.

  • Base Sequence: 2,5,8,5,2,5,8,5,22, 5, 8, 5, 2, 5, 8, 5, 2
  • 1st Derivative Sequence: 3,3,−3,−3,3,3,−3,−33, 3, -3, -3, 3, 3, -3, -3
  • 2nd Derivative Sequence: 0,−6,0,6,0,−6,00, -6, 0, 6, 0, -6, 0
  • 3rd Derivative Sequence: −6,6,6,−6,−6,6-6, 6, 6, -6, -6, 6

Integration:

We saw how differentiation works. Let’s take a look at integration, the opposite of differentiation. Here, instead of looking at the differences between two consecutive terms, you will have to take the sum of all consecutive terms and write a sequence of sums. Our first sequence is…

  • 3,7,11,15,19,23,27,31,35,393, 7, 11, 15, 19, 23, 27, 31, 35, 39

As we integrate this sequence, we take the sum of all of the number, which in Algebra, is called the series. The first term is 33 since the sum of the first term alone is 33. The second term is 1010 since the sum of the first two terms is 1010. The third term is 2121 since the sum of the first three terms is 2121. The fourth term is 3636 since the sum of the first four terms is 3636. You see where I’m going? Each term after, you add the next term to the previous sum. You don’t just take the sum of each set of two consecutive numbers and write the sequence there. It’s also the same as taking the sum of all the numbers.

Sums of Terms:
 1. 3=33=3
 2. 3+7=103+7=10
 3. 3+7+11=213+7+11=21
 4. 3+7+11+15=363+7+11+15=36
 5. 3+7+11+15+19=553+7+11+15+19=55
 6. 3+7+11+15+19+23=783+7+11+15+19+23=78
 7. 3+7+11+15+19+23+27=1053+7+11+15+19+23+27=105
 8. 3+7+11+15+19+23+27+31=1363+7+11+15+19+23+27+31=136
 9. 3+7+11+15+19+23+27+31+35=1713+7+11+15+19+23+27+31+35=171
10. 3+7+11+15+19+23+27+31+35+39=2103+7+11+15+19+23+27+31+35+39=210
  • Base Sequence: 3,7,11,15,19,23,27,31,35,393, 7, 11, 15, 19, 23, 27, 31, 35, 39
  • Sequence of Sums: 3,10,21,36,55,78,105,136,171,2103, 10, 21, 36, 55, 78, 105, 136, 171, 210

To check our work, we can find the sequence of changes, which if we did, we would yield nine of the same numbers. 10−3=710-3=7, 21−10=1121-10=11, 36−21=1536-21=15. These are the same numbers as the base sequence’s.

As you see, for every operation there is in mathematics, there’s always an inverse. For differentiation, there’s integration. The term for integrated functions is antiderivative. If you ask, there is no higher order integration like we do for higher order differentiation. The derivatives are for finding the rates of change. The antiderivatives are for finding area, arc length, and continuous averages.

In Calculus, integration is much harder than differentiation. It’s for the following reasons:

  • Since the derivative of a constant is always 00, and arbitrary constant, CC, must be added for every time you take the antiderivative.
  • Some of the rules of differentiation, such as the product rule and quotient rule, do not apply to integration. Integration has its own rules.
  • Not all functions have an integral that can be found analytically.

Linearity:

In both differentiation and integration, when you take the derivative or antiderivative of the entire expression, you also take the derivative or antiderivative of each term individually. And if the term is scaled by a coefficient, the scalar stays.

Here are three sequences to work with:

  • Sequence A: 0,1,4,9,16,25,32,37,40,410, 1, 4, 9, 16, 25, 32, 37, 40, 41
  • Sequence B: 2,4,6,8,10,13,16,19,22,252, 4, 6, 8, 10, 13, 16, 19, 22, 25
  • Sequence C: 1,10,2,9,3,8,4,7,5,61, 10, 2, 9, 3, 8, 4, 7, 5, 6

As we differentiate all three sequences for one order of derivatives and integrate them, we also want to combine Sequence A and Sequence B while Sequence C can be multiplied by 44.

Sum/Difference Rule:

Just to give it out already, here are the derivative and antiderivative sequences to both Sequence A and Sequence B:

  • Sequence A:
    • Base Sequence: 0,1,4,9,16,25,32,37,40,410, 1, 4, 9, 16, 25, 32, 37, 40, 41
    • Derivative Sequence: 1,3,5,7,9,7,5,3,11, 3, 5, 7, 9, 7, 5, 3, 1
    • Antiderivative Sequence: 0,1,5,14,30,55,87,124,164,2050, 1, 5, 14, 30, 55, 87, 124, 164, 205
  • Sequence B:
    • Base Sequence: 2,4,6,8,10,13,16,19,22,252, 4, 6, 8, 10, 13, 16, 19, 22, 25
    • Derivative Sequence: 2,2,2,2,3,3,3,3,32, 2, 2, 2, 3, 3, 3, 3, 3
    • Antiderivative Sequence: 2,6,12,20,30,43,59,78,100,1252, 6, 12, 20, 30, 43, 59, 78, 100, 125

We’re going to calculate the sequence of changes and sequence of sums of the combined sequence anyway, just to show how the approaches are the same.

First, let’s combine both base sequences. And we combine them by corresponding terms.

Sum of Sequences:
  • The first term of Sequence A is 00, and the first term of Sequence B is 22. 0+2=20+2=2, so the first term of Sequence A+B is 22.
  • The second term of Sequence A is 11, and the second term of Sequence B is 44. 1+4=51+4=5, so the second term of Sequence A+B is 55.
  • The third term of Sequence A is 44, and the third term of Sequence B is 66. 4+6=104+6=10, so the third term of Sequence A+B is 1010.
  • The fourth term of Sequence A is 99, and the fourth term of Sequence B is 88. 9+8=179+8=17, so the fourth term of Sequence A+B is 1717.
  • The fifth term of Sequence A is 1616, and the fifth term of Sequence B is 1010. 16+10=2616+10=26, so the fifth term of Sequence A+B is 2626.
  • The sixth term of Sequence A is 2525, and the sixth term of Sequence B is 1313. 25+13=3825+13=38, so the sixth term of Sequence A+B is 3838.
  • The seventh term of Sequence A is 3232, and the seventh term of Sequence B is 1616. 32+16=4832+16=48, so the seventh term of Sequence A+B is 4848.
  • The eighth term of Sequence A is 3737, and the eighth term of Sequence B is 1919. 37+19=5637+19=56, so the eighth term of Sequence A+B is 5656.
  • The ninth term of Sequence A is 4040, and the ninth term of Sequence B is 2222. 40+22=6240+22=62, so the ninth term of Sequence A+B is 6262.
  • The tenth term of Sequence A is 4141, and the tenth term of Sequence B is 2525. 41+25=6641+25=66, so the tenth term of Sequence A+B is 6666.

As we have calculated, the base sequence for Sequence A+B is 2,5,10,17,26,38,48,56,62,662, 5, 10, 17, 26, 38, 48, 56, 62, 66. Now let’s find the first derivative sequence of Sequence A+B. Let’s try Approach 1 and Approach 2.

Approach 1:
1. 5−2=35-2=3
2. 10−5=510-5=5
3. 17−10=717-10=7
4. 26−17=926-17=9
5. 38−26=1238-26=12
6. 48−38=1048-38=10
7. 56−48=856-48=8
8. 62−56=662-56=6
9. 66−62=466-62=4
Approach 2:
1. 1+2=31+2=3
2. 3+2=53+2=5
3. 5+2=75+2=7
4. 7+2=97+2=9
5. 9+3=129+3=12
6. 7+3=107+3=10
7. 5+3=85+3=8
8. 3+3=63+3=6
9. 1+3=41+3=4

If we took Approach 1, where we find the sequence of changes for Sequence A+B, we should get 3,5,7,9,12,10,8,6,43, 5, 7, 9, 12, 10, 8, 6, 4. If we took Approach 2, where we add the sequence of changes from both Sequences A and B, we should get 3,5,7,9,12,10,8,6,43, 5, 7, 9, 12, 10, 8, 6, 4. It seems that both sequences are the same. This is one of the rules of differentiation. If you take the derivative of a sum or difference between two or more terms, it’s the same as the sum or difference between the derivatives of two or more terms.

As for sequences of sums, let’s try both approaches:

Approach 1:
 1. 2=22=2
 2. 2+5=72+5=7
 3. 2+5+10=172+5+10=17
 4. 2+5+10+17=342+5+10+17=34
 5. 2+5+10+17+26=602+5+10+17+26=60
 6. 2+5+10+17+26+38=982+5+10+17+26+38=98
 7. 2+5+10+17+26+38+48=1462+5+10+17+26+38+48=146
 8. 2+5+10+17+26+38+48+56=2022+5+10+17+26+38+48+56=202
 9. 2+5+10+17+26+38+48+56+62=2642+5+10+17+26+38+48+56+62=264
10. 2+5+10+17+26+38+48+56+62+66=3302+5+10+17+26+38+48+56+62+66=330
Approach 2:
 1. 0+2=20+2=2
 2. 1+6=71+6=7
 3. 5+12=175+12=17
 4. 14+20=3414+20=34
 5. 30+30=6030+30=60
 6. 55+43=9855+43=98
 7. 87+59=14687+59=146
 8. 124+78=202124+78=202
 9. 164+100=264164+100=264
10. 205+125=330205+125=330

If we took Approach 1, where we find the sequence of sums for Sequence A+B, we should get 2,7,17,34,60,98,146,202,264,3302, 7, 17, 34, 60, 98, 146, 202, 264, 330. If we took Approach 2, where we add the sequence of sums from both Sequences A and B, we should get 2,7,17,34,60,98,146,202,264,3302, 7, 17, 34, 60, 98, 146, 202, 264, 330. Just like with differentiation, we got the same sequence. It’s also one of the rules of integration. The antiderivative of a sum or difference between two or more terms is equal to the sum or difference of the antiderivatives of two or more terms.

  • Sequence A+B:
    • Base Sequence: 2,5,10,17,26,38,48,56,62,662, 5, 10, 17, 26, 38, 48, 56, 62, 66
    • Derivative Sequence: 3,5,7,9,12,10,8,6,43, 5, 7, 9, 12, 10, 8, 6, 4
    • Antiderivative Sequence: 2,7,17,34,60,98,146,202,264,3302, 7, 17, 34, 60, 98, 146, 202, 264, 330

Constant Multiple Rule:

Looking at the other linearity rule, we already decided to multiply every term in Sequence C by 44, both before and after Calculus strikes. Here are the facts:

  • Sequence C:
    • Base Sequence: 1,10,2,9,3,8,4,7,5,61, 10, 2, 9, 3, 8, 4, 7, 5, 6
    • Derivative Sequence: 9,−8,7,−6,5,−4,3,−2,19, -8, 7, -6, 5, -4, 3, -2, 1
    • Antiderivative Sequence: 1,11,13,22,25,33,37,44,49,551, 11, 13, 22, 25, 33, 37, 44, 49, 55

So if we are going to multiply the sequence by 44, we’ll have to multiply every term by 44. The sequence we generate is 4,40,8,36,12,32,16,28,20,244, 40, 8, 36, 12, 32, 16, 28, 20, 24. Notice that as we scale up all numbers by the same multipliers, so will the results. For instance, 10−1=910-1=9, but 40−4=3640-4=36. 44 is a product between 11 and 44, 4040 is a product between 1010 and 44, and 3636 is a product of 99 and 44. The same works with addition, where if we add 44 to 4040, you get 4444, which is a product of 1111 and 44. What this means is that if you applied the scalar before using Calculus, both the derivative sequence and the antiderivative sequence are scaled by 44 in comparison to the base sequence’s derivative and antiderivative sequences had they not been scaled. Of course, if you multiply every term by 44 after differentiating or integrating, you’ll get the same terms.

This is exactly what the constant multiple rule is. The derivative of the product between a constant and an expression is the same as the product between the constant and the derivative of the expression. And the antiderivative of the product between a constant and an expression is the same as the product between the constant and the antiderivative of the expression.

  • Sequence 4C:
    • Base Sequence: 4,40,8,36,12,32,16,28,20,244, 40, 8, 36, 12, 32, 16, 28, 20, 24
    • Derivative Sequence: 36,−32,28,−24,20,−16,12,−8,436, -32, 28, -24, 20, -16, 12, -8, 4
    • Antiderivative Sequence: 4,44,52,88,100,132,148,176,196,2204, 44, 52, 88, 100, 132, 148, 176, 196, 220

Final Review:

For just operating on sequences alone, here’s what we learned:

  • The derivative of a function is the rate of change.
  • If the function is constant or at a critical point, the derivative is zero.
  • If the function is increasing, the derivative is positive.
  • If the function is decreasing, the derivative is negative.
  • If the function is increasing at a faster rate, the derivative is positive and increasing.
  • If the function is increasing at the same rate, the derivative is positive and constant.
  • If the function is increasing at a slower rate, the derivative is positive and decreasing.
  • If the function is decreasing at a faster rate, the derivative is negative and decreasing.
  • If the function is decreasing at the same rate, the derivative is negative and constant.
  • If the function is decreasing at a slower rate, the derivative is negative and increasing.
  • Higher order derivatives can be taken by taking the derivative of the function’s derivatives. The derivative of the base function is the first derivative, the derivative of the first derivative is the second derivative, the derivative of the second derivative is the third derivative, and so on.
  • The antiderivative of a function is the opposite of the derivative. It’s the sum of all points from a given start.
  • The derivative of the sum is the sum of the derivatives.
  • The derivative of the difference is the difference of the derivatives.
  • The derivative of a product of a constant and a function is the product of a constant and the derivative of the function.
  • The antiderivative of the sum is the sum of the antiderivatives.
  • The antiderivative of the difference is the difference of the antiderivatives.
  • The antiderivative of a product between a constant and a function is the product of a constant and the antiderivative of the function.

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