Science,English,and math 5th

【What should we do so as to understand the differential and integral calculus?】

《What is the differential and integral calculus?》

Even if it’s not outstanding as unsung hero, it has played an active part in our every day life.

《Math is s language》

When communicating with a foreign people, we need a language like English. When asking a way on somewhere or introducing ourselves, we can managed to do with a gesture, but trying to enjoy talking with others in English, it seems to be hard.

Or when translating the Japanese into the English, there is the one which we can do easily and other...

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【What should we do so as to understand the differential and integral calculus?】

《Math is a language》

...and other which we can’t easily. If coming across foreign languages which we can’t translate into the Japanese easily, we’ll make a new Japanese world. Math is the same.

When buying three things of which each price is 100yen, if translating it into the language on math, it’s 100×3＝300.

But a difficult Japanese language is hard to express in language on math. Adding the differential and integral calculus is equal to that the Japanese in which we can express in the language on math spreads

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【What should we do so as to understand the differential and integral calculus?】

《Math is a language》

A numerical equation looks dull, but the one who understands the numerical equation accumulates original meaning and and imagines the result on the numerical equation means.

【An order when being born and the one when learning】

Which was the first to be born?

《Its history was opposite》

The differential and integral calculus is we learn the differential calculus first in high school and in the next we do is the integral calculus. It seems that almost all of high schools adopt the plan.

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【An order when being born and the one when learning】

《Its history was opposite 》

This book is expressed with the same order, but when looking at the history, the one which was born first was the integral calculus.

The way of thinking on the integral calculus was born at the time of Ancient Egypt. Then origin of the way of thinking on the integral calculus was used in order to divide land fairly.

On the other hand, the differential calculus is the one finding all the change on a thing, so it is hard to imagine. As to the speed, there is a speedometer at present. It is useful, but the...

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【An order when being born and when learning】

《Its history was opposite》

But the speed is invisible actually, on the other hand, the integral calculus was finding the value on area, so it is easy to imagine.

《The differential calculus which was easy to calculate is taught first》

Why is the differential calculus taught first in high school? Its answer is simple. The way on thinking and its image are hard to understand, but its calculation is easier than the one on the integral calculus.

If using the way on calculating the differential calculus, the calculation on the integral calculus...

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【An order when being born and the one when learning】

《The differential calculus which is easy to calculate is taught first》

If using the way on differential calculus, the calculation on integral calculus is also easy comparatively, so the differential calculus is taught first in high school.

Without using the calculation on differential calculus, it’s few on calculating the integral calculus unexpectedly, but the way on calculating is mainly taught in the class on math in high school. It isn’t interesting very much, so lots of students hate math. The author said it’s regrettable.

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【An imagination on the differential calculus】

After dividing minutely and thinking

《Dividing minutely is the differential calculus》

The author said he was going to show each image on the differential calculus and integral calculus from now on.

It has a Chinese Character, 微 in the Japanese, so dividing minutely and thinking. What we divide? For example, it’s distance and time.

I’m sure we watch the display on TV and smart phone everyday, even if it’s a beautiful screen, if expanding it’s gathering of shining light.

One dot and the other usually look like the same, and another next to does

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【An image on differential calculus】

《Dividing minutely is the differential calculus》

Putting into practical use on the fact of being lots of similar colors next to, and reducing the size on an image is called compression on images.

《If the color next to changes so much, it’s an outline》

As to lots of images, next dots are usually similar colors, contrary to it, if choosing a place where the next color is different altogether, it’s a technical skill called contour drawing 輪郭抽出.

Drawing information on color which is expressed with numerical value of a place and its next and subtracting.

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【An image on differential calculus】

《Dividing minutely is the differential calculus》

Subtracting small one from big one on numerical value for information of color, and if the result is big, the color changes, in short it’s recognized the place where the color is changed is the outline.

Strictly speaking, it isn’t done the differential calculus, for strictly speaking, the differential calculus is done, it means dividing infinitely small.

Dividing infinitely small and seeing through its nature and dealing it with. It’s a way of thinking on the differential calculus.

【An image on the...】

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【An image on the integral calculus】

Dividing minutely and collecting. After that thinking it of.

《Dividing minutely and collecting. It’s the integral calculus》

The integral calculus is an opposite calculation to the differential calculus. The differential calculus is dividing, and its opposition means gathering. Not only gathering but using with the differential calculus. Please imagine with the next example.

For example, we soon find the value on area of square or circle by formula which we learn in algebra.

On the other hand, let's suppose we drew a big picture in school ground.

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【An image on integral calculus】

《Dividing minutely and collecting. It’s the integral calculus》

What should we do so as to find the value on the area of drawing picture in the school ground?

《A value on area which we can’t find in a formula》

As to the value on area which isn’t simple geometrically, how about using a photograph on digital camera? We plan to take a photograph a thing which we recognize its size beforehand. For example, a piece of paper of 10 centimeters.

At first expanding the photograph so as to being able to see the shining point like the differential calculus.

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【An image on the integral calculus】

《A value on area which we can’t find in a formula》

When expanding so big, we can judge whether a single shining dot is inside or outside on the picture.

Then we count the number of the dot inside the picture. If being based on the thing which we recognize its size, the number on dot is equal the value on area, so we can find the value on amorphous shape 不定形.

The more minutely the dot is, the more precise the picture improves, we can somehow understand it. Collecting or counting dots is the integral calculus, for dividing minutely and gathering.

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『Column, what is divided minutely by the differential calculus?』

There is a tool called 道筋カード in ドラえもん. There is a sticker showing a direction, which is showed with an arrow. If putting the sticker on a toy, the toy moves ahead as the sticker showed.

In a story onドラえもん, the stickers were put on a toy of truck, go ahead, turn right, and go ahead, in turn and a message was sent to しずちゃん who was a heroine in the story.

The author said he thought how pleasant tool it is.

In fact, showing route to the house of しずちゃん minutely, go ahead, turning right is equal to the differential calculus.

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『Column, what is divided by the differential calculus?』

The author expressed that dividing something minutely is the differential calculus, the route to the house was divided into minutely and changing each place to other one as time passed.

On the other hand, collecting the stickers which show each place to other by the time passes is equal to the integral calculus.

Putting the sticker on a toy on hand and making it moves as we intend, it’s a dreamy tool, and it doesn’t come true yet, but putting on a sensor on a person and recording changes on speed or direction as time passes, or ...

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『Column, what is divided by the differential calculus?』

...but putting on a sensor to a person and recording changes on speed and direction as time passes, or installing a program on a robot and making the robot walk as program shows.

Those things seem to be able to do with the present technical skill and it’s just the changes as time passes, so can they reach the house of しずちゃん?

No. Without information where to start, or direction when starting, they can’t reach. We call the first situation an initial rule. In short, the initial rule and changes as time passes are correct information,

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『Column, what is divided minutely by the differential calculus?』

I made a mistake. The last sentence in the last response is like the next.

The initial rule and changes as time passes is equal the route to the house as amount of information. Then even if the initial rule was in the wrong a little, margin of error is accumulated and reaching the house of しずちゃん is impossible.

Actually sending the message to しずちゃん with the stickers on arrow alone seems to be harder unexpectedly.

【Coordinates 座標 and coordinate axis】

A place is expressed with graduation and axis

〈A coordinate〉

The author...

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【Coordinate and coordinate axis】

《A coordinate》

The author said he was going to express a coordinate before describing the differential and integral calculus.

The coordinate is a thing which we express a place with an appropriate axis and graduation.

When saying your locker is the third place from the left, starting point on counting was from the end on the left, and there is an axis toward the right, so a single locker is equal to one in the graduation. It’s established in inconspicuous way.

The point which we start to count is called a starting point. When showing the place on the...

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【Coordinate and coordinate axis】

《A coordinate》

When showing the place on the locker, it’s the second place from the upper place, and the third place from the left. Then the starting point is upper left, and axes toward toward right and down are established.

《The x-axis and y-axis》

We usually say the second place from the upper place, the third place from the left everyday life, but we name each of axes and call them x or y and showing a point in a plane somewhere in math.

Meaning or width on the graduation depends on what we think then. It’s changeable. It it’s the locker we don’t ...

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【Coordinate and coordinate axis】

《The x-axis and y-axis》

If it’s the locker, it doesn’t need fraction and nor decimal. On the other hand, if it’s speed or time, it needs a value on real number from minus to plus infinitively theoretically.

As to the axis, we can choose it as we like. For example, the time from an apple fell was x, and the distance on the apple fell was y, a parabola is expressed.

【A thing which is expressed with dots on a plane】

Specifying two numbers

《From a number to a position, from the position to the number》

The position on the locker was specified in the last...

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【A thing which is expressed with dots on a plane】

《From a number to a position, from the position to the number》

The position on the locker was specified in the last time, on the contrary, if pointed out the locker, we can specify the place from the locker, the place from the left, and the place from the upper place.

When establishing x-axis and y-axis on a plane, and pointing out a place, it shows us information on a x and a y on the plane.

《A thing which is expressed with a plane of x and y》

The plane where x-axis and y-axis are set up is called xy plane. The dot on the plane means...

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【A thing which is expressed with dot on a plane】

《A thing which is expressed with xy plane》

The dot on the plane is the value on x and y.

It shows a point from x and y.

Two of them, x and y are expressed with the point.

Two of typical interpretation on the dot of the xy plane is expressed until now, and there is one more interpretation. It’s finding the y from the x and an equation in regard to the x and the y.

It’s the pattern on function and is mainly dealt with in the differential and integral calculus. We try to understand on the function deeply in the next, the author said like that

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【What is the function?】

Finding the connection between sets 集合

《Finding a relation between them》

We’ve gone into the function just now.

The function is closely connected with the differential and integral calculus. The coordinate and function express the same thing. Without being aware of the function enough it means that we can’t understand the coordinate enough as well.

For example, an orange, a cake, a beef, a cucumber, a juice, a TV, and an apple.

We’re going to classify those things in fruit, vegetable, home appliance, drink, meat, and confectionery.

《Function is a thing which...》

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【What’s the function?】

《Function is a thing which connects between sets》

We call a thing which connects between sets the function. The set is gathering. If adopting the example I’ve expressed before, the apple and orange are classified with fruit, and the fruit is the set.

It doesn’t always mean it has to respond one to one. Something applies for other, the orange does for fruit, it’s the function. When expressing with a numerical equation, taking its initial, it’s expressed with ∫（x）, and it means it’s the function for x.

【A function which is expressed with a simple equation】

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【A function which is expressed with a single equation】

At first, a liner function

《Variable 変数 and fixed number》

As to the function, it is important to understand connection between things. Which one applies to other?

In usage, if saying, y=x squared ＋ x ＋1, the character on the left side is the function for the character in the right side, y is the function for x.

Then two things alone appeared, so we don’t have to give a notice, for there is low possibility to be misunderstood, but as it’s unidiomatic, if we trying to be exact, we express y is the function for x, next to the equation.

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【A function which is expressed with a single equation】

《A variable and fixed number》

If its equation is y=a x squared, it is expressed y is the function for x in unidiomatic, but we can’t judged whether it’s y is the function for the x or not it’s y is the function for the a from the equation alone.

Without writing down, y is the function for the a or y is the function for the x beside the equation, it isn’t clear.

When y is the function for the x, the x is variable, so its value changes. On the other hand, the a doesn’t change, so we call it the fixed number.

An equation in which...

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【A function which is expressed with a simple equation】

《A variable and a fixed number》

As to equations in which we call a simple, quadratic, cubic equation, whether or not a radical root 累乗根 is used. A raised to higher power. Multiplying by the same number.

For example, a ＋ b＝ c, there is no radical root, so it’s simple, and we call it a simple equation. In the subtitle, I made a mistake. I said single equation, but it’s not single but simple.

And the next equation, a squared ＋ b squared＝ c squared. It’s a quadratic equation, and a cubed ＋ b cubed= c cubed is a cubic equation.

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【A function which is expressed with a simple equation】

《Being expressed with a graph》

An equation of y = x is expressed with a graph. When the value on x = 1, then the value on y also is 1. The graph on x-axis is a horizontal axis, and y is vertical, so the value on the equation is expressed with a straight line extending upper light, and dividing the coordinate in two exactly.

Both of the value on x and y starts from zero, so it starts from the point of intersection on y-axis and x axis.

When expressing the function with the graph, we can understand a relation between x and y at a glance.

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【Quadratic function which is expressed with a curve】

A typical function which is expressed with graph by a curve.

《A curve》

A quadratic function is expressed with an equation like y= x squared, and when expressed with a graph, it resembles an orbit as if we threw something, so it’s called a parabola.

To be concrete, what kind of the quadratic function is? The simplest one is the value on area of square. The value on area of circle is also the same.

As to the square, the length on four side is the same, when squaring the value on the side, we can find its answer.

Then x is the...

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【A quadratic function which is expressed with a curve】

《A curve》

Then x is the length on the side of the square, and y is the area, so its equation is y = x squared, and the graph is expressed with the curve. Then x is the length on the side, it’s on the condition that x is bigger than 0.

《We don’t have to express exactly》

Expressing the graph on quadratic function like the curve precisely is hard, but we would hardly be asked to express exactly so much. For example a likeness 似顔絵 which catches a character accurately is to understand easier a person who it is than a photograph.

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【A quadratic function which is expressed with a curve】

《We don’t have to express accurately》

Grasping the characteristic and expressing a graph easily as if it were precise, it’s the one who is in high level on the function.

Though its character is different by each graph, but for example, as to a simple function, the straight line itself is the characteristic, and if there is a special information of passing a point on intersection of x and y, or passing a point, the information is added to the graph, so it becomes accurate.

If it’s the quadratic function, the shape on symmetrical...

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【A quadratic function which is expresses with a curve】

《We don’t have to express exactly》

If it’s a quadratic function, a symmetrical mountain or valley is one of characteristic on the information.

In the next information, where is the top of the mountain, or where is the bottom in the valley?

In the last information, y intercept or x intercept. The place where the function passes the y-axis or x-axis on the graph.

If adding the information is added to the function on the graph, even if it’s not good at describing more or less, it will be an exact graph.

【Expressing a graph from an...】

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【Expressing the graph from an equation】

If being able to understand the equation on function, we can express its graph.

《Expressing two kinds of graph》

As being able to understand both on the equation and graph, let's express a graph actually, the author proposed in the book like that.

1 y = 4x ＋ 6

2 y = 2x squared

He proposed we describe the two kinds of graph on the equation, but I can’t express them here actually, so I’m going to express his sentence in the book as it was.

As to the 1, when x = 0, y = 6. The place where its straight line goes across is an intercept, especially...

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【Expressing the graph from an equation】

《Expressing the two kinds of graph》

...especially we can understand it’s the y intercept with regard to its straight line.

As to 2, its value is squared so its graph becomes symmetric between the y-axis. Its shape becomes the one like valley and its bottom is goes across both point of 0 on the y and x axis.

《Why does it become the curve?》

Why does it become the curve? When calculating, we can understand, as to an equation which is squared, when its value is 1 or 2, it doesn’t change so much, but for example when 10 is squared, it becomes 100, 11...

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【Expressing the graph from an equation】

《Why does it become a curve?》

...11 is squared, it becomes 121, the bigger the value on x, the bigger the value on y when x changes every time.

The change on x is expressed minutely at will, so when connecting the value which responds to y, it becomes a gentle curve.

【What is an inclination?】

How much does it go ahead? How much does it go up?

《The inclination on a slide is minus》

A phrase on inclination is frequently used everyday life, but in math it becomes more definitely.

When going toward x-axis some space, how much does it goes up for y?

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【What is an inclination?】

《The inclination on slide is minus》

It has been fixed beforehand. As to a straight line, wherever looking at the value, it’s the same.

When x = y, if x goes ahead 1, y goes up the same, so its inclination is 1.

When y = − 3x, if x moves forward 1, y goes down 3. Going down 3 means going up − 3, so its inclination is − 3.

It’s complicated. Though I can express it in English, to my sorrow, it doesn’t ring a bell for me very much.

Then what about y = 4?

When expressing an equation which doesn’t include x on a plane showing with x and y, the y has nothing to do...

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【What is an inclination?】

《An inclination on slide in playground is minus》

...then the y has nothing to do with the value on the x, the function is a straight line parallel to the x axis. Then even if the x changes, the change on y is zero, so its inclination is also zero.

《Inclination changes in a curve》

If thinking of a slide in playground as an image, dividing the height on the slide by horizontal distance is its inclination. It’s minus.

I’m afraid it’s hard to understand, for myself who express this sentence can’t very much. If showing with some number concretely, it’s easy to grasp.

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【What is an inclination?】

《Inclination changes in a curve》

What about a slide which changes inclination when goes forward? Then its inclination isn’t uniform, it isn’t the same exactly whenever goes forward. It changes when the place on the slide changes.

【Asking for an inclination】

Calculating each point on the coordinate alone, we can understand an angle on the slide.

《It’s all right from a point on A to a point B, or from the point on B to the point to B》

As to its inclination, if dividing distance on moving forward by other distance on moving forward, we can find its value.

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【Asking for inclination】

《It’s all right from the point A to the B or its opposition》

When thinking of two points, A （3,8）B （2,4）, moving forward from B by 1, and going up by 4, there is the point of A, so its inclination is 4.

Then we happened to think which is left, A or B?

But without thinking anything, distracting the value on B from A on the y-axis, distracting the value on B from A on the x-axis, and dividing the value on A by the value on B, we can find the value on its inclination.

Actually even if the two points are lined in any kind of order, when calculating it with the...

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【Asking for an inclination】

《It’s all right distracting from the point B from the point A or its opposition》

...when calculating with the equation, its minus is counterbalanced with fraction smoothly, and we can find an exact inclination.

《Calculation fraction of fraction》

Can you find an inclination on the point A（a half, one third ）and B（two thirds, one）?

The author said and its equation is expressed with fraction. I find it complicated, so I’m going to omit it. When calculating fraction in English alone, it’s hard to understand, so I hate it.

But the author said, it’s answer is 4.

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【Asking for an inclination】

《Calculation on fraction of fraction》

The author said we should be careful not to be confused with the equation which seems to be complicated because it’s expressed with fraction, for its calculation itself is the one which we learn in elementary school, so if calculating with composure, it’s all right, but I don’t feel like doing it.

【What is inclination on a point in curve?】

Inclination on curve changes when its place changes.

《Inclination on curve》

As a slide in playground appeared in the explanation on image of the inclination, the slide isn’t always ...

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【What is inclination on a point in curve?】

《Inclination on curve》

…the slide isn’t always definite like a straight line, but then how should we find the inclination on curve? At first, what’s a kind of direction on inclination of the curve?

Please imagine something gentle and suddenly being something sharp like a slope in a skiing ground, or a roller coaster.

What will happen if the slope changes its inclination suddenly? When a person was sliding on the slide, the inclination on the slide changed suddenly. What happened to the person then?

As you imagine, the person should move...

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【What is inclination on a point in curve?】

《Inclination in curve》

...the person should move forward a direction immediate before the inclination changed. We call the direction the inclination on the spot in curve.

The inclination on curve isn’t definite, so without pointing out on the spot clearly, we can’t express its inclination precisely. As we suppose function on x as curve here, if pointing out the x concretely, we know the position on curve accurately.

《An inclination on the spot of x》

As a result, we know there is information not only on position but on inclination in curve with...

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【What is inclination on a point in curve?】

《An inclination on the spot of x》

...so we know there is information not only on position but on inclination with regard to the x in curve, but it doesn’t always mean that there are two piece of information on any kind of the x.

For example, there is a function which is expressed with an equation, y = one xth. When the x is zero, there is no inclination nor position on the x.

The author said he was going to express in detail in the next chapter.

【A graph on absolute value】

Does the inclination exist or not then?

《A graph on y =Ⅰ x Ⅰ 》

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【A graph on absolute value】

《A graph on y = absolute value on x》

As to the subtitle in the last response, its not multiplication on 1. I’ve tried to express absolute value on x, but it was the one like that.

This time a graph on absolute value appears. As to the absolute value, if it’s a positive number, it remains as it is, and if it’s a negative number, the mark on negative changes into positive.

In short, as to the absolute value, positive number 3 is equal to 3, and negative number 5 is expressed with 5, so we make a graph on both of them.

But function y = x on absolute value...

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【A graph on absolute value】

《A graph on y = absolute value on x》

But, as to the function y = absolute value on x, when the x = 0, the y = 0. Then what about its inclination?

I have to express on a graph of x = y in absolute value, for the author expressed with the graph, and without looking at the graph, it’s hard to understand.

The value on y is expressed with positive number, and the value on x is both of positive number and negative number, so its shape is v,

As to the absolute value on positive number, it goes up to upper right from 0, on the other hand, as to the absolute value on...

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【A graph on absolute value】

《A graph on y = absolute value on x》

...as to the absolute value on negative number it goes up to left upper.

I’m afraid an explanation on order is mixed up, for I should have expressed the situation on the graph at first.

There is a horizontal axis, it’s x axis which extends from negative number to positive number.

On the other hand, as to a vertical axis, y-axis is starts from 0 and it’s positive number alone, for it’s absolute value, so its shape is v of which bottom is 0. Positive number on x goes up right upper and negative number on x goes up left upper.

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【A graph on absolute value】

《A graph on y = absolute value on x》

Then what about its inclination on the point of 0?

As to the inclination on a place in graph, we suppose the graph was cut there, and the direction on the one who slides on the slope jumped out from the slope.

As to absolute value on x, when thinking as x = 0, there is a problem. When being in the point of x = 0, should we think it starts from x＞0, or should we think it starts from x＜0?

The direction the one which slid on the slope is different if we adopt a different choice.

In fact, the direction from which the one...

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【A graph on absolute value】

《A graph on y = absolute value on x》

In fact the direction from which the one jumped out shows accurate inclination is the same inclination even if looking at from both left and right. If there is a discrepancy between the left and right on the inclination, the inclination isn’t fixed.

In short, there is no inclination at the place.

《The inclination doesn’t exist sometimes》

When the direction jumping out of the place where the graph is cut both to left and right corresponds, we call it smooth in math term. The inclination on graph doesn’t exist in the place...

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The inclination doesn’t exist in the place where there isn’t smooth.

【Function which shows its inclination】

Where is the sharpest place?

《Wanting to express the inclination in easy word》

As to the inclination on the place in curve, we express the direction which jumped out when it was cut there. When understanding an optional place how much it was tilted, we feel happy often.

For example, if being able to understand the place where is the sharpest in the ski slope or in a mountain trail, aren’t you happy?

There is the information not only at the place on x but its inclination in a curve.

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【Function which shows its inclination】

《Wanting to express its inclination in easy way》

As to the function, y = ∫（x）, the place on x is ∫（x）. As to its place, we can express with ∫（x）, so it is convenient for us to adopt an easy way of saying on its inclination on the x. It’s expressed with ∫´（x）.

As an inclination is different in each curve, the function which shows its inclination, ∫´（x）could have made from ∫（x）, so instead of g （x）which has nothing to do with ∫（x）, we named ∫´（x）which is derived from ∫（x）

How should we find ∫´（x）from ∫（x）?

《Making a function which shows inclination》

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