past perfect simple (part 1)

 Used to describe: 

• A completed action in the past before the specific moment in past. 
• A completed action in the past before another action in past. 

Form: (had + verb in past simple)

• I/He/ She/It/You/We/ They had loved
• I/He/ She/It/You/We/ They had not/hadn't loved
• Had I/He/ She/It/You/We/ They loved?

* for irregular verbs in past we use second form 

Examples:

• By 2000, he had raised several millions pounds for charity. 
• U2 had performed in 1985 Live Aid concert before they played in the live 8 concert in 2005. 

past continuous (part 1)

 Used to describe: 

• action not completed in the specific moment in the past: I was reading newspaper yesterday at 7 am.
• action not completed in the the past when another one has occurred: i was watching tv yesterday when the phone rang.
• to make descriptions in the past: the tree were moving gently in the breeze. 

The form:

• I/he/she/it + was living - was not living ¿was I/he/she/it living?
• you/we/they + were living - were not living ¿were I/he/she/it living?

examples:

Mama was cooking dinner when Isabella arrived. 

Was mama cooking dinner? No she wasn't. 

You were eating dinner when papa opened the door.

Were you eating the dinner? No you weren't eating the dinner. 

past simple (part 2)

1) Four years ago my wife and I went to Escorial as tourists. While we were there we visited two sites. The first was the "Escorial Palace" with a museum inside. The second was a great mount called "Pico de Abantos". It was a really fun trip!

2) A few weeks ago my daughter and I visited museum of Guadalajara in Spain. We saw many antique items there, including table cutlery, weapons, and tools from primitive civilizations. It was an awesome walk!

3) A few years ago, my friend and I went to the great mountain called 'Siete Picos' in Madrid. We spent three days there. While we were there, we were cooking over a fire and we were sleeping in the open air. I remember how we encountered a wild goat there. It was a wonderful weekend!

4) Two years ago, my wife, my daughter, and I visited Seville in Spain. While we were in Seville, we visited my sister Natalia and her husband. Then, in the evening, we went to the beach - not to swim, as it was winter. We were just watching the sunset. It was so beautiful.

One man, whom we can call Joe, met another one in a casino. Another man was spending all his money until Joe said to him, 'Man, just put all your money on number two'. The other man did this and finally won a lot of money. This man was so happy. He wanted to give half of his money to Joe. But Joe said to the man, 'No, thanks, I don't need money. But if you want, you can give me a 'blank check.' As the man was so happy, he gave him the paper, and it was a mistake. Joe was a liar. He was 'working' with the casino staff, making the man win, but then used the blank check to take out much more money. Okay, so Joe took all this money and went to spend it with a prostitute. In bed, after having sex, he was telling her: I told this man that my lucky number is two, and it was true. The number two is my favorite number. It's because many years ago I was driving my car in the countryside, on a forest road. Two black cats jumped into the road, and I had an accident with my car. Then I was walking in the forest until I saw a very big house. I went into the house looking for a phone. It was a wonderful house. There, I saw those two cats. Suddenly, two beautiful women came into the room. They were identical. At first, they were scared. But then we were talking and becoming friends. I was in love with both of them, and they were too. I wanted to marry them so I could have this big house of my own. As I was very clever, I decided to do something. I told them that I have a brother who is identical to me. So, one weekend, I was in the house with one of these women, and the next weekend, it was still me, pretending to be my brother, who I told her was exactly like I am. So I could marry one of them, and my invented brother could marry the other one. I had everything I wanted: a big house, a lot of money, and two beautiful women. But suddenly, those women realized that it was a lie. They saw me changing my clothes to appear as my brother. One of them said, 'We have only one man!' The other one said, 'He is mine. They wanted to fight, but then they changed their minds. One of them said, 'Let's share him. One half for me, and the other half for you.' They took a chainsaw, went into Joe's room, and cut him in half. So they killed him, and one of them had one half, and the other had the other.

Four years ago

A couple days ago

Yesterday

Last weekend / week

A few weeks ago

Last month

•  https://www.youtube.com/watch?v=NH2GfNTfqsE
• exercise irregular verbs: https://www.perfect-english-grammar.com/irregular-verbs-exercise-1.html

past simple (part 1)

Used to describe: 

• completed actions in the past
• completed actions in the specific moment in the past 
• action that occurred while another one was in process. 

form:

each person have a same form for past simple:

• i/he/ she / it/ you/ we / they lived
• i/he/ she / it/ you/ we / they didn't live
• did i/he/ she / it/ you/ we / they live?

Also, like a short answer for affirmative form can be - yes i did

Some irregular verbs:

• go - went
• get - got
• make - made

time expressions:

• ago, yesterday, last week...

examples: It is very frequently to find this tens in stories:

• The boy started running, he stopped and listened. 

exercise

• https://www.youtube.com/watch?v=PgsG98vByiw

• https://www.perfect-english-grammar.com/past-simple-exercise-1.html

What is derivative of a function PART 3 (y=x^2)

Let's try to understand derivatives a bit more through their graphical interpretation. In the first part, we saw how we can represent changes in a function using a straight line over a specific interval. We express this change as the difference in $y$ divided by the difference in $x$, or $\Delta y / \Delta x$

We can also examine what happens to $\Delta y / \Delta x$ as the interval shrinks toward a specific point. We can do this manually by moving the value of $b$ closer to $a$. In this case, we are talking about the derivative of $y$ at point $a$

Another important observation is a key property of the straight line that represents these changes. In the limit, as $b$ approaches $a$, this line has only one intersection point with $f(x)$.

Note: In the limit, this line has a specific name: the tangent line.

Not all functions have this property, but when they do, we can use it as an alternative definition of the derivative. We can find the tangent line and then associate its slope with the derivative of the function.

Now, let's apply this concept:

Find the intersection between $y = x^2$ and a straight line $y = mx + n$.

We set their equations equal to each other:

$$x^2 = mx + n$$

which gives us the quadratic equation:

$$x^2 - mx - n = 0$$

We cannot solve this equation directly because we do not have values for $m$ and $n$. However, we know that the tangent line must have only one point of intersection with $y = x^2$, which means the quadratic equation must have exactly one solution.

For a quadratic equation , a unique solution occurs when the discriminant is zero:

$$m^2+4n=\mathrm{0}$$

In this case, the solution for $\mathrm{x\; is:}$

$$x = \frac{m}{2}$$

This means that the slope of the tangent line at any point $x$ is $m = 2x$

So, for $f(x) = x^2$, the derivative is: f '(x) = 2x

vocabulary used:

Let's try to understand a bit more
through the graphical interpretation
a straight line
the interval shrinks toward a specific point.
to apply this concept
the discriminant is zero

$$\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

present continuous (part 1)

 Used to describe: 

• Actions that are happening now: Paul is sending Sandra a WhatsApp. 
• Actions prolonged in the present: I'm winning this game of tennis. 
• Monotonous and repetitive actions: We are always buying a new clothes. Something you doing in your life. 
• Concrete plans in the future: We are coming home on Monday. 

Verb forms:

• I am living / She is living / they are living
• I am not living / He is not living / You are not living
• Am i living?/Is it living/Are we living?

Contracted form: 

• i'm, you're, she's, we're, they're

frequently time expressions:

• now, at the moment, at present, tomorrow, this weekend...

Spelling rules: 

• give - giving, study - studying, die - dying
• plan (one-syllable) - planning...

I thing is easiest than present simple because it clear when you should use it. However, all the time, i'm thinking about use present simple and present continuous in same context. I can say I learning English right now it is focused on action. When i say I learn English it is about a fact of learning English. 

https://www.youtube.com/watch?v=rFdhrR6Dpco

https://www.perfect-english-grammar.com/present-continuous-exercise-5.html

present simple (part 2)

Things I Don't Understand About the Present Simple in English

Sometimes we use the present simple even when the action is happening in the present. For example:

"The first little pig wants to build his house very quickly.
So he builds his house with straw."

This is from The Three Little Pigs. The whole book is written in the present simple (the simplest tense for a children's book).

The first sentence is clear—wants is a stative verb, so it naturally stays in the present simple.

But the second sentence describes an action that seems to be continuous: he builds his house. The pig is building his house right now, so why isn’t it in the present continuous (he is building his house)?

There are two reasons for this:

1. It describes a simple fact, not a process. The focus isn’t on the act of building but on the result—what the pig is doing as part of the story.

2. It follows the storytelling convention of using the present simple for sequences of events. He wants to build > So he builds.

This isn’t unique to English. In Spanish, for example:
"Él construye su casa de paja."
This may be more appropriate than "Él está construyendo su casa de paja."

In Russian, however, the distinction between present simple and present continuous is not as clearly defined. For example:
"Он строит дом."
This phrase can mean both "He builds a house" (present simple) and "He is building a house" (present continuous), depending on context.