Johnny is told to appreciate Sara
Johnny did very well in school the last semester. There were two main reasons for it. The first reason was that his parents had bought him an expensive bike but he would have had to pay for it if he did not get an average above 85%. The second, and perhaps more important reason was that Sara, his girlfriend of two years used his interest in the bike to make him a Trig whiz.
Johnny’s mom liked Sara and told him to do something to show his appreciation of his smart girlfriend. His mom was persistent and asked him what he was going to do. She hoped it was something special for the girl who helped him so much. Johnny didn’t see it as a big deal that his girlfriend hung around with him, and in the process helped him with Trig. Mom insisted that he should do something – may be take Sara on a date or a romantic trip. She even offered that he could borrow her car.
Finally, Johnny talked to Sara about their going for a trip to a romantic place. Sara told him that her Nana would let her go only for a day trip – no overnight stay even though she knew that they’d been seeing each other for two years.
Where to go for a dinner date?
Then, Johnny and Sara had to decide where to go. It had to be a decent drive. Maybe they could have dinner and spend some time together, but where? They started looking for restaurants. There were all kinds of eateries with different types of foods. It was difficult to choose one. Suddenly, Sara made a silly face.
Sara: This is weird. I don’t believe that there is this restaurant called Johnny and Sara’s Place. The place is named after us. That’s romantic enough. It has to be good although there are no reviews. It’s on a dead end on McLinton Road. It should be a short drive from here. When do you want to go?
Johnny: That sounds romantic – a restaurant named after us.
Johnny checked with his mom to see when he could borrow her car and then asked Sara if the next day was okay. Sara agreed. They decided that he would pick up Sara from her home at 5 pm the next day.
The next day, Johnny drove to Sara’s home and met her Nana. After that, they drove away. Sara had a new smart phone in her hand. Her dad had given it to her as a reward for her achievement in the last semester. Sara’s average was 98 %.
Sara: I brought this cell phone because it has a GPS in it. We can monitor all our movements with it – just like you could do with your bike. I know your mom’s car has a GPS. I still brought this because with this I can save our co-ordinates and send the data to my laptop.
Soon they turned onto McLinton Road and kept going. Sara touched the phone screen several times to record the GPS co-ordinates.
Johnny and Sara’s Place
At the end of the road there was a small restaurant with a sign that said ‘Johnny and Sara’s Place’. Johnny parked the car and they went in. The place was tiny. It had four tables to seat a maximum of 16 people.
They sat down and a lady came with the menus. She said, “Welcome to Johnny and Sara’s Place. I am Sara and I will be happy to take your orders.”
In subsequent conversation the waitress said, “When I finished high school, my boyfriend Johnny and I opened this restaurant. I attend the front and Johnny cooks. We like the slow paced life as the restaurant never gets too busy.”
The menu listed mostly burgers, sandwiches and soup of the day. They ordered the soup and two burgers. The food was delicious. They chatted away and ate. The waitress came to tell them about the desserts.
Waitress: Johnny makes a very good apple strudel.
Sara: Thanks, we’ll take it. We also want some tea. By the way, I am Sara and this is my boyfriend Johnny. We were looking for a short romantic rendezvous and found out about this place. What could be more romantic than Johnny and Sara going to Johnny and Sara’s Place.
Waitress: Gee, that’s beautiful.
The strudel’s on the house
The waitress brought the dessert and the tea. She also brought the cook who was her husband and introduced him by saying, “Johnny, these are Johnny and Sara. They came here because this is Johnny and Sara’s place.” The cook welcomed them and urged them to come again as it was their place.
Johnny and Sara enjoyed this date. Johnny took Sara home. They kissed each other goodnight and Sara said she would come by the next day.
GPS data of the car movement
As promised, Sara came over to Johnny’s place the next day. This time she had her laptop with her. She had recorded the GPS data at every major intersection that had a traffic light. The graph showed how many kilometers they had travelled towards West and towards North (Fig.C1.1).
Johnny: Sara this looks like what we did with the bike. Are you trying to teach me Trig again?
Sara: No. Doesn’t McLinton Road look nice? It goes more West than North at the start but at the end it goes mostly North without going much towards West.
Johnny: Why didn’t you draw a continuous line connecting the dots to show the road?
Sara: The road was not continuous. There were lots of bends at different places. You can’t see them in this small map.
Sara got the map of the road from the Internet, enlarged small parts of it and then showed it to Johnny.
Sara: See, there are lot of bends in the road. There is no continuity.
Johnny: But nobody is going to fall or even hit anything just because of the small bends.
Sara: For practical purposes you are right but to describe the shape of the North-West movement relationship by a continuous function, there cannot be any bends or breaks.
Johnny: If someone were to smoothen over the bends, can you write it as a continuous function?
Sara: I suppose for now we can say that the function is continuous but we should read about this concept carefully.
What function did the movement fit?
Johnny: What would the North-West relationship be if West movement was x and the North movement was y?
Sara: I struggled with it myself. Then my mom helped me. She figured out the best function to fit this x-y pair relationship. See the picture (Fig.C1.2).
Johnny: You took out the dots. Is this because now we are calling it a continuous function described by the equation y = x2/10?
Johnny’s mom came in asking about their date. After listening to their visit to that small and inexpensive restaurant, she was not impressed. She had forgotten that a long time ago, she dreamed of having a place of her own along with Johnny’s dad. May be that’s why she didn’t see the romantic angle of their date.
Sara had a lot of time on her hands. She had a volunteer job to help refugee children with their language and social skills but that would start in about two weeks. Until then, what could be better than visiting her boyfriend more often? So she did just that. This time she had her laptop again.
Sara: Johnny, remember the graph I showed you yesterday. It followed the function y = x2/10. I did one more thing with the GPS readings we had.
Johnny: What’s that?
The derivative – slope of the function
Sara: First, tell me if the curve was y = x2 and our car moved a teeny-weeny bit in the direction of x, what will be our ratio of y to x movement?
Johnny: How much are we moving?
Sara: Say, the length of an ant. Let’s call it a for the ant. We were at x and now we are at x + a, right. The function remains y = x2.
Johnny: So y would now be at (x+a)2 instead x2. So the change in y will be (x+a)2 minus x2. If you divide this by a, you will get the slope.
Sara: Is it okay if I write x2 +2ax + a2 instead of (x+a)2 ?
Johnny: Of course then the y movement will be x2 +2ax + a2 – x2. The plus and minus x2 will cancel out and we will have 2ax + a2 as the difference. We divide it by ‘a’ which is the movement in the x direction. So the slope will be 2x + a, right? But if the value of x is 10 kilometers, can`t we just forget about the length of the ant (which is really miniscule compared to 10 km) and just say the slope is 2x.
Sara: No. In Math you can’t ignore the length unless it is zero. If we had used the example of a grain of sand which is smaller than the ant, still we could not ignore it. It comes back to the continuity issue. If the function is continuous, we could assume the increment in x to reach to zero and then say that the slope is 2x. The length of an ant or a grain of sand can’t be zero. In fact any length that you can measure is not zero. In book terms we could have written the change in y as δy and change in x as δx.
Then δy/δx = ((x+ δx)2-x2)/ δx
and we get δy/δx = 2x + δx
Then we could say that when δx approaches zero, he limit of δy/δx becomes dy/dx = 2x.
That ’s how we define the slope of a continuous function in Calculus and you call it a differential or a derivative of y with respect to x or dy/dx. That’s what Calculus is all about; rise divided by fall which is the slope of any variable with respect to another.
Johnny: But we said that McLinton Road followed the function y = x2/10, not y = x2.
Sara: We could do the calculation for y = x2 times a constant and see what happens. I could write δy/δx = ((x+ δx)2-x2)/ δx x c and then get the slope to be (2x + δx) x c and dy/dx to be 2x x c.
Here c =1/10 so the slope would be 2x/10. This is what I drew for dy/dx at different values of x. Its instantaneous slope at any given value of x is 2x/10 or x/5.
Johnny: This is getting confusing. This graph is a straight line.
Sara: Yes. It is a straight line in which the North/West slope increases with the increasing distance towards West. It is what I said before,” Look at how McLinton Road goes more West than North at the start but at the end it goes mostly North without going much towards West.” The function describing McLinton Road was y = x2/10, its first derivative was x/5. The slope of this line at any given point is x/5. That’s the same value you got when you did the calculation for dy/dx.
The second derivative
Johnny: So we are done with the derivate of y = x2/10. Okay. I suppose, we could also get the derivative of y = x/5.
Sara: Remember that y = x/5 graph is a straight line. The slope of a straight line does not change with x. It’s constant. In this case you can see that it will be 1/5.
Johnny: What do you call the slope of a slope of a function?
Sara: It is called the second derivative or d2y/d2x.
Johnny: That means d2y/d2x = 1/5 which is a line without any slope or you can say with a slope of zero. Does that mean for the McLinton Road curve y = x2/10, the slope of the slope of the slope or third derivative d3y/d3x will be zero? Wow, that’s neat.
Johnny paused for a while as if to digest it all but then fired more questions.
Johnny: What if the function was a sum of expressions, how would you determine the derivatives? Like in Algebra we learned the quadratic equation y = ax2 + bx + c.
Sara: That’s simple. We can take the derivative of each part and then sum them. dy/dx for ax2 is 2ax, for bx it is b and dy/dx for c is zero. We can sum up the derivatives of each of the three components to get dy/dx = 2ax + b + 0.
Johnny: What is the derivative of y = x3 ?
Sara looked in the book and said: For any value of n, the derivative of y = xn is nxn-1. Here is a interesting one, the book says that dex/dx = ex.
Johnny: What, is e the Eurler’s constant here?
Sara: We could prove it using e as an the sum of an infinite series but let us leave it for now.
Sara thought that Johnny was impressed but seemed somewhat lost about the meaning of all the derivatives. She thought for a few minutes before saying anything.
What do the derivatives mean?
Sara: Remember when we go in a car, we read the distance traveled on the odometer. Then we also talk about the speed of the car which is the distance traveled per hour or the first derivative of the distance traveled with respect to time. The second derivative is acceleration which is the rate at which your speed increases. Some car companies advertise how fast they can increase their acceleration. That would be the third derivative.
Johnny: Now, that makes sense. Otherwise, you had me lost.
Sara: Here is a joke I read on the internet about derivatives.
“U.S. President Richard Nixon, when campaigning for a second term in office, announced that the rate of increase of inflation was decreasing, which has been noted as “the first time a sitting president used the third derivative to advance his case for re-election.”
Johnny: That’s funny. The guy was a crook. Now that you mentioned the third derivative, I read in the newspaper today, “The decline in the pace of home sales in the Vancouver region appears to have sped up significantly…”.
Johnny’s car is parked. When he starts it and pushes on the gas paddle, after time t it would have covered the distance (d) given by d = 0.5 at2 where a is 3 meters/sec2. The speed of the car after time t is given by the first derivative (ds/dt). How fast will the car be moving 10 sec after it is started?
Solution: d = 0.5 at2. Therefore, ds/dt = 0.5 a x 2 t = at
With a = 3 meters/sec2 and t = 10 sec, ds/dt = 30 meters/sec which is 108 km/hour.