Why was Sara teaching Calculus to Johnny?
Johnny and Sara kept hanging around together as they were in love. Of course, they would also occasionally work with the Calculus book. Johnny asked why Sara started teaching him Calculus.
Sara: There are three reasons. First reason is that I was going to go over the calculus book anyway, and I thought it would be good to do it together. The second reason is that I knew you were going to ask me to teach you Calculus anyway.
Johnny: I didn’t ask you. What makes you say that I was going to?
Sara: Here, lover boy give me a kiss. I can read your mind.
Johnny kissed her and said: Okay, then tell me what I am thinking right now.
Sara: You are thinking of two different things. One thing you are thinking of what is the third reason.
Johnny: Yes. You said there were three reasons. You gave only two. What is the third reason?
Sara: You are fun to work with and I learn better when we are working together.
Johnny: You are a mind reader. I was thinking of the third reason. Now tell me the other thing on my mind.
Sara: You were going to ask me for another date.
Johnny: I love you Sara. Let me ask mom if I can borrow her car again to go for a ride again.
Johnny went to talk to his mom.
Johnny: Mom, can I borrow your car again? I want to take Sara out for a date. Remember, you said before that I should do that.
Mom: Not today. How about tomorrow?
Johnny asked Sara if she could go for a date the next day. Sara agreed readily.
Trip to Mountwin
They looked at places nearby and decided to go to the town of Mountwin because they thought it would be a scenic drive. To the West of their city were two very similar mountains which people often called Twin Mountains. To the West of these mountains was the township of Mountwin.
The next day, Johnny borrowed his mom’s car, picked up Sara in the afternoon and off they went. The road ran between the two mountains. Each mountain had a large smooth curved base. The scenery was beautiful with lush greens, tall trees and small lakes. On the way they saw a park which was next to a lake. They decided to get out and enjoy for a while. They also played Frisbee at the park. Then they drove off towards Mountwin.
Johnny: Sara, what are you doing?
Sara: Nothing much, just clicking on my phone to record our GPS co-ordinates. I won’t bother you. I will just click once every few moments. Let’s keep enjoying the beautiful mountains around us.
They kept driving until they reached Mountwin. It was a very small town. The main market was just one large plaza. The town had many houses and a small school with a playground and a recreation center.
The main plaza in Mountwin had a restaurant. Johnny and Sara decided to check it out. The place was small but it also had a dance floor. They ordered the food. While eating they found that there was a small computer on which customers selected songs to play. There was no DJ but people could dance when their song was playing.
Johnny: Sara, do you want to dance?
Sara: Not really. Instead, can we go back to the park we visited earlier? I liked that place. We could hang there for a while.
Johnny asked the waitress for the bill. Suddenly Sara interrupted.
Sara: Let’s split the bill.
Johnny: No, let me pay.
Sara: You paid the last time too.
Johnny: I didn’t pay it the last time. My mom wanted me to take you out and she gave me her credit card. I used her card. Frankly, she was not happy that we ended up going to a very cheap restaurant. So she gave me her card again today.
Sara: Why should she pay?
Johnny: She loves me and maybe she likes you too.
Johnny paid the bill and the love birds drove back to the park next to the lake. They hung around there for a while and then drove back. Johnny took Sara home and they kissed goodnight.
Sara woke up and had breakfast with Nana. Nana asked her how the date went. They kept chatting while Sara did things using her laptop. Sara’s phone rang. It was Johnny asking her if she could come over for a while. Sara went over to Johnny’s house. She carried her laptop to show him something.
Analysis of recorded GPS co-ordinates
Sara: When we left the park to go towards Mountwin, the road was heading from East to West. So I started recording our GPS co-ordinates.
Johnny: The road wasn’t just to East to West. It was wavy.
Sara: Here is a part of this wave I recorded. This is just a graph showing our drive towards West on the horizontal axis and movement to North or South on the vertical axis. Oh, I changed the scale a little bit so that the total kilometers to the West come out to be 2π (equals 6.28).
Johnny: Trickster, so you made the angles in radians because 2π radians equal 360°.
Johnny: This looks like the y = sin x graph we did in Trig.
Sara: It could be if there was continuity in the road.
Johnny: McLinton Road had no breaks or major kinks and hence we had assumed it to be continuous. Why can’t we do the same for this road?
Sara: I guess we could. Here is the graph for y = sin x.
Johnny: Did you record more of the co-ordinates?
Sara: Yes, but after this the road got curvy in all sorts of weird ways. That’s why I showed you only this part. Another thing, I also measured the ratio of our Northward movement to the Westward movement for first quarter of the wave. I wrote down the slope at each point in the graph.
Analysis of slope of sin x
Johnny: You said the left side is for only the first quarter of the wave and that would be from x = 0 to π/2. Looks like the slope was very large in the beginning and then it decreased as we reached the end of this graph. Why did you stop here? Why not measure for the whole wave?
Sara: I did and then drew these slopes at any given points as a line graph. This is on the right side.
Johnny: The slope also makes a wave. I have seen this wave somewhere in Trig.
Sara: Yes, the graph of the slopes looks like y = cos x.
Johnny: The slopes of the y = sin x graph gave you a curve for y = cos x. By definition of the slopes, and assuming that our continuity assumption was true, that means:
d sin x/dx = cos x.
Johnny checked in the Calculus book and said: Hey, that’s correct. I would have never thought that way. Calculating the slope thing works.
Johnny kept staring at the graph for a while and then started talking: I see that you drew dotted vertical lines where the graph for d sin x/dx became zero at the two points. One time it was zero when sin x was maximum which was at the end of the first quarter. The second time it became zero when the sin x graph became flat at the bottom and was at its minimum value.
Sara: That’s interesting and makes sense. When the value of a function plateaus, its slope will be zero. Look, we have a chapter on Maxima and Minima. It says that the first derivative is zero at the points of maxima or minima.
Slope of y = cos x
Johnny: I wonder if the same thing would happen if we looked at slopes of y = cos x.
Sara: I calculated the slopes from the graph for y = cos x the same way. On the left is the first quarter. The slopes at each point are written in the graph.
Johnny: These are all negative.
Sara: Not if you continue after the second quarter. On the right is a graph for the instantaneous slopes of the whole curve of y = cos x.
Johnny: This graph is confusing. It looks like the graph for y = sin x but upside down.
Sara: Check in the book. Does it say that d cos x/dx = – sin x? That’s what the graph says.
Johnny: Yes. One more thing we didn’t do the first derivative of tan x.
Sara: Remember how tan x can approach infinity at some angles. Therefore, it is harder to show it on the graph but the book says d (tan x)/dx is sec2x.
A few days later, Johnny visited Sara and they told her mom and dad that they had been going over Calculus for the last few days. Both the parents were happy. Sara’s mom and dad then explained to them several applications of calculus in physics and engineering. Her dad went on to talk about its applications in economics and investment strategies especially those involving option trading. Her mom also explained how Calculus had helped understand strategies of how much of a drug should be used and how often. Basically, anywhere you had to consider a change, Calculus was useful for measuring the rate of the change and its interpretation.
Determine d2(sin x + cos x)/dx2.
Solution: d(sin x +cos x)/dx = cox x – sin x
Therefore, d2(sin x+cos x)/dx2 = d(cosx)/dx -d(sin x)/dx = -sin x – cos x.