Monday, July 20, 2015

Fun Cartoon Calculus: The Fundamental Theorem of Calculus

Written by Rebekah


Once upon a time, someone came up to you and asked you to compute the area under a curve. You were told that if you accomplished this feat, you would be awarded $1 million dollars.

So you grabbed some markers and made some pictures.


The area you were looking for was between the points a and b. So naturally, this is what you drew.

Inside the area you drew rectangles (see (a)). Each rectangle had a width of Δx (Δ means you’re dealing with little pieces), and a height of whatever f(x) is. This then means that the area of each rectangle is:

Area = Base x Height = f(x) Δx  

You then added the areas of each rectangle together to get the approximate area under the curve.

Using math notation:

Area under curve ≈ Σf(x) Δx

But you weren’t satisfied with the results, so you decided to make the rectangles even thinner (see (b)). This way, you’d be even closer to the exact area under the curve.

But then you got an idea!

Why not make Δx so small, that it practically reaches zero (see (c)). Of course, this would mean that you would have an almost infinite amount of rectangles. So you decided it was time to get some new notation. The small widths that are almost zero would now be called dx instead of Δx, and the sum of these pieces would now be written as:


The a and the b mean that the little rectangles are being summed from a to b.

Basically you decided to use ∫ (which looks like an “s” for ∫um) instead of Σ.


Your idea was fantastic and you were extremely excited. You even decided you would call this new way of calculating almost-infinite sums “integrating.” 

But all of a sudden, you realized that there was one huge problem: how were you going to calculate this huge sum (i.e. integral)?!

So you cracked your fingers, grabbed some more markers, and made yet another drawing.

“Hmmm…” you wondered, “what if there exists a function that describes the area under the curve?” So naturally you decided to call this function A(x). The next logical step was to call a small increase in area ΔA. Because ΔA was so close to the area of a small rectangle (like in your first picture) you immediately realized that 

                               ΔA ≈ f(x) Δx


Then, you did the following…

                               ΔA ≈ f(x) Δx
                               ΔA/Δx ≈ f(x) <---divided both sides by Δx

You then decided you’d let those little pieces get so close to zero, but never actually reach zero. So…

                                                   dA/dx = f(x)

And then it clicked. Your eyes opened wide and you gasped in astonishment: “dA/dx is the derivative of A(x)!” you exclaimed.

Now all you had to do was find a way to undo the derivative dA/dx, so that you would end up with the function A(x). Of course, this would mean you would have to treat f(x) like a derivative, so that you could do the same thing on the other side of the equation (remember, you must perform equal operations on both sides of an equation in order to maintain both sides equal).

You decided to call the act of undoing a derivative “finding the antiderivative.”  


So… 

Antiderivative of dA/dx = antiderivative of f(x)

This had tremendous implications. And you summarized your conclusions as follows:

“A(x) (area under a curve) =  

∫f(x) dx = 

Sum of All Rectangles (i.e. Integral) = 

Antiderivative of f(x)”

Now all that was left was figuring out how to calculate antiderivatives. So you started simple.

“The derivative of x² is 2x, so this must mean that the antiderivative of 2x is x²,” you reasoned. But then you realized something very important—the derivative of x² + 1  was also 2x. In fact, the derivative of x² + 2 , x² + 3 , and even x² + 157 will always be just 2x.

So, you decided you would say that the antiderivative of 2x is simply x² + C (C for any constant).

In order to not get confused later, the antiderivative of f(x) would be denoted F(x) + C.

∫f(x)dx = F(x) + C (a capital F implies antiderivative). 
So based on this notation, A(x) = F(x) + C

But what then was C in your problem? 

That was simple.

When the function is equal to A(a), this means the area is still zero.

So… A(a) = 0 = F(a) + C

Solving for C we get that C = - F(a); thus, A(x) = F(x) + C becomes A(x) = F(x) -F(a).

Because in your problem you were looking for the area from a to b, this would translate to: total area under curve = A(b) = F(b) – F(a).

At last, you had finished! You jotted down your results, and presented them to the guy who had offered you $1 million dollars.

The result was as follows:


The man handed you a check, and asked “do you have a name for this thing you have invented and/or discovered?”

With a glimmer in your eyes, and a huge smile on your face you answered and said “yes, it’s called the ‘Fundamental Theorem of Calculus.’”
                    
                              THE END


This post was

Sunday, July 5, 2015

Skier on a Snowball (Physics Problem Solved)

Written by Rebekah

Let’s solve the following physics problem:

“A skier starts at the top of a very large, frictionless snowball, with a very small initial speed, and skis straight down the side. At what point does he lose contact with the snowball and fly off at a tangent? That is, at the instant he loses contact with the snowball, what angle does a radial line from the center of the snowball to the skier make with the vertical?”


Let’s begin by making a more detailed picture.

As you can all see, our drawing is pretty involved, so let’s talk a little about it.  First notice that we are looking for the angle Θ.  Because we basically have two parallel lines cut by a transversal, we can use Θ in our body diagram.  Then we can break the force acting on the skier into its components (notice the coordinate I chose for this).

So then, this means that the force acting in the y direction is: 

ΣFy = ma = mgcosΘ - n

Loosing contact implies that the normal force, n, equals  0

So...

ΣFy = ma = mgcosΘ

But remember that the formula for centripetal acceleration is v²/R (R being the radius) . 


Rewriting the equation for F = ma, we get 

mgcosΘ = mv²/R

We can divide both sides by m, and then solve for  

v² = RgcosΘ <-----This will come in handy later.



From this drawing we can easily see that cosΘ = h/R


Solving for h, we see that: h = RcosΘ<---This will come in handy later.

But how does all this come together? Through conservation of energy!

Conservation of energy implies:

0 + mgR = ½mv² + mgh
        gR = ½ + gh<---Let's plug in the values we got earlier for v²and h
        gR½RgcosΘ + gRcosΘ
          1 = ½cosΘ + cosΘ
          1 = 3/2 cosΘ
      2/3 =  cosΘ
arccos(2/3) = 48.2°

So there you have it, the answer is Θ = 48.2° (isn't that nice ☺)

Monday, August 4, 2014

Algebra "Coin" Problems--5 Simple Rules


Posted by Rebekah and Esther

The dreaded algebra coin problems have come to Planted by Rivers. Okay, okay, so they really shouldn't be dreaded. The truth of the matter is, if you know 5 simple rules, then solving them should be pretty easy.


Here are the rules of the game.

Rule #1:

"Coin "Art by Rebekah
Know how much each coin is worth.

Penny: 1 ¢                                          

Nickel: 5 ¢

Dime: 10 ¢

Quarter: 25 ¢

Half Dollar: 50 ¢


Important note: Because we are working with cents and not dollars, we want to make sure we convert everything in the problem to cents. This will ensure we get rid of any pesky decimals (you’ll see what I mean).


Rule #2: Choose variables (letters) to represent the number of coins.

In other words, if I am working with dimes and quarters, I could use “d” to represent the number of dimes and “q” to represent the number of quarters. It usually helps to use the first letter in the coin’s name (although you could use any letter).


Important note: Don’t forget that each letter represents the NUMBER of coins!


"Coin "Art by Rebekah

Rule #3: Learn what phrases like “4 times as many nickels as dimes” means.

Okay, so to some people this may seem pretty straightforward. First choose a letter to represent the amount of nickels (let’s use “n”), and then choose a letter to represent the amount of dimes (let’s use “d”).

So you should end up with an equation that looks like this: n = 4d

But why not d = 4n? If you’re still mystified, keep on reading! (If not, just skip to rule #4)

Now, let’s start simple. Let’s say we have the phrase “Enoch has 4 times as many balls as Daniel.” So then, if “x” represents the amount of balls Daniel has, Enoch must have 4x balls. So, Enoch’s number of balls = 4x

But what if we say, “Enoch has 4 times as many balls as Daniel has cars”? Well, let’s choose the letter “b” for number of balls and “c” for number of cars. Hmmm…the only real difference between this phrase and the above phrase seems to be that now the concentration is between two objects and not two people. Taking this into consideration, one can see that the correct equation for this problem would be: Enoch’s Number of Balls = 4c or b = 4c.

But look! We can also say: “4 times as many balls as cars,” and the equation still remains b = 4c.

So then, phrases like “4 times as many nickels as dimes,” must mean n = 4d.


Rule #4: Know what “more than” and “less than” mean.

"Coin "Art by Rebekah
Yes, yes, I know—they’re never phrased so nicely. Usually they’ll go something like this: “there are 7 more nickels than pennies” or “there are 7 fewer nickels than pennies”.

Now, let’s use “n” to represent the number of nickels, and “p” to represent the number of pennies. As a result, “there are 7 more nickels than pennies,” must be

n = p + 7 and “there are 7 fewer nickels than pennies” becomes n = p – 7.


WARNING: Although n = p + 7 is the same as n = 7 + p, n = p - 7 is not the same as n = 7 – p. This is because subtraction is not commutative, and as a result, 7 - p could result in a negative number of nickels! 


Rule #5: Do the Algebra!

The best way to explain this is to do actual problems. :) (which I’ll try to post next monday)

To be continued…^_^!



Sunday, August 3, 2014

Science Sunday: DNA Replication (Part 3)--Enzymes

Posted by Rebekah

 Please click here for “DNA Replication (Part 2)

Note: this post describes DNA replication as understood today.

While reading this post, it is important to note that DNA replication in eukaryotes is somewhat different than in prokaryotes (although they do have a lot in common). However, as I mentioned in my last post (yes, it’s been a long time), much of the knowledge we have concerning DNA comes from research involving bacteria. As a result, Lord willing I will be concentrating on DNA replication in E. coli.
OriC Up Close

Origins of DNA Replication

Origins of DNA replication are crucial in that they attract replication enzymes.

But how do they work?

In E. coli, the origin of replication (which is known as oriC) is characterized by an adenine and thymine rich sequence. A-T bonds require less energy to denature than G-C bonds; hence, making it a logical design. This sequence is approximately 245 base pairs (bp) long, and is partitioned by three 13-bp sequences which are then followed by four 9-bp sequences (these are called13-mer and 9-mer sequences respectively).

Replication Enzymes

So, how does it all begin?

An enzyme called DnaA attaches to the 9-mer repeats; The DNA then bends, and the AT-rich 13-mer repeats hydrolyze (break). Thus, resulting in an open complex where the double stranded DNA has begun separating.   

And this is where the drawing I left in my last post comes in handy…



An enzyme called DnaC (not pictured) caries another enzyme called DnaB to oriC. DnaB is a kind of helicase protein, which separates the two complementary stands of DNA by hydrolyzing the hydrogen bonds connecting complementary nucleotides.

This to me (yes, this is my opinion here), reminds me of a zipper: helicase is analogous to a slider and the complementary strands are analogous to the two chains of teeth (See here to learn about the structure of a zipper).

Now, in order to keep the two strands from reannealing (joining again), proteins called single stranded binding proteins (SSB) attach to the unwound strands of DNA.

Unfortunately, all this unwinding causes torsional strain on the DNA; which in turn can lead to supercoiling (kind of like a rubber band when twisted too much). Not surprisingly then, there is another enzyme that relieves this strain—topoisomerase. Topoisomerases do so by catalyzing the cutting and rejoining of the “supercoiled” DNA; hence, causing the DNA to rotate and remove the coil.

But which enzyme is responsible for the synthesis of new DNA daughter strands?

The DNA polymerase III (pol III) holoenzyme!


Note: holoenzymes are enzymes with lots of proteins (as well as other compounds) that help it (the enzyme) do its job.

However, in order to begin work, DNA polymerase needs a 3′-hydroxyl (-OH) group (Lord willing I’ll try covering DNA’s molecular structure later). In order to fix this problem, DnaA, several proteins, and an enzyme called primase unite at oriC and form a complex called the primosome. Primase then synthesizes an RNA primer that provides the very much needed 3′-hydroxyl group. Simply put, an RNA primer is a short stretch of RNA (somewhere between 12 and 24 nucleotides long), and “RNA” (ribonucleic acid) is a lot like DNA, but it uses the nucleotide uracil (U) instead of thymine (T). (Note that there are also several other differences between DNA and RNA not mentioned here).

But how are these RNA primers turned into DNA?

Well, pol III finishes its job once it runs into the RNA primer; subsequently leaving a single-stranded gap between the last DNA nucleotide (of the new daughter strand) and the first RNA nucleotide of the primer. In turn, DNA polymerase III is replaced by an enzyme called DNA polymerase I (pol I), which is attracted to the DNA-RNA single-stranded gap.  DNA polymerase I is special in that it is capable of exonuclease activity. This means it can remove the RNA nucleotides one at a time. Note that, as DNA pol I removes the RNA nucleotides, it replaces them with the necessary DNA nucleotides. All this is done in the 5’ to 3’ direction.

Once the primer has been completely removed, however, there remains a DNA-DNA single-stranded gap. In order to close this gap, an enzyme called DNA ligase steps in, and forms the phosphodiester bond necessary to close this gap.

The Replisome
We have now learned about many of the enzymes associated with DNA replication in E. coli; nonetheless, it would be erroneous to assume that these enzymes act independently from each other. In fact, research now indicates that these proteins and enzymes are all part of larger protein complexes called replisomes.


To be continued… 


                                                                                Sources
Reece, Jane B., et al. Campbell Biology. 9th Global Edition. “Many Proteins Work Together in DNA Replication and Repair.” Boston: Pearson, 2011. 357-365. Print.
Sanders, Mark F., and John L. Bowman. Genetic Analysis: An Integrated Approach. 1st ed. N.p.: Benjamin Cummings, 2012. Print.

WARNING: Due to several reasons, I do NOT recommend Campbell Biology for your homeschool. However, due largely impart to its prevalent use in colleges and universities (and even Wikipedia), I chose to use it as a reference.  
                                                         
                                                                      

Tuesday, April 8, 2014

More Birthday Cake Pictures--and Hopefully More Posting



Posted by Esther


Wow, we haven't posted here for over eight months now; however, we're planning on changing that soon! Lord willing, we want to revive this blog by posting an array of topics that will hopefully, be of interest to readers. We're also planning on posting more often, as that has been a struggle ever since we started this blog.

Well for now I decided to post some pictures of some birthday cakes I made recently. As I mentioned earlier in another post, we like to make our cakes from scratch at our home; and since I love to bake and cake-decorate, I'm usually the one to make my siblings' birthday cakes.

Here are the pictures (they're from 2013-2014):




This red guy--I'm not sure what kind of creature he is--is from a computer game my brother (or should I say brothers) likes to play.

             This cake was for my youngest sister's first birthday--which we recently celebrated!


This is a penguin from a movie my younger siblings love--his name is Scamper (just in case anyone is curious ;) ). His beak was made out of a carrot that was covered in orange frosting, and the top of his hat was simply a piece of curly-leaf lettuce--strange, I know :) . Oh, and his eyes and eyebrows were made of raisins.

 

I'll leave you all with this famous piece of scripture from Matthew, that I was just recently studying; although it has nothing to do with what I posted above, I find it too beautiful not to post.  

{Emphasis is mine}.

Now when the tempter came to Him, he said, “If You are the Son of God, command that these stones become bread.” But He answered and said, “It is written, ‘Man shall not live by bread alone, but by every word that proceeds from the mouth of God."


Matthew 4:3-4
 

Well, good bye for now!