subject: how far we've come
style: mathematical
source: Charlie Brown
"Some days you think maybe you know everything...
some days you think maybe you don't know anything...
some days you think you know a few things...
some days you don't even know how old you are."
If y=x^2, find y'.
For us, this is a one-step problem. This is mental math. We practically take it for granted by this point. But what if it weren't ? What if you had to try to explain how to solve this problem, to someone who only knew the four basic operations of mathematics ? How would you do it ? To show how much we have learned throughout all mathematical career, to prove how much it actually took us to get to where we are, and to demonstrate the complexity of our minds, I will write out all of the tiniest elements of one's thought process in solving the above problem. Here goes.
1. Suppose "y" and "x^2" can be called expressions.
2. Suppose the equal sign (=) signifies that the expressions on either side of it are equivalent.
3. Suppose the combination of the three elements forms an equation, and more specifically, a function.
4. Suppose a variable is a letter that stands for a number of undetermined (but not necessarily indeterminable) value.
5. Suppose "x" and "y" are variables.
6. Recall that to multiply two numbers, A and B, you add A to itself (B-1) times.
7. Suppose that to square a number means to multiply it by itself.
8. Suppose the "2" in superscript is referred to as an exponent, and signifies that you are squaring the (unknown) value of x. Thus, x^2 is equivalent to x times x.
9. Suppose "y" here represents the name of a function which consists of x^2.
10. Suppose that to "plug in" values to this function means to figure out what the value of x or y would be when you assign the other a particular value.
11. Suppose "y'" is a symbol for the derivative of the function y.
12. Suppose the term "derivative" refers to a separate, second function which expresses the rate at which the first function changes y-values as you plug in x-values.
13. Suppose a "term" is a collection of numbers and variables multiplied together, such as x^2.
14. Suppose y' can be found, when the variable y is isolated on either side of the equal sign, and when the other side consists only of one term (which contains no variables other than x) by subtracting one from the exponent of x, and multiplying the constant of x by the original exponent of x.
15. Suppose that when two numbers or variables are placed immediately beside each other with no symbol in between, you are multiplying them.
16. Suppose the "constant" of x refers to the number you are multiplying it by.
17. Suppose that when no constant appears beside x, you can assume that the constant is implied to be 1.
18. Recall that 2 minus 1 equals 1. Change the exponent of x to 1.
19. Suppose that when the exponent of a variable is 1, you can leave the exponent out and it will be implied that you mean the exponent to be 1. Take away the exponent.
20. Recall that 2 times 1 equals 2. Write a 2 immediately beside the x.
21. Suppose that when you have a constant multiplied by a variable, the constant should always precede the variable in order to practice good mathematical form. Confirm that the 2 is in front of the x.
22. Conclude your work, and resolve that y'=2x. Box your answer.
Phew. Did I miss anything ?
I took a lot away from that lesson in CP^2 in which we learned to write technical documents.
I hope this has taught you never to take calculus for granted. xP
My head hurts. Does yours ?
style: mathematical
source: Charlie Brown
"Some days you think maybe you know everything...
some days you think maybe you don't know anything...
some days you think you know a few things...
some days you don't even know how old you are."
If y=x^2, find y'.
For us, this is a one-step problem. This is mental math. We practically take it for granted by this point. But what if it weren't ? What if you had to try to explain how to solve this problem, to someone who only knew the four basic operations of mathematics ? How would you do it ? To show how much we have learned throughout all mathematical career, to prove how much it actually took us to get to where we are, and to demonstrate the complexity of our minds, I will write out all of the tiniest elements of one's thought process in solving the above problem. Here goes.
1. Suppose "y" and "x^2" can be called expressions.
2. Suppose the equal sign (=) signifies that the expressions on either side of it are equivalent.
3. Suppose the combination of the three elements forms an equation, and more specifically, a function.
4. Suppose a variable is a letter that stands for a number of undetermined (but not necessarily indeterminable) value.
5. Suppose "x" and "y" are variables.
6. Recall that to multiply two numbers, A and B, you add A to itself (B-1) times.
7. Suppose that to square a number means to multiply it by itself.
8. Suppose the "2" in superscript is referred to as an exponent, and signifies that you are squaring the (unknown) value of x. Thus, x^2 is equivalent to x times x.
9. Suppose "y" here represents the name of a function which consists of x^2.
10. Suppose that to "plug in" values to this function means to figure out what the value of x or y would be when you assign the other a particular value.
11. Suppose "y'" is a symbol for the derivative of the function y.
12. Suppose the term "derivative" refers to a separate, second function which expresses the rate at which the first function changes y-values as you plug in x-values.
13. Suppose a "term" is a collection of numbers and variables multiplied together, such as x^2.
14. Suppose y' can be found, when the variable y is isolated on either side of the equal sign, and when the other side consists only of one term (which contains no variables other than x) by subtracting one from the exponent of x, and multiplying the constant of x by the original exponent of x.
15. Suppose that when two numbers or variables are placed immediately beside each other with no symbol in between, you are multiplying them.
16. Suppose the "constant" of x refers to the number you are multiplying it by.
17. Suppose that when no constant appears beside x, you can assume that the constant is implied to be 1.
18. Recall that 2 minus 1 equals 1. Change the exponent of x to 1.
19. Suppose that when the exponent of a variable is 1, you can leave the exponent out and it will be implied that you mean the exponent to be 1. Take away the exponent.
20. Recall that 2 times 1 equals 2. Write a 2 immediately beside the x.
21. Suppose that when you have a constant multiplied by a variable, the constant should always precede the variable in order to practice good mathematical form. Confirm that the 2 is in front of the x.
22. Conclude your work, and resolve that y'=2x. Box your answer.
Phew. Did I miss anything ?
I took a lot away from that lesson in CP^2 in which we learned to write technical documents.
I hope this has taught you never to take calculus for granted. xP
My head hurts. Does yours ?
