A girl named Alice fell asleep during a discussion about the difference quotient.

She did not think it could possibly have anything to do with real life. While dreaming she saw a white rabbit. The white rabbit told her she shouldn't fall asleep during math class. The rabbit told her that even the most abstract topics sometimes produced solutions to challenging real world problems. He also told her of the power of the number zero, and the great things could be accomplished if she understood inffinitesimals. But beware said the rabbit, when doing these computations you must be very careful. The rabbit showed her the following calculation:

Lets begin by saying that x = 1 and y = 1.

Ok, said Alice. Then what? The rabbit showed her the following equation.

x² - y² = x² - xy:

He asked Alice if the equation represented a true statement. Alice frowned. She didn't like math. She studied the equation hard. She didn't know what the rabbit was asking. Then the rabbit helped her out. This is a problem from

MAT0024 he told her. You studied this problem two semesters ago. You liked math back then, he told Alice, you even got an A in that class. Just plug in

x = 1 and y = 1 and see if it is true. She wrote the following:

x² - y² = x² - xy:

Alice plugged in x = 1 and y = 1 and obtained

1² - 1² = 1² - 1 = 1

and that results in

0 = 0:

Yes!, it is a true statement Alice proclaimed. Ok, good said the rabbit. If x = 1 and y = 1, then the equation is a true statement. But do you really, really believe it is a true statement Alice? Yes, of course she replied. Ok good, but now I want you to factor the equation and then plug in x = 1 and y = 1. Alice was good at factoring. On the left side of the equation she used the di_erence of squares formula and on the right side of the equation she factored out an x. Just like the rabbit told her, she factored, then cancelled like terms, then plugged in the x = 1 and y = 1. this is what she got.

x² - y² = x² - xy:

Factoring the left hand side using the difference of squares and factoring x from right hand side, we get

(x + y)(x - y) = x(x - y):

Dividing both sides by x - y we get

(x + y) = x:

Since x = 1 and y = 1, we get

2 = 1:

Hey! said Alice, how did I end up with 2 = 1 when I was using a true statement? Was my factoring wrong, she asked. Nope, said the rabbit. The rabbit then hopped away. Wondering what type of math magic awaited her in a world where 2 = 1, Alice followed the rabbit down the hole.

While sliding down the rabbit hole, Alice's speed was given by the function

f(x) = 4x²;

where x is in seconds and f(x) represents feet. It took Alice 20 seconds to slide all the way into wonderland. If a human's instantaneous velocity is more than 80 feet per second upon entry into wonderland, he/she will be killed. Did Alice survive the trip into wonderland? Make the following calculations:

1. Calculate Alice's average velocity over the intervals [19:5; 20]; [19:9; 20]; [19:99; 20], and then estimate Alice's instantaneous rate of change at x = 20.

2. Calculate the difference quotient for f(x) and let h = 0; x = 20 (make sure you simplify the difference quotient before plugging in these values).

What does this say about Alice's instantaneous rate of change at x = 20?