Scribe Post: Day 42

Today we learned the derivative rule for an exponential function.


f(x) = a ^x
f(x+h) = a ^x+h

f ' (x) = lim f(x+h) - f(x) /h
h->0

= lim a ^x+h + a ^x /h
h->0


When you graph the parent function and its derivative on your calculator you'll find that the graphs are very similar. The y-intercept of the functions is different. This must mean that there a constant, C, to multiply with that varies the y-intercept.

f ' (x) = C a^x


In the table of values, the values of both of the graphs are doubling. In Y3, if you put y2/y1, you will find the constant value. When x = 0, the output is the same.

Try different values of a, and find the constant value , C.

* this is supposed to be my table, sorry if it looks kind of weird *

A C f(x)

2 0.693 f ' (x) = o.693 (2 ^x)
3 1.098 f ' (x) = 1.098 (3 ^x)
e 1.000 f ' (x) = 1 (e ^x) * this is a special number *
4 1.386 ...
5 1.610 ...
6 1.791 you can see the pattern here, so I don't want to type all these eq'ns =)
7 1.941 ...
8 2.077 ...
9 2.197 ...

We are finding the derivative of the parent function at x=0, and that gives us the constant.

_________________________________________________________


f(x) = e ^x
f' (x) = e^x
e ^x is a special number because it is its own derivative.

_________________________________________________________

Now back to the table. If you plot this on your calculator, the graph looks similar to a square root function, but if you use the stat key on your calculator you can find the accuracy it fits on the graph. A second that fits much much better it the natural log , Lnx.

From this fact, we have found that Ln x is the constant of the derivative at!
eg.
ln2 = 0.693
lne = 1

___________________________________________________

f '(x) = lim a ^x+h + a ^x / h
h->0

= lim a^xa^h - a ^x/ h
h->0
* we used the law of exponents above*

= lim a ^x ( a^h-1 ) / h
h->0
* a ^x was factored out *

= a ^x * lim (a^h-1 ) / h
h->0
* a ^x was taken out of the limit b/c it will not be affected by h*

f ' (x) = a ^x * lim (a^h-1 ) / h
. h->0

lim (a^h-1 ) / h is the derivative of the parent function at x = 0.
h->0
_______________________________

p ^x has a slope of 1 at x = 0.

lim p ^x - p ⁰ / (x-0)
x->0

p ^x - p ⁰ / x = 1

p ^x - p ⁰ = x

p ^x = x + p ⁰

p ^x1/x = (x+1)^1/x

p = (x+1)^1/x * graph this on your calculator *

As x -> 0 it approaches e.

e = lim (x+1)^1/x

Homework for tonight is 4.2, odd questions. Due for wednesday, I don't if that means to hand in =( ... does it? Anyway, the next scribe is Lindsay.


hehe I thought this was funny...

thats
. like
. my brain
after


. math


sometimes.