graphing nDeriv(, IVT, RoC

Greetings Kids!

So, it appears as if I chose a good day to have the most siblings/step-siblings of all of you, and if any of you were hiding anything, it's your loss.

As it turns out, we didnt go over much in class, mostly just how to graph a numerical derivative using our calculator, the IVT (Intermediate Value Theorem), and Rates of Change. Based on the fact that I spaced making this post until the night before class, I won't have time to download the software and do the whole screenshot deal, but it's a nice thought. Use your imaginations.

GRAPHING THE NUMERICAL DERIVATIVE:

1. Input the funtion of which you wish to see the derivative into the Y1 entry line.
2. In the Y2 entry line, follow the following (teehee!) progression of buttons:
MATH
8
VARS
→Y-VARS
1
1
,
X
,
X
)
3. Now that you have punched this in, hit the ENTER button. On the resultant graph, you now have the original function and its derivative.
4. Note: When you see in the Y2 entry line “nDeriv(Y1,X,X)” I suppose you may want to know what this all means. This means you are finding the numerical derivative of the Y1 function for the variable X in terms of X. Instead of using nDeriv( to find an individual slope at a point, we are instead using it to graph the slopes of all tangent lines to the curve, THE DERIVATIVE!

INTERMEDIATE VALUE THEOREM

The Intermediate Value Theorem is stated as follows:

If a and b are any two values on a function, f, and f is differentiable then f’ takes on all values between f’(a) and f’(b).

As Marchetti said, that’s a lot of mathspeak, which is something most of us don’t understand. Basically, if a function is differentiable, then a tangent line exists at all points. If I’m wrong on that, then correct me. Basically, it says that if the function can have a derivative between a and b, then the function will.

Rates of Change:

There are two methods for finding Instantaneous RoC, finding the RoC for any point on the function, and finding the RoC at a specific point. Here they are:


Inst. RoC for a Function (Defn. of Derivative):

f'(x)=(f(x+h) - f(x))/h

Inst. RoC at a Point, “a”

f'(a)=(f(a+h) - f(a))/h

I think that’s about it, thank you all for your time and I apologize dearly for my lack of punctuality. The homework, in case you’re looking desperately at this, is pg 140 # 29, 28, 33, 31, and 32 (yes, in that order).