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March 8th, 2011, 10:57 AM   #1
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Linearization :( HELP!

Using a calculator or computer, generate a graph like the one shown below by graphing y=e^(2x)-1 and y=2x for -0.5 < x < 0.5. (What is the relationship between y=e^(2x)-1 and y=2x?)

So...I'm given a graph with 2 lines here (1 curved and 1 straight - would that be the tangent line or something?) but I can't post the graph.

1. Use your graph to estimate to one decimal place the (largest) magnitude of the error in approximating e^(2x)-1 by 2x for -0.5 < x < 0.5.
error:

2. Is the approximation an over- or an underestimate when x > 0? (Enter over, or under.)

THANK YOU SO MUCH IN ADVANCE FOR HELPING ME!
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March 9th, 2011, 09:15 PM   #2
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Re: Linearization :( HELP!

Hey,

I should be able to help you a little. I'm not fully sure what the question is asking for when it says, "(largest) magnitude", but nonetheless ---
[note, if interval really is -0.5 < x < 0.5 where x cannot be 0.5, you will have to change this up a bit]


is the linearization of:

at a = 0.

Now, let and

Then the tangent line, L(x) at a = 0:


The "max" amount of error from point where a = 0 would be at either or . Both have the same max error.
The max error is found by:


So with an error of +/- 0.5 at x = a = 0:

Says that with an error of 0.5 in x, there is a max error of 1 in y AT .

Now, if by the question "magnitude" does it mean error from at , then you would simply take the absolute value of the difference of and at :


I say take it at 0.5 as is greatest at within the interval [-0.5, 0.5]

So this implies that there is an error of value e at = 0.5 in . Considering that , it is an underestimate. If the interval however is indeed -0.5 < x < 0.5 where x cannot be 0.5, then error would approach e.

I'm not sure if I did all this correctly, but hopefully I did a descent job helping you out anyway.
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