<p>Langrange error bound is the difference between the sum of an infinite series and the sum of a finite series.</p>
<p>Euler's Method
This might take a little longer to explain, but it is fairly easy once you understand what to do (I taught this to my Calc teacher :-D).
Since the slope, dy/dx, shows the ratio of how much the function increases to how much x increases, and since f'(x) = dy/dx, dx*f'(x) = dy. A small change in x, multiplied by the slope is the amount the function, increases. Easy to understand, right?
Now, Euler's Method takes advantage of this in order to sketch a curve. So what you do is:
For a given point (1,2) on x^2, we want to approximate what value of th function x=3, using Euler's Method with step x=1.</p>
<p>This might be daunting, but don't let it be. All that you do is find the change in y after the function has traveled to x=2, and add that value to the f(1) to find the new coordinate (2, f(2)) So this is what we'd do mathematically:
Find dy=dx<em>f'(1)
=1</em>(2x)
=2
So the function went out one and up 2, so our new point is: (1+1, 2+2) = (2, 4)
So let's find for (3, f(3))</p>
<p>dy=dx<em>f'(2)
= 1</em>2*x
=4
(x+dx, y+dy) (that is all we are doing!)
(2+1, 4+4)
(3, 8)</p>
<p>Obviously the smaller your dx's are the more accurate the curve will be.</p>
<p>Newton's Method
This is a method to find a root of a function. I won't derive how the equation works right now, but if you plug in values to this equations: x[sub]n+1[/sub]=x[sub]n[/sub] - f(x[sub]n[/sub])/f'(x[sub]n[/sub])
Do this thee times, and then you should have a very accurate value for one of the function's root.</p>
<p>**Logistics Growth<a href="something%20with%20limits%20and%20a%20P(t)%20=%20something">/b</a>
Most MC questions ask what is the max pop. that the logistic function represents, which is pretty simple because dP/dt must be greater or equal to zero.</p>
<p>Slope Fields
For these MC questions you plug in values for x and y, and see if they match up to the slope at that point, or the value of dy/dx</p>