This week we took some more AP tests, so I'll do some problems from them.
Example 1:
Find the limit as x goes to infinity of (20x^2 - 13x + 5)/(5 - 4x^3)
The answer is zero because you use you're limit of infinity rules:
1) if the degree of the top equals the degree of the bottom, then the limit is the coeffients of the two made into a fraction
2) if the degree of the top is greater than the degree of the bottom then the answer is plus or minus infinity (depending on the coeffients given).
3) if the degree of the top is less than the degree of the bottom then the answer is always zero (perfect example is this problem).
Example 2:
Find the limit as h goes to zero of (ln(2+h) - ln(2))/(h).
This problem is the definition of a derivative. So you would just take the derivative of ln(2), which using the formula dy/dx ln(x) = 1/x the problem would equal 1/2.
Example 3:
If y=e^(-x^2), then y''(0) equals:
So you take the first derivative:
e^(-x^2) * (-2x)
Then take the second derivative:
e^(-x^2) * (-2x) * (-2)
Then plug in 0 for x.
e^0 * (-2)(0) * (-2)
Equals:
Zero
Linearization:
I have no idea what this is. I probably do know, I just don't know what to do with it when I see it in a problem.
Example:
And example of this would be like number 33 on AB Practice Exam 1 - Part B (calculator).
It tells you to look at a graph and then find the local linerearization of H(x) near x=3.
But in all honesty, I have no clue what that means. Help?
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for linearization,find the equation of the tangent line and plug in the x value (decimal).
ReplyDeleteFor a linerization problem, you're going to be asked to "approximate." You'll also know a problem is a linerization problem if it has a decimal in it.
ReplyDeleteLinerization is just finding a tangent line and plugging the decimal in for x.
linearization is just finding the tangent line. You first take derivative solve for x, then plug your x back into get a y value. then set up your new equation
ReplyDeletelook for the word "approximate". it means it's a linearization problem. find the equation for the tangent line, then plug in the x value and solve! that's all :)
ReplyDeleteWith linearization, always find the equation of the tangent line. You will need it!
ReplyDeleteThe derivative of ln(2) is 0
ReplyDeleteLn(2) equates to .69314
Therefore the derivative is zero