Sunday, February 11, 2007

4.2 Concavity

Concave up - All tangent lines are below a curved graph
Concave Down - All tangent lines are above a curved graph

You can tell whether a graph is concave up or down by looking at its second derivative:
f''(x) is Positive(+), the graph is concave up and its slope increases
f''(x) is Negative(-), the graph is concave down and its slope decreases

Point of Inflection:
When the concavity of a graph changes

to find where the graph is concave up and where it is concave down, you may use
The Number Line Test:
in the number line test, you find the point of inflection, then you plug in numbers that are larger or smaller than the x value in the point of inflection to find the correct signs to use the rule above.

ie. y=x^5
y'=5x^4
y''=20x^3
Now, you must make the y=0 to find the point of inflection, so if you do that, the x must equal 0 to make the y value also equal 0.
that is our point of inflection (0,0)
so we start to plug in numbers like 1 and -1 which are larger than 0 and smaller than 0
the results we get are for 1 and -1 are 20 and -20 respectively. and since, 1 results in a positive number, we can say, by using the rule, that the right side of 0 must be concave up and vice versa for the negative .

Maximums and Minimums:
If the second derivative is positive, we get a minimum
If the second derivative is a negative, we get a maximum

ie.
f(x)=x^3-3x^2-24x+32

f'(x)=3x^2-6x-24=0
3(x^2-2x-8)=0
3(x-4)(x+2)=0 x=4,-2

f''(x)=6x-6
for values 4 and -2 plugged into the second derivative equation, we get for 4, positive value which indicates a minimum, and for -2, a negative value which indicates a maximum.

for additional reference go here --> here--> no, here
"As long as Calculus is taught in school, there will be prayer in school."

Graphing rational functions is a pain in the asymptote.

This isn't really a joke, it supposedly happened in a UK GCSE exam some years ago, but it may amuse you:
Q: how many times can you subtract 7 from 83, and what is left afterwards?
A: I can subtract it as many times as I want, and it leaves 76 every time.

Danica Thou Art Next!

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