3/6/11

What to doooo, what to do...Guess I'll just continue giving example problems.

Ex. 1) What is the x-coordinate of the point of inflection on the graph of y=1/3x^3+5x^2+24 ?
*Okay, first let's identify our key words in this problem and see what we actually have to work with
*POINT OF INFLECTION is the key word..which means 2nd derivative! So all you have to do for this problem is follow the guidelines for the 2nd derivative test which says you take the 2nd dervative, set it equal to zero, solve for x, set up intervals, and check numbers on each interval to see where the interval is negative/positive. If the signs change, that means there IS in fact a point of inflection. If the signs DO NOT change, guess what, there's NO point of inflection
*So when you take the 1st derivative of the equation they gave you, you should get:
x^2 + 10x
*And when you take the 2nd derivative you should get 2x+10
*Set it equal to zero and solve for x:
2x+10=0
2x=-10
x=-5
*Set up intervals:
(-infinity, -5)u(-5, infinity)
*for the 1st interval you can plug in -6 and you should get a negative number; for the 2nd interval you can plug in 0 and you should get a positive number; therforeeeeee, there IS a point of inflection at x = -5, so that's your answer!

Ex. 2) If x^2 + xy = 10, then when x = 2, dy/dx = ?
*Again, let's first find our key word(s)...which is DY/DX!!!! And all that means is they want you to take the derivative of the equation they gave you (using implicit differentiation, by the way, because there is a y in the equation) And thennnnn, they want you to plug in 2 for x afterwards
*So first let's differentiate this problem..You should have this before you simplify:
2x + (x)(dy/dx)+(y)(1) = 0
*Then simplifying that all the way you should wind up with this:
dy/dx = (-2x-y)/x
*Now in order to plug back into that equation you have to get a y-value also..because you were only given an x-value. So to do that, you plug your x, 2, into the original equation to get the y-value:
(2)^2 + 2y = 10
4 + 2y = 10
y = 3
*Now plug your x and y-values into the dy/dx equation above:
[-2(2)-3]/(2)
=(-4-3)/2
= -7/2 and that's the answer they wanted