2/26/11

Okayyyy, so for this blog I'll just go over some problems that we normally see on the multiple choice no calculator section..

Ex. 1) 1/2 S e^t/2 dt =
*(1/2 times the integral of e^t/2)
*Okay so to solve this integral we must first integrate e..but remember to leave the number, 1/2, out in front until the end
*So to integrate e^t/2, it's going to be e^t/2 / 1/2...where 1/2 is the derivative of the exponent of e
*So you should have this 1/2[e^t/2 / 1/2], which simplifies to:
1/2[2e^t/2]
*And distributing the 1/2 through you get e^t/2 + c...*don't forget the +c because you weren't given bounds for this integral

Ex. 2) At what point on the graph of y=1/2x^2 is the tangent line parallel to the line 2x-4y=3?
*First, since they say "tangent line" you're going to have to derive the 1st equation they gave you, and you should get that y'=x
*And in order to find where the graph of that^ is parallel to the line (2nd equ.) you must find the slope of the 2nd equation and then set that slope and the derivative you just found equal to each other
*So let's find the slope of 2x-4y=3
-first subtract the 2x over to get -4y=-2x+3
-now divide 4 by everything and you should get y=1/2x-3/4, which means your slope is 1/2
*Now set your derivative equal to your slope:
x=1/2 (so that's also your x-value...to find your y-value you must plug the x-value into the original equation y=1/2x^2..so you get that y=1/8
*Since you found your x and y, now you have a point!
(1/2, 1/8) is your final answer

Ex. 3) The average value of cos x on the interval [-3,5] is
*AVERAGE VALUE is the key word in this problem! All you do for this problem is simply plug into a formula..which is this:
1/b-a Sa,b f(x)...where -3 is a and 5 is b
*So plugging in you get: 1/5-(-3) S-3,5 cos x dx
= 1/8 [sinx] l-3,5 ...where the "l" is the bar
Fundamental Theorem of Calculus:
1/8 sin(5) - [1/8 sin(-3)]
(*Since the negative for -3 would become -sin(3) that means the subtraction sign would become a + sign)
*So now you have 1/8 sin(5) + 1/8 sin(3) and that's your final answer..or you could put it as a fraction like this:
(sin5 + sin3)/8