[adwight]

Its that time of the year again, AP Calculus Final is in a few weeks, and I've been given practice exams to go over.  I can get help for these, so this is why I'm asking.  They may be simple, but I couldn't figure them out.  Any help is appreciated.

1.  (integral of) x sin(2x) dx

a) -x/2cost(2x) + 1/4sin(2x) + C
b) -x/2cost(2x) - 1/4sin(2x) + C
c) x/2cost(2x) - 1/4sin(2x) + C
d) -2cost(2x) + sin(2x) + C
e) -x/2cost(2x) + 1/4sin(2x) + C

2. If the region enclosed by the y-axis, and the curve y = (square root)x, and y = 2 is revolved about the y axis, the volume of the solid generated is.

a) 32pi/5
b)16pi/3
c)16pi/5
d)8pi/3
e)pi

3.  If x^2 + y^2 = 25, what is the value of d2y/dx2 at the point (4,3)

a)-25/27
b)-7/27
c)7/27
d)3/4
e)25/27

[Night Hammer]

eww math

[FireCrack]

AP calculus = hell.


I was taking it this year but then i dropped it because i was only getting 60 even after scaling, that said i did drop it once all studying was finished and it was just review, and now i've reviewed anyways (just an easier exam) so...

1) int(x sin(2x) dx)

Integration by Parts

int(x dv) = x v - int(dx v)

so

int(x sin(2x) dx)
=
-x 1/2 cos(2x) - int( -1/2 cos(2x) dx)
=
-x 1/2 cos(2x) + 1/4 sin(2x)

Massive arithmitic errors, had to redo that 5 times but yeah E


2) isnt that an infinite reigon? or am i notreading the question right?


for 3 the awnser is D unless my 5 second solution blows. You get the second derivative then sub in the y' from the first derivative to match it wiht X

[Anaz]

EDIT: oops

[phreak]

Originally posted by adwight
Its that time of the year again, AP Calculus Final is in a few weeks, and I've been given practice exams to go over.  I can get help for these, so this is why I'm asking.  They may be simple, but I couldn't figure them out.  Any help is appreciated.

1.  integral(x sin(2x)*dx)
Integration by parts, u=x, dv=sin(2x) dx

a) -x/2cost(2x) + 1/4sin(2x) + C
b) -x/2cost(2x) - 1/4sin(2x) + C
c) x/2cost(2x) - 1/4sin(2x) + C
d) -2cost(2x) + sin(2x) + C
e) -x/2cost(2x) + 1/4sin(2x) + C

2. If the region enclosed by the y-axis, and the curve y = sqrt(x) is revolved about the y axis, the volume of the solid generated is. I need an interval, it goes from 0 to what?  The integral is pi*integral(y^4*dy,0,?)

a) 32pi/5
b)16pi/3
c)16pi/5
d)8pi/3
e)pi

3.  If x² + y² = 25, what is the value of d²y/dx² at the point (4,3)

f'(x) = (y'=x/y)
f''(x) = (y''=(y-xy')/(y²)) = (y''=(y²-x²)/y³)

a)-25/27
b)-7/27
c)7/27
d)3/4
e)25/27

[Bobboau]

1) E

2) infinite, unless there is an interval

3) not sure of the syntax here, you want the second derivitive of y divided by the first dirivitive of x^2 ?

[adwight]

My bad, the 2nd problem is bounded at the top by y=2, but I figured that one out anyways.  Thanks guys.  The second one the answer is A, because of Phreak's help I got.  I hate not being able to retain all of this stuff...