MATH 136 CALCULUS 1
FALL 2011
MATH 136 CALCULUS 1
FALL 2011
MORE TEST FOUR REVIEW
MORE TEST FOUR REVIEW
Here’re some more problems. Solutions to these are on my website.
1. Using n=6, approximate

4
2
( x  16)dx with left endpoints,
2
Here’re some more problems. Solutions to these are on my website.
1. Using n=6, approximate

4
2
( x 2  16)dx with left endpoints,
carefully drawing the necessary rectangles on a graph of the function
being integrated.
carefully drawing the necessary rectangles on a graph of the function
being integrated.
Without drawing the new rectangles, calculate another
approximation using right endpoints.
Without drawing the new rectangles, calculate another
approximation using right endpoints.
Which method do you think gives the better approximation for this
integral? Why?
Which method do you think gives the better approximation for this
integral? Why?
2. During the recent Team Milkshake Drinking Championships,
sponsored by JackInTheBox, Team USA drank at a rate of r (t )
ounces per minute. Explain what

85
0
r (t ) dt would mean, including
what units it would have.

2
1
2
e
(

85
0
r (t ) dt would mean, including
3. Integrate.
6
 2 x)dx
x4
x2  x
dx
x3
c)

e)
 sin( x)e
g)
 x (ln( x
1
ounces per minute. Explain what
what units it would have.
3. Integrate.
a)
2. During the recent Team Milkshake Drinking Championships,
sponsored by JackInTheBox, Team USA drank at a rate of r (t )
cos( x)
b)
d)
dx


5
1
(| x  2 | 1)dx
9 x 2 ln( x 3  8)
x3  8
xdx
f) 
2x 2  3
dx
dx
3
 24)) 4
a)

2
1
2
e
(
6
 2 x)dx
x4
x2  x
dx
x3
c)

e)
 sin( x)e
g)
 x (ln( x
1
cos( x)
b)
d)
dx


5
1
(| x  2 | 1)dx
9 x 2 ln( x 3  8)
x3  8
xdx
f) 
2x 2  3
dx
dx
3
 24)) 4
4. If f ' ' ( x)  x  cos( x) and f ' (0)  2 and f (0)  4 , find f (x ) .
4. If f ' ' ( x)  x  cos( x) and f ' (0)  2 and f (0)  4 , find f (x ) .
5. The acceleration of a trachybrycheodon (an extinct prehistoric
2
ancestor of the jellyfish) is given by a(t )  12(t  t ) meters per
second. Assuming its initial velocity and position are zero, find its
velocity at the moment it returns to its starting place.
5. The acceleration of a trachybrycheodon (an extinct prehistoric
2
ancestor of the jellyfish) is given by a(t )  12(t  t ) meters per
second. Assuming its initial velocity and position are zero, find its
velocity at the moment it returns to its starting place.