© Scarborough
Math 131
Fall 2007
Quiz 8 KEY
(20pts) NAME (printed neatly): ________________________________
(10pts) Section Number (circle correct section): 502 (10:20am)
503 (11:30am)
506 (4:10pm)
Quiz Grade: ______________
100
1. (20pts)
∫ ln ( 5x
2
)
− 6 + 2 ⋅ 3x dx = 0
100
2. (25pts) The marginal revenue for gadgets made by the Valley plant is
RV '( x ) = 82.5 x dollars per gadget for x gadgets produced and sold for 0 ≤ x ≤ 50 ,
and the marginal revenue for gadgets made by the Panhandle plant is
4
1
RP '( x ) = 3x + x 3 dollars per gadget for x gadgets produced and sold for
2
0 ≤ x ≤ 50 . Find and interpret the area between the two curves where the
Panhandle marginal revenue is less than the Valley marginal revenue.
2
y1 = 82.5x
4
1 3
x
[an appropriate window would be [0, 50] by [0, 8000]
2
use intersect on calculator to find intersection points of (0, 0) and (27, 2227.5)
y2 = 3x 2 +
27 
 2 1 4 

82
.
5
x
−
)  3x + x 3  dx =
∫ (
2 
0 


4
27 
1 3
2
82
.
5
x
−
3
x
−
x dx =

∫
2

0 
7
 82.5 2
1  3  3  27
3
x − x −  x  0 =

 2
2  7  

7
13
2
41.25 ( 27 ) − 273 −   ( 27 ) 3 − 0 ≈ 9919.607143
27
The Valley plant made $9919.61 more in revenue than the Panhandle plant when
production increased from 0 gadgets to 27 gadgets.
© Scarborough
3
∫
3. (25pts)
ln ( x )
1
3
∫
ln ( x )
1
ln( 3)
∫
x
x
Math 131
Fall 2007
Quiz 8 KEY
dx =
dx =
Let u = ln ( x )
du =
udu =
0
1
dx
x
ln( 3)
1 2
u
=
2 0
1
1
( ln(3))2 − ( 0 )2 =
2
2
1
( ln(3))2
2
If x = 1 , u = ln(1) = 0 .
If x = 3 , u = ln ( 3)
OR
∫
ln ( x )
x
dx =
∫ udu =
du =
1 2
u +C =
2
2
1
ln ( x ) ) + C
(
2
3
∫
1
ln ( x )
x
Let u = ln ( x )
dx =
3
2
1
ln ( x ) ) =
(
2
1
2
2
1
1
ln ( 3) ) − ( ln (1) ) =
(
2
2
2
1
1 2
ln ( 3 ) ) − ( 0 ) =
(
2
2
1
( ln(3))2
2
1
dx
x
2