Name: __________________
1
Class:
Find the derivative of the function
mentally :
f (x) = ln (6x 2
3
4
8
Find
dy
dx
9
Find
dy
dx
+ 7) ln x .
using implicit differentiation.
x
Find the derivative of the function :
10 Find
g (x ) = ln | 2x using implicit differentiation.
2x + 3y = 4.
3).
Find the derivative of the function :
f (x ) = ( x
Date: _____________
dy
dx
2
2y = 8.
using implicit differentiation.
1|.
11x + 3y = xy
4
Find the derivative of the function :
11 Find
f (x ) = e
4x
7
dy
dx
using implicit differentiation.
ln 4x
e
5
x
Find the derivative of the function :
4x
2
7x + 1
x
h (x ) = e
12 An employment research company estimates
that the value of a recent MBA graduate to
an accounting company is
.
V = 4e
6
Find the derivative of the function :
s (x ) = e
x
7
y = 7
4x
2
2.
2
If $28000 is invested in a savings account
yielding 3% per year, compounded
semiannually, how fast is the balance
growing after 3 years?
PAGE 1
where V is the value
a number of years of
experience, and g is
grade point average.
de
dg
when g = 2.
2
+ 6g
3
of the graduate, e is
prior business
the graduate school
If V = 290, find
Name: __________________
Class:
13 Locate all extrema in the graph :
Date: _____________
19 I want to fence in a rectangular vegetable
patch. The fencing for the east and west
sides costs $4 per foot, while the fencing
for the north and south sides costs only
$2 per foot. I have a budget of $80 for
the project. What is the largest area I
can enclose?
20 Hercules films is deciding on the price of
the video release of its film "Son of
Frankenstein". Its marketing people
estimate that at a price of p dollars it
2000p
can sell a total of q = 200000 copies. What price will bring in the
greatest revenue?
14 Find the exact location of all absolute
extrema of the function
with domain [ 3
f (x) = x
21 The Chocolate Box Co. is going to make
open topped boxes out of 9'' 12''
rectangles of cardboard by cutting squares
out of the corners and folding up the
sides. What is the largest volume box it
can make this way?
14x
7, 7 ].
15 Find the exact location of all the
absolute extrema of the function
f (x) = x
with domain ( 5, 4
12x
3
).
22 Fair Weather Airlines will accept only
bags for which the sum of the length and
width is 57 inches, while the sum of
length, height, and twice the width is 114
inches. What is the largest volume of the
bag that it will accept?
16 Maximize P = xy with
2
4x + 5y = 40.
23 Calculate
d y
dx
17 Minimize
:
2
y = 4x
2
+ 17x .
S = x + y
with xy = 36 and both x and y > 0.
2
24 Calculate
18 For a rectangle with area 100 to have the
smallest perimeter, what dimensions should
it have?
PAGE 2
d y
dx
:
2
y =
12.
x
Name: __________________
Class:
29 Combined SAT scores in the United States
could be approximated by
2
25 Calculate
d y
dx
:
2
8
y = x
T (t ) = .
2
d y
dx
0.5245t
954 ( 0 :
2
y = 4e
(x 8)
0.01478t
2
2
26 Calculate
Date: _____________
x.
3
+
10.32t +
t 23 )
in the years 1970 1993. Here t is the
number of years since 1970, and T is the
combined SAT score average for the United
States. Find all points of inflection of
the graph of T, and interpret the result.
2
27 Calculate
d y
dx
:
2
y = 9x
10
28 The graph of a function
x
2
(x
2
24)
+ 2 lnx .
30 In 1965 the economist F.M. Scherer modeled
the number, n, of patens produced by a
firm as a function of the size, s, of the
firm (measured in annual sales in millions
of dollars). He came up with the following
equation based on a study of 448 large
firms :
f (x ) =
is given.
n =
3.79 + 139.87s 21.88s
2
3
+ 1.538s .
2
a. Find
d n
ds
Find the coordinates of all points of
inflection of this function ( if any ) .
PAGE 3
.
2
s = 2
ANSWER KEY
Practice for Test 4 061
1.
29.
6
( 6x 3)
(
)
4
3
2. 4x ln x + x +7
( )
3.
x
2
( 2x 1 )
4x
7
4. 28e4x x6 ln 4x + e
(
)
7
x
2
x
e
2
( 4e ( x 2) 2x ( e
( x 2)
4x
6.
4x 7x+ 1
x
1
8x 7 5.
2
1
2
4x
))
2
2
7. $911.67
2
3
8. 9.
x
1
( 11 y )
( x 3)
10.
11. y
12. 1.16
13.
14.
15.
( 2,0 ),( 1,20 ),( 1,0 ),( 2,20 )
( 7, 245),( 7,245)
( 9, 2187)
16. 20
17. 12
18. 10,10
19. 50
20. 50
21. 82
22. 27436
23. 8
24
24.
x
25. 3
48
4
x
26. 4e ( x 8)
27. 990x
12
2
2
x
28. ( 2, 80 ),( 2, 80 )
ANSWER KEY ( 11.83,880.85)
30. 25.30
Page 1