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Math 76O-250 Calculus for Business
Self-marking Practice quiz 4
(Relative Extrema , Anti-derivatives)
Submit over the internet by 11:59 PM Sunday 28 October 2017 Maximum value toward semester grade is 4 pts
Methods will be discussed in class; CAUTION: Prof McFarland makes new tests each semester.
(b)(4 pts) Find both CVs (critical values) for f(x) in [1a]
above. Then test each CV by entering numbers in the
12 empty boxes of the table below. Scoring: 1 point for each
CV (top two answer boxes) ; 1 point for each of the remaining 2 rows below
Note that the
table at the
right is a 1st
derivative test
test value
CV
test value
CV
test value
x
-2
+1
+4
f '(x)
0
0
Type of CV : Relative Maximum, Relative Minimum, Ledge
Max
Min
Ledge
Max
Min
Ledge
[2]
Let f(x) = x3 - 3x2 - 9x + 1
with the domain of f restricted to: 0x+2 :
Compare carefully with [1] above.
(a)(1 pt) Enter BOTH CV's for the above f(x) into the
2nd row of the table below, as well as a test value between.
(b)(4 pts) Fill in the remaining empty boxes in the 3rd, 4th, and
5th rows of the table below.
Thus, you will display both a graph test and
a 1st derivative test for these two CV's.
Scoring : 1 point for each answer in bottom row ; ½ point each for other table entries
smaller CV
Test Value
larger CV
x
+1
f (x)
f '(x)
Type of CV : Relative Maximum, Relative Minimum, Ledge
Max
Min
Ledge
Max
Min
Ledge
[3]
An open-topped box is to be made from an
6 cm by 8 cm square of cardboard, by removing a
smaller square from each corner, and folding up the
resulting flaps, as in the animation below.
What size of cut-out square will produce the box of greatest volume?
(a)(2 points) For the problem enclosed above, let "x" be the edge length of
cut-out squares, as shown, and let V be the volume of the resulting box.
Enter integers below (such as 3, 0, or -2), expressing the relationship between
"x" and "V". Thus, if you (incorrectly) thought V = (10 - x)(x + 2),
then you would multiply out the product and enter "0", "-1", "19", and "20" in the boxes below.
V (x) =
x3 +
x2 +
x +
(b)(3 points) Put aside (for the moment) any physical restrictions on x,
and find the two values of x at which V ' = 0 ;
you will need the quadratic formula to find these numbers. Enter them as CVs in the
table below, and complete the 2nd derivative test below, thus answering the
question in the problem above. Scoring : ½ point for each of 6 correct table boxes below
Note that the
table at the
right is a 2nd
derivative test
smaller CV
larger CV
x
V "(x)
Type of CV : Relative Maximum, Relative Minimum, Ledge
Max
Min
Ledge
Max
Min
Ledge
As to those "physical restrictions" : cut yourself a 6" by 8" rectangle and use the larger CV above to build a box ; what happens?
(unscored)
[4]
(a)(3 pts) If q represents the number of lamps sold,
and if p represents the price of each lamp, and if p and q are
related by the equation :
q = 60 - 0.1p , then calculate the
elasticity of demand when p = 200. [See sample problem]
elasticity of demand =
(b)(2 pts)) The gross national (GNP)
product of Singapore was $100 billion in 2000, and $200 billion
in 2010. If Singapore's GNP is growing exponentially over time,
what should it be in 2020? [See sample problem]
Singapore's GNP = $
billion
[5]
(3 pts) Find f '(x) and f "(x) if : f(x) =
x x
AFTER finding these derivatives, let "x" = 1, to simplify your entries below.
Use the SCORE CHECKER below to be sure answer to problem [5b] is correct.
Use this f "(x) to find the only 2nd
order CV for f(x), that is, CANDIDATE. Then test your
candidate as to whether or not it is indeed an IP
(inflection point) by filling in the table below (accurate to within 0.0001).
As you use the Substitution Method, you will alter
the integrand so that u' appears as a factor. Identify your substitution variables
u and y and their derivatives in the table at the right.
Finally, let C = 0, x = 2, and enter F(2) =
Scoring : 3 pts for F(2) above ; 2 pts
for for table (right)
For example, if this question were to find
8x3 dx ,
then F(x) = 2x4 + C ,
and you would enter F(2) = 32
u =
x
x
3x2 - 8
3x2
x2
du dx
=
1
3x2 - 8
18x
6x
2x
y =
u
u2
2( )3
dy du
=
u
x
3
[9]
Find
(1 - x) e x
dx = F(x) + C ;
As you use Integration by Parts to find
F(x), you must choose expressions
u, v, u', and v'
from alternatives in the table at the right.
Finally, let C = 0, x = 1, and enter F(1) =
Scoring : 3 pts for F(1) above ; 2 pts
for for table (right)
For example, if this question were to find
8x3 dx ,
then F(x) = 2x4 + C ,
and you would enter F(1) = 2
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x
x
- 2 x -3 ] dx .
Finally, let C=0 , x=1 , and enter F(1) :
dx (integrand is 
3x2 - 8
