IVIr. Orchard 's Math 142 WIR
Test 3 Review
Week 12
1. A runner's speed was measured at 2-second intervals are given in the table below. The
runner steadily increases her speed the entire time.
(a)
(b) Give a lower bound for the distance traveled by the runiwr using n = 6 during this
time.
L!-IS
2- ( 0 1 J 2 f 4 '2
0
°
+i)_ sf s .~ 1- t
0
'2) -=-
:: &iJJ-r~~tJ
2. Evaluate the following functions at the given point.
(a) f(x , y) = 0.9x
+ 0.5y- 0.05xy
at (0, - 1)
e. 9' ( o j+
G
J( -I) ;;
6
o5 {o) {-t)
J- C?. 5]
(b) g(x , y) = ln(x
Cj
+ y- 7)
at (e, 7)
[e / 7) ::: IYl (e +7 - 7}
(c) W(a , b, c, d) =
a(l+;}-d
2
-=-
ln fe )
~[J}
at (0.7, 8, 1, - 7)
0 . 7 { t +g) - {- 7)1.
)_ (I)
:::f_iL)S~
Mr. Orchard 's Math 142 WIR
Test 3 Review
Week 12
~+ 50 and the total
o sweaters is $760. What is the total cost of making 64 dog s~eaters?
cost of making 16 d~
3. The marginal cost fun ction for dog sweaters is given by C'( x)
l{!!
'-~ cl;:"'
X
l[;
::: 16 -r-;tso ;>(~ t--c
({x):=
,1(
7lo .:;_ { ( 16) ~ 1b(tt) f!Ou if1i +c
) (()
=
10
t-
50x-'-/x ~
-: ' 1() ';( '- ioup
C( b v-).::_ /o (6' ~) t
+
c
lbo fi'f +206
-)1!£y6J
4. Find the domain of t he following functions of two variables.
(a) f(x , y ) = J5 x
+y
5 x+ '( .> o
--~~
-?__y·
2
~~ tJt / ~· vf? S
Ufl41 Pv.ft7 rvofs
./1/o
1/
I
tJ 7 I
I cI
c; r
~
•. /
. (J I) It
t. n
(b) g(x , y) = ln( x - y- 10)
f,' v e (/ v m 6r:r)
!
/-A ff''l
Mr. Orchard 's Math 142 \;\fiR
Test 3 Review
Week 12
5. Find the exact value of J~ 2 - J4- x 2 dx by finding the area of an appropriate geometric
region.
~·a/t · d/{ /rf
ol rt~~t!;'v ~ .' }_
6. The graph of g(x) is given. Find the exact value of f~:3 g(x). ·
T{ (2__''/""
+ /(z.)
1f
-
?
lrr +-I]
-2
Mr. Orchard 's Math 142 WIR
Test 3 Review
Week 12
8. A manufacturer makes gadgets and widgets. The weekly demand and cost functions for
the consoles are
p = 300 - 2x + 2y
q = 225- X+ 6y
C(x , y) = 900 + 90.x + 120y
where x represents the weekly demand for gadgets; y represents the weekly demand for
widgets; p and q represent the price (in dollars) of gadgets and widgets (respectively) ;
and C ( x , y) is the cost function.
(a) Find R(x, y), the weekly revenue fun ction.
~ (x,y)=P
X+- f.Y == (300-lx 1ly)'n-
~ '7&0X-2' t2Xy
2
f-2-lSy
:=.-~ :1-2- ) ' ~ 2x
1._
62s-x ftt)r:
- ~ tbyL=
+ 6y1..
t
xy]
(b) Find P(x, y), t he weekly profit function.
Ptx,y) = (<.,(x,y)- {(x, y/)60 'I tJ-2
_ )_/Of(
s
t- /o Sy
2
xy - (9tN) f 70 y: f 12 DY)
byz. fYY - 9' 00
-2x r ty-,_ f-
- L;x-1..
""
l
-
(c) \i\That is the profit earned when t he manufacturer makes 4 gadgets and 8 widgets?
p{~ g) : : J-1
-
0 {'f) f-
IDs ( 8) -2 ( '-1 ( +- ~ { f5 )'- + tf' 5
·-?·OO
Test 3 Review
Mr. Orchard's Math 142 vVIR
Week 12
9. The supply and demand functions for a commodity are given by
D(x) = 53.2- 1.2x
P(x) = 0.2x 2 + 10
(b) vVhat is the consumers' surplus?
S((5~- 2
'0
I 'T.lf x -
-/:2y..) -
-/:~do~x : :
12.
0xL)
L
D
-=-
~'
. Lf
- D
Test 3 Review
Mr. Orchard 's Math 142 WIR
Week 12
10. Find the exact values of the following integrals using The Fundamental Theorem of
Calculus.
1
(a) f 1 T'dx
·;
= 'j. 'i
I
_/
:= J
1'
- { -1)
<f
=I -
T?)l
I -=---Lf?J
Mr. Orchard 's Math 142 \1\TIR
Week 12
Test 3 Review
12. Find all first-order partial derivatives of the following functions.
(a) f(x, y) = x 2 + 2x 3 y 4
-
~x 3 - y
2
-1-~(x~ y):: 2x'- -~ 2y.; ·- ~x· 2 - · ~x- - 0
rr: (x,v) :
(~
i ( X~y)
2 "', t- b '/- ,_7'.,. -
xY
-7
~ 0 + 2 1-". 4- y 3 - 0 - I
=-
I+> (r,y} -= ~x /1 -} ]
(b) h(x , y , z) =
t ::: y' X
_]
,---h)x; f / 'l) :::... -
1'~ (x, y, r) .,
j
-
2:
J
X- 2 ~- I
x-' z
-'7
13. If f(3) = 15, f'(x) is continuous, and }~ 4 f'(x)dx = 15, what is the value off( -4)?
('
~ 6:) ~~
J_lf
I
='
+(~) -+(- ~)
lS= }S - 1/-'f)
+'f-tf}
::-0
Mr. Orchard's Math 142 vVIR
Test 3 Review
Week 12
14. Use u-substitution to find the following antiderivatives.
-::::- 2"
1
/II I 'f. +-/)<{
+-
~
Mr . Orchard's Math 142 WIR
Test 3 Review
Week 12
15. Speedometer readings (in feet per second) for a motorcylce at 12-second intervals are
given in the table below:
t
60
v( t)
33
(a) Estimate the distance traveled by the motorcycle during this time using a left hand
sum with n = 5.
-
(b) Estimate the distance traveled by the motorcycle during this time using a right
hand sum with n = 5.
\I~ {x)
16. Find the derivative of ;;:1" 1 2"' dx
_
_
£
)
1
,,
-{1
1.
X
d
)l.
Mr. Orchard's Mat h 142 WIR
Test 3 Review
Week 12
17. Find the average value of f(x) = x(x- 1) on t he interval [-1 , 2].
I
\
- -.. j
2 - {-;)
"1
X{;<-1
._,
)
J;X :::
~ ·~ (4- ~))
I
3
\
2-
J (x
·t
,
1
-i)cAx =
-1
2
~(
~ ~ {~ i ))
(
-
:c j
1
Mr. Orchard 's Math 142 WIR
Test 3 Review
vVeek 12
19. Find the area below the x-axis and above y = (x - 2) 2
=- )( 1._
-
4.
'-1 ~
r Lf
-
'f
·=- X
2
Lf Y
-
20 . Use a left sum and a right sum witl@ectangles of equal width to approximate J~ 1 e- x dx
2
L H5 : ~ (e- 'f e-l~
Sub r" 11 tfrvvtl :
/.Lf{;,'2
I - C'-1)
'-1-
J
-
2
R~ S: { (e- ~ r 1 +t"- \ e-)~::-><_j__:_-+---=--,--r---:--t--Lr'!{1. Lftb]
[(,)
e
-(
)
I
L
L
1
e
-
II
I
I
-y
e e
-
I