Download Exam with Answers Key - Calculus and Analytic Geometry | MATH 221 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! QUESTION 1
A draining conical reservoir
(120)
Water is flowing at the rate of 50 m®/min from a shallow concrete
conical reservoir (the cone points downwards) of base radius 45 m and height 6m. Given that the
volume of acone is V = inr?A,
(a) At what rate (cm/min) is the water level changing when the water is 5 m deep?
(b) How fast (in cm/min) is the radius of the water’s surface changing then?
[10 points]
{10 points]
ANSWER
“ok
ay 45 4
av
Fa = 7 be
Ve sar h
xe ta).
v= 45 | S|
Ve $a(Eh kh
3
= Lk
av
Te * Bah) ah
gos DE xf
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QUESTION 2 ¢ /20)
It costs you ¢ dollars each to manufacture and distribute backpacks. If the backpacks sell at x
dollars each, the number sold is given by
a
n=——— +6(100~2)
where a and b are positive constants. Use calculus to find the selling price that will bring a maxi-
mum profit (justify that itis a maximum).
{Hint: profit per backpack is the difference between selling price and cost!]
ANSWER
Re - C) n
= OFC) ea: + bllco-x9)
= At bl00-x) VEC)
Maximize POX)
Plogso > benyvce tbh twoo-x) =0
= xX: Sot +
2
wk Pox) = At b (s0- =)
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(4)
QUESTION 5 ( 140)
For the function
2
x
f@)= 34
(a) What is the domain of f(x)? Is f(z) even, odd, or neither? [5 points]
(b) Find and simplify the first derivative of f(x). {5 points]
{c) Find the critical points of f(x). Where is the function increasing/decreasing? [5 points]
(d) Determine if the critical points are local maxima, minima, or neither. Which ones are also
global extrema, if any? [5 points]
(e) Where is f(x) concave up/down? Are there any inflection points? If so, find them. [5 points}
(f) Are there any asymptotes? Find their position. [5 points]
(g) Use ail the above information to sketch the graph of y = f(x). {10 points]
ANSWER ¥ Lo. ‘ (28)
00 aye -20-)'- (-2x) 20 =!
fo en e rr) el
domain is Ce, ju gull ) | QF) ya
} ven
fo) = ft, » fw = aos) 81", Geht
“eae
la)= Ix t-p- x) | — bx OP oop
(xJ= Dx B-Y~ se 2 Sign git by
rn Nomura «20 = 9 2
(
_ (% wy? rn 1.
a sail
Fo) =0 at x= , urslfrad at x * | ZO IALY
' wy te sg» 4 fly p \ 3 j
rane . oY —ik, *® U © concave ng > OV = down,
a . 7 = invessins Aty edi Te concavity eagle bab
7 | 7 ‘> \y = Aes ‘Hos ace wel jn at to not inf heli
-} oO } x peinh, vt
(f} hin = =} So boieated argeeg
O is 4 hed max sn f "changes Sigh bn a “ ye
{
j bso |
oe = oA fF con inulars a ; "
- jf bef max, Tne fm, a = bin A= 0 verte expept
meee aa , aa x7 atx = \ as
(continue on the next page if you need more-space) 71-71" ye
Page 6 of 8 (next poy -)
ANSWER
(9 (com tinued) Am ee = +e ver tice asyrpirt
- 0
A+-] 4) at Ya-] |
Jin, 1a
(continue on the next page if you need more space)
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