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Math 215 Fall 2011
Exam 1
Name: Lab section:
Instructions:
- The exam consist of 6 problems for a total of 68 points. Please look through the exam booklet and make sure it has eleven pages. The last page is blank and is to be used as scratch paper.
- The exam duration is 90 minutes.
- No calculators are allowed.
- For multiple choice problems there is no partial credit. For all other problems, show all your work to receive full credit.
- Make sure your answers are clearly marked (circled or boxed).
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Points
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Total
You may find some of the following formulas useful (but probably not all of them)
- sin^2 (x) + COS^2 (X) = 1 and cos(2x) = cos^2 (x) sin^2 (x)
1 cos(2x)
- sin(2x) = 2sin(x)cos(x) and sin^2 (x) 2
- cos2() x^ = ----'--'^1 + 2
- cos(IT/3) = 1/2 and sin(IT/3) = .j3/2.
- cos(IT/4) = .;2/2 and sin(IT/4) = .;2/2.
- cos(IT /6) .j3/2 and sin (IT /6) 1/2.
- cos(O) 1 and sin(O) O.
- esc (x) = - esc (x) cot _(x).
- ft sec(t)_ sec(t) tan(t). - to tan(O) sec 2 (O). - t. cot(r) - csc^2 (r).
- Arc length function: Length of curve from (x(a), y(a), z(a)) to (x(t), y(t), z(t)) is s(t) J~ _Ir'(u)ldu.
- T(t)_ = I~:~gl' N(t) = I~:m" B(t) = T(t) x N(t).
- proJba^.^ = (b.a)jbj2 b^ , compba^ =^ b'aTbT
' , k)
- (a, b, c) x (d, e,.f) = det ~ t c = (bl ce, -(al - cd), ae - bd). ( del
- a x b = -b x a, a x (ab) = (aa) x b = a(a x b).
- ft(r(t). uet)) (r'(t). u(t)) + (r(t). u'(t)) , ft(r(t) x u(t)) (r'(t) x u(t)) (r(t) x u'(t)) ,
ft(f(t)r(t)) = (f'(t)r(t)) + (f(t)r'(t)).
- Second derivative test: D = Ixxlyy f;y.
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iii) Find the area of the triangle whose vertices are A, B, and C.
@D b) 12V c) 2 d) 4 e) None of the above.
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,Ar<&. = i Ifg;- ,fc 1:= ~ I(Z/0- Z^ ) x(~ 6; +)/. tl</2/ -2~ /"2'? I
I t^ (^1 t^ 1.1^ '1^ t^ t
-== lli"^ 2'/Z)t/2^ -==-^ L^ Vll+f'(l~/l
L
; -t J 6-/1' --~ .L.r&.17. -:.^ b {(
iv) Find the distance from the point C to the line containing A and B.
a) ~ b) 2V v'29 c c) 12)
~ e) None of the above.
A B
are~ l t.-.~"f5&. ~ -4- 1.48 I h ;:- bĀ« (/-.- fA (",~ dis+ = h/
(2 1b^ 12fi: Jd
t So, (^) h:: =^ T2fi 1481. ~ (^) ~ bD U 'l'+o To;' (^21) fiB <
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Problem 2 (5+5=10 points): Consider the space curve given by the parametric equations
x - 2t - 2, y e + 3t + 2, Z = 2t + 2.
i) What is the unit tangent vector to this curve at the point (-2,2, 2)?
rIo = (-Zf--Z I fL"-3f+2 ,,2f+^2 > r-(~):' /-2 2 2) ::=-J +:::- 0 C I / r'{f) ~ <. -2 (^) f 'If + (^3) , 2 /
rl{O) ~ (. -2 "3 2? I' /
_ (^) ,-/-1 I )2) I =) (^) .~\ (^) ",-/- 2 I (^) (^3) I "2 > TID)~ V (
ii) \Vrite, but do not evaluate, an expression for the length of the curve between the points (0.0,0) and (-2,2,2).
r(t); (u/o/> ) t= -/
r:{t) ~ (-7. 2' 2~ -,. -f.: 0 I f
[~I{f)1 = Jt~ {?t~3t ~ f
L '" f "'{Sf {2f+3/ el+
.-(
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Problem 4 (5+5+5=15 points): Let a point on Earth close to London, England be expressed by the coordinates (x, y), where x and yare the longitude and latitude, respectively, of the point. Suppose that the
temperature of a place close to London is given by the function T(x, y) 5 sin x ~ + 110.
i) The coordinates of the city of Greenwich, England are (0,101/2). In which direction should one walk
from this point such that the temperature will increase the fastest? (You answer should be of the form "In the direction of the vector .")
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\71~F ( S- (<>S )(' J -~ >
r; -~ ') QT (OJ ~) = (^ / "Z.
(2" 1l. d"" ([..,,, J /zv,
(Jec/cv. (~/ -{]
ii) What is the directional derivative of the function T(x, y) at Greenwich, in the direction towards Aberdeen,
Scotland, which has coordinates (2, 115/2)? ...::> _ 1 I- .-> , -L. LL -:.. 411'..t /,.t"ec~- /.., d'''< (/P..... e> V .... .r= (1 7)
v= (~/ 7) / ua^ I
-L (:2 7> .. <~ -J. >
,e I / 2 l)~ T(~)- (^) vt;"
'21 __ _
- )^
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- -(^ {IO^ -^2 -^ ZlfB
o '-----
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iii) Suppose that the coordinates of an African swallow migrating south t hours after it has taken flight are
given by (x(t), yet)), where x(t) = (t - 3)3, y(t)
t
How fast it the temperature changing for the bird when it is directly over Greenwich (assume that the altitude of the bird does not affect the temperature)? You may use the facts that
x'(t) 3(t-3f, y'(t) -1/2,
and (x(3), y(3)) (0,101/2), which are the coordinates of Greenwich,
dT dT^ dk^ + 1I<~ ---- -== ----^ < c{t Jx^ JA-^ d~^ d+
'::: ~ ~'it/
dTI _ b/O^ +(-~){-i)^ I.^ 1-^ rhv A~ -t"
Problem 6: (3+3+3=9 points) Consider the function f(x, y) +8y2 x^2 + 4. The first order partial derivatives of this function are:
fx(x, y) = -2x, fy(x, y) = _4 y 3 + 16y.
The second order partial derivatives or this function are:
lrx(x, y) = -2, fyy(x, y) = -12y2 + 16, fxy(x, y) = fyx(x, y) 0.^ 'I
Classify each of the following points as the location of a local maximum, local minimum, saddle point, or none of the above, V+(o(o) ~(O/ 0 ') i) (0,0)
a) Local maximum (^) t.lo,') Fo/",</- f;./,A" ~2){I(') - 0 0 -32 LD b) Local minimum <:!l: Sad~lle poi~ d) None of the above
ii) (0,2) ;l (v{~)^ '=^ <~^ -f'S^ 'a'L")^ =c.q,^ 0)
~lmaxiI~ b) Local minimum {!.Ad^ ~,A1)^ Ā£./",2)'^ '"^ f:-7{12^ ,'/-+^ /6)^ =^ {,4^ > c) Saddle point d) None of the above (^) **{>ex_** { Ā°1 ))~ - L <:. 0
iii) (0,2/ v3)
a) Local maximum b) Local minimum (^) o ,,<P(OI '~)_ (O - 4-{~) + Ib{3) -Ā£
~
c) Saddle point (^) iJ3 I 3.[j T
~ne7The~
