Test 1: Multivariable Calculus

Math 212Ron Buckmire

Friday October 14 20059:30pm-lO:30am

Name:

Directions: Read all problems first before answering any of them.Questions 2-4 are all related, but different. There are 7 pages inthis test. This is a one hour, open-notes, open book, test. Nocalculators. You must show all relevant work to support youranswers. Use complete English sentences and CLEARLY indicateyour final answers to be graded from your "scratch work."

60

40

20

-20

-40

-603

-1 -1-2 -2

-3 -3

I No. [ Score I Maximum I

I Total [100

1 302 203 204 30

BONUS 5, ,

1. Equation of Planes, Vector Operations. You are given the following three points iJthe plane:

A=(1,2,3) B = (2,2,5) c = (-1,3,4)

(a) (6 points.) Find the vector v which starts at A and points to B, and the vector w whicJstarts at A and points to C.

(b) (4 points.) Find V. w. Explain in complete sentences what this tells you about the anglbetween the two vectors and why.

" al\ ol c$ Clre Or--th6j~e::c 1/ I"~e ,b e..J-v.rt'tP -t\1tr" t'S qo ~

1"ht' QntjJe

,.-)::: (21 2..J5) - ( I; 2) $) (J J 0,'4

V-; -A AV -= r + 2

Q :: (- A. -:. (--lj >, '1} - ( 'J'2J3) = (-'-; I JI)...... ""VJ -; - 21 +) +1

(c) (10 points.) Find iJ x w. Explain in complete sentences what this tells you about thearea of the triangle ABC.

/\

1 J t/

-})

0 2

}

-jI

) 2/

+- t( ! O JI 0 2 - 'I -2 I 2 I

..2. I J

~ ~vxw::

~( )

1\ 1\

: J - '2- -J ( I - (-1))+ k. II\. A"

:; ..;21 "';J -r k.

:. (-1.)-5,1)

Are fA of b AB( :: 1) if )([;j \ :: 1.J4 + 2S +-i- -2. 2(d) (10 points.) Find the equation of the plane that contains all three points A, B, and C.

We {(f\ 0 (!J (- L I-?; i) i.> I1~M~ f -t-b the/p { 0..£\ C!. C6'r-. -\-t:dl\j1\-,

- '2 ( ;( ~ I) - 5 (y - 7.) tJ{ l-~) = 0- '2.)l-+ 1. - 5 Y+10 -+"C - 3 ~ 0

-1-X - 5y -+ 7;. -t '1 =-0

:: J .J3o-2

2. Level Sets, Vertical Slices. For the rest of the exam we will be considering the surfacez = f (x, y) = X3 - y3.(a) (10 points.) Identify which of the following graphs represents the level sets f(x, y) = kor different vertical slices (x = k or y = k) of the function for k = -2, -1,0,1,2.

Figure 1, Figure 2

! I/'Tl- ,. !. .1 .1 .~~/ i ,I fjJ "/r-j ::

.,-.,

Figure 3

\\\ ~\ \ \ I\\~\\ ~\~ \

\ ~ \~ \\

~ \\\

.\\\\

CLEARLY LABEL WHICH GRAPH REPRESENTS HOLDING WHICH.';

VARIABLE CONSTANT AND FULLY EXPLAIN YOUR CHOICE BELOW.

F"i5",.e1 i> I t~ 5t,f s ~:. I(. . S v-tr-M/-y ::. 0

{yx.. : 0

A-I- -the Dr-i~;1\ 1hQ. r'"k of Ch4i1je of 1 uJi*' Xt\\? Cir '1- -: 0 j 5 tloe sa ""e IH ()1\e (V1ov-e5

vel J-iU~l)))o f{tifJ :: D. A 5\1""("" q.Y)Ui' j:.(j1:

~ 1!3\-;.c>.1:>'1-~ ~y I

Fl~ 1:

4

3. Tangent Plane Approximation.

(a) (10 points.) Find the equatio f ththe point (x, y) = (1,2). no e tangent plane to the surface f(x, y) = x3 - y3 at

t (Vi-):: l3- 'l:) '" - 1 {.. '" 3)( 'I. +'y "- - :l Y~-f (x,~) F::" T(x,y) -;: f ( l, '2..) + f)C. (1/2.)( It -t) + Fy (I, 'L){Y-l)

T(>(,y):: - 1 + 3(j{ - I) - IL. (y- '2.)

(b) (10 points.) Use this tangent plane approximation of the surface at this point to esti-mate the value of (0.9)3 - (1.99)3. [Note: I know the exact value is -7.151599. I'm lookingfor an estimate of this using the tangent plane approximation.]

T(o.'I,I.

4. Gradient, Directional Derivative.

(a) (15 points.) The gradient of a function f(x, y) evaluated at a point (xo, Yo) is a vectorpointing in the direction of the maximal rate of change of this function f (x, y) at the point(xo, Yo).

In what direction would you go from the point (1,2) to follow the maximal rate of changeon f(x, y) = X3 - y3? What is the magnitude of this maximal rate of change?

V-F ()(t 'f) : (3 X~ -> y3).-J r ( )

- (3 - l2)\l1 1/2.. - ,/1.F ~Ol.\ ~o 11\ -the &il'ec.""-"" 31'- 1'2.) ihat -\$"/Ae.. tJ,ir-e di ot\. 0 f fV\M /M"( r 4k of cA...t\j' d [: ,

~ij {( 1,2-) r :: ~ 1- 12 :: vq f 144- ~ ViE] --(b) (10 points.) What would the rate of change have been if you went in the direction

~ 47 3"7 7W = 5Z- 5J.

Q:E:;

ddVj.c;j -: ( 3) -(2J - ( 1: ) ~] )

.5 :5

:~... 4~-

5

J'2.. +3~- -:5 5

(c) (5 points.) What is a vector direction you can move in if you want the rate of change off(x, y) = X3 - y3 at (1,2) to be zero?

---rhe dire cHb(\ -Ib""()~ if\. l.J.>olll,:9.!Je> u i/t,. -f'r,o.f 11t=(1/z.) . V :.. 0 r..AeJc v=(v'j~)

.3 V, - J '2..v2..-::0

> -v . ( /1.J 3 )

6

BONUS QUESTION. Continuity, Set Theory. (5 points.)

Consider 9 : ]R2 -t ]Rwhere g(x, y) = f(x, y) = X3 - y3. Describe the domain of the functionx-y x-y

g(x, y). What kind of set (open, closed, et cetera) is it? Is the function g(x, y) continuous onthis domain? EXPLAIN YOUR ANSWER THOROUGHLY, EXTRA CREDITPOINTS ARE HARD TO GET.

~e J 0'" a j" i s(x,,):fK2 \ { ., =.() . ArU (JlljJ P ~ir.si to- R L. e't.