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Name:
MATH 215
MIDTERM I
This Exam contains 5 problems. The problems are worth 12 points each. Each part of a problem counts equally. On problems 3, 4 and 5 you can get partial credit. Hence, explain yourself carefully on these problems.
NO CALCULATOR. 1 TWO-SIDED 3in. BY 5in. NOTECARD OK. CHECK YOUR SECTION IN THE TABLE
Sec. Time Exam rm. Professor GSI ME
20 9-10 1210 Chem Angela KUBENA Harlan KADISH
30 10-11 1400 and 1800 Chem Harry D’SOUZA Robin LASSONDE
40 11-12 1400 and 1800 Chem Harry D’SOUZA Giwan KIM
50 12-1 1400 and 1800 Chem Harry D’SOUZA Holly CHUNG
60 1-2 170 Denn Zuoqin WANG Timothy FERGUSON
70 2-3 182 Denn Zuoqin WANG Crystal ZEAGER
269 Denn Extended time 5-11pm
1
Problem 1. TRUE/FALSE QUESTIONS. NO PARTIAL CREDIT. CIRCLE TRUE OR FALSE. IF YOU THINK A STATEMENT DOESN’T MAKE SENSE, CIRCLE FALSE.
(a)
∂ ∂x
∫ (^) x 2
x^3
y t
dt = −
y x
if x > 0 TRUE / FALSE
ANSWER: TRUE
(b) (^) ∫ dx x^2 + x^5
x
4 x^2
+ C TRUE / FALSE
ANSWER: FALSE
(c) The formula ~n · (~r − ~r 0 ) =< 0 , 0 , 0 > determines a plane. TRUE / FALSE
ANSWER: FALSE
(d) The expression ~r(t) =< sin(et), cos(et) >, t ∈ [0, ln(1 + 2π)] parametrizes a circle. TRUE / FALSE
ANSWER: TRUE
(e) The cross product of ~a and ~b equals the area of the parallelogram determined by ~a and ~b.
TRUE / FALSE
ANSWER: FALSE
(f) The gradient of a function is perpendicular to the level sets of the function. TRUE / FALSE
ANSWER: TRUE
WORKSPACE:
Problem 3. YOU CAN EARN 0, 2 OR 4 POINTS ON EACH PART. A SCORE OF 2 POINTS WILL BE AWARDED ONLY IN THE CASE OF A SMALL MISTAKE.
(a) Find the domain of the vector function ~r(t) =< (^1) t , 2 + t, ln(1 − t^2 ) >.
(b) Evaluate the integral (^) ∫ 2
1
t
, tet, tet
2
dt
(c) Find fxy if f (x, y) = xyex
(^2) y .
ANSWERS: a: (− 1 , 0) ∪ (0, 1)
b: < ln 2, e^2 , e^4 / 2 − e/ 2 >
c: ex^2 y[1 + 2x^2 + x^2 y + 2x^4 y]
WORKSPACE:
Problem 4. IF YOUR ANSWER IS CORRECT, YOU GET FULL CREDIT. IF ANSWER IS INCORRECT, YOU CAN GET PARTIAL CREDIT IF YOU EXPLAIN YOURSELF CAREFULLY.
(a) Find an equation for the tangent plane to z = yex
(^2) y at (1, 2 , 2 e^2 ).
(b) Find the gradient of f (x, y, z) = cos( πxyz ) at (1, 1 , 2).
(c) Find the rate of change of f (x, y) = yx in the direction of a unit vector parallel to ~u =< 1 , − 1 > at (2, 2).
(d) If z = f (x, y) = arctan xy , x = s + t, and y = t^2 , find ∂z∂t at (s, t) = (1, −1).
ANSWERS: a: z = 2e^2 + 4e^2 (x − 1) + 3e^2 (y − 1)
b: < −π/ 2 , −π/ 2 , π/ 4 >
c: − 1 /
d: 1
WORKSPACE:
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
Problem Points Score
TOTAL 60
