MULTIVARIABLECALCULUSEXAM1FALL2012Name…：多变..

MULTIV ARIABLE CALCULUS
EXAM1
F ALL2012
Name:
Honor Code Statement:
Directions:Complete all problems.Justify all answers/solutions.Calculators are not permitted.Best of luck.
(1)Find a unit vector that is perpendicular to both2i+j−3k and i+k.
(2)If a·b=a·c and a=0,does it follow that b=c?Explain.
Date:October11,2012.
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(3)Give an equation for the plane containing the point(9,5,−1)and perpen-
dicular to i−2k.
Now give the set of parametric equations for this plane.
MULTIV ARIABLE CALCULUS EXAM1F ALL20123 (4)State the Cauchy-Schwarz Inequality.
(5)Is the following set closed?Give reason for your answer.X={(x,y,z)∈
R3|0≤x≤1,0≤y≤1,0≤z<1}.
(6)Is the function f(x,y)=xy−7x8y2+cos(x)differentiable at every point
in its domain?Why,or why not?
(7)Given that f(x,y,z)=e ax sin(y)+x2sin(y)+x10y3−e bx cos(z).Note that
f(x,y,z)is class C∞(i.e it is smooth).Find f zxx.
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(8)Determine several(say3with c≥0and3with c<0)level curves of the
given function f(and make sure to indicate the height c of each curve).
Use this information to describe the graph of f,be as specific as you can.
(A sketch would be appropriate,but not required.)
f(x,y)=xy
MULTIV ARIABLE CALCULUS EXAM1F ALL20125 (9)Show that the following limit does not exist.
lim (x,y)→(0,0)(x+y)2 x2+y2
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(10)Find a“good”linear approximation to the function f(x,y)=e x+2y at the
point(0,0).
Use this approximation to estimate the value of the function at(0.1,0.1)
MULTIV ARIABLE CALCULUS EXAM1F ALL20127 (11)Find the gradient f(a),where f(x,y)=e xy+ln(x−y)and a=(2,1).
(12)Find the matrix of partial derivatives of the function f(s,t)=(st,t sin(s),se t).
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(13)A rectangular stick of butter(that is a right parallelepiped with square
base)is placed in a microwave to melt.When the butter’s length is6 inches and its square cross-section measures1.5inches on a side,its length is decreasing at a rate of0.25inches per minute and its cross-sectional edge is decreasing at a rate of0.125inches per minute.How fast is the butter melting(i.e.at what rate is the solid volume of butter turning to liquid) at that instant?。