University of Pittsburgh MATH 0240 (Calculus III) Midterm 1, Fall 2014
Transcript of University of Pittsburgh MATH 0240 (Calculus III) Midterm 1, Fall 2014
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Department of Mathematics
University of Pittsburgh
MATH 0240 (Calculus III)
Midterm 1, Fall 2014
Instructor: Kiumars Kaveh
Last Name: Student Number:
First Name:
TIME ALLOWED: 50 MINUTES. TOTAL MARKS: 100NO AIDS ALLOWED.WRITE SOLUTIONS ON THE SPACE PROVIDED.PLEASE READ THROUGH THE ENTIRE TEST BEFORE STARTINGAND TAKE NOTEOF HOWMANY POINTS EACHQUESTION ISWORTH.FOR FULLMARKYOUMUST PRESENT YOUR SOLUTION CLEARLY.
Question Mark
1 /20
2 /20
3 /20
4 /20
5 /20
6 /1
TOTAL /100 + 1
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1.[20 points] Find the point on the line r(t) = (t, 1− t, 2+ t) which has min-imum distance to the point P = (1, 0, 2). Calculate this minimum distance.
(Hint: if Q is a point on the line, look at the vector !PQ.)
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2. [20 points] Show that the function
f(x, y) =
!x2y
x6+y2(x, y) "= (0, 0)
0 (x, y) = (0, 0)
is not continuous at (0, 0). (Hint: Look at two different curves passingthrough the origin.)
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3. Consider the curve:
r(t) = sin(t)i+ sin(t)j+√2 cos(t)k, 0 ≤ t ≤ 2π
(a) [15 points] Find the unit tangent and unit normal vectors T(t) andN(t) to r(t).
(b) [5 points] Find the curvature κ at any point r(t).
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4. [20 points] Find local maximum, local minimum and saddle point(s) ofthe function
f(x, y) = x4 + y3 − 3y + 4x+ 5.
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5.
(a) [10 points] Find the derivative of the function f(x, y, z) = (y ln(z))+x2
in the direction of the unit vector u = 1√3(1,−1, 1) at the point (2, 1, 1).
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(b) [10 points] Consider the quadric surface:
(x− 1)2 + 3y2 − z2 = 1.
By looking at the cross sections/traces (along xy, or yz or xz coordinateplanes) determine what type it is (i.e. an ellipsoid, hyperboloid, ellipticparaboloid, hyperbolic paraboloid or a cone). Next find the equationof tangent plane to this surface at the point (1,−1,
√2).
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6. [1 point] Draw a cartoon showing yourself writing this test!
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