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FINAL EXAM
Math 51, Spring 2001.
You have 3 hours.
No notes, no books. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck!
Name
ID number
- (/50 points)
- (/50 points)
- (/50 points)
- (/50 points)
- (/50 points)
Bonus (/20 points)
Total (/250 points)
“On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.”
Signature:
Circle your TA’s name:
Kuan Ju Liu (2 and 6)
Robert Sussland (3 and 7)
Hunter Tart (4 and 8)
Alex Meadows (10)
Dana Rowland (11)
Circle your section meeting time:
11:00am 1:15pm 7pm
- Let the function f :
R^2 − {
→ R^2 have components f 1 and f 2 as described by
f
x y
f 1 f 2
(x^2 )
y
xy^2
(a) Note that the function f is not defined at the origin; this is because the component f 1 is not defined there.
Is this discontinuity in f 1 removable? Justify your answer.
(b) Find the Jacobian matrix for the function f at the point
x y
- Let the functions f and g be given by
f (t) =
f 1 (t) f 2 (t) f 3 (t)
(^) and g
x 1 x 2 x 3
(^) = x^21 x^32 x 3
(a) Write down an equation for ∇g.
(b) Suppose that f 1 (t) = sin t, f 2 (t) = cos t, f 3 (t) = t^2 , and consider the composition g ◦ f. Use the chain rule to find an expression (in terms of t) for
dg dt
(c) Suppose instead that you do not have formulas for the components of f ; instead, you are given only that
f (0) =
(^) , and dg dt
Find the value of
df 3 dt
(b) Use the result of part (a) to show that if the vectors
{ ∂f ∂x 1
∂f ∂xn
are dependent at a point −→a ∈ Rn, then we can draw the same conclusion – that there must exist some non-zero vector −→v with
Df,−→a (−→v ) =
(Hint: Recall the relationship between the dimensions of the row space and the column space of a matrix, and then use the result of part (a).)
- (a) Consider the function
f
x y
x^2 − y^2 x^2 + y^2
f 1 f 2
Find and identify all critical points of the function h = ‖f ‖^2.
- Consider the function
f
x y z
(^) = x + y + z
(a) Find the point which achieves the absolute minimum value of f on the surface x^2 + y^2 = z.
(b) Find the points which achieve the absolute minimum and maximum values of the function f on the curve which is the intersection of the surfaces x^2 + y^2 = z and y + z = 1.
