Description
1. (10 pts) Find a recursive formula for a sequence of points (x0, y0),
(x1, y1), . . . , (xn, yn), whose limit (x∞, y∞), if it exists, is a point of
intersection of the curves
x2 + y2 = 1
x2
(x + 1) = y2
.
2. (10 pts) Let f : R2 −→ R be the function f(x, y) = cos(xy)+x3+y2.
Find a recursive formula for a sequence of points (x0, y0), (x1, y1), . . . ,
(xn, yn), whose limit (x∞, y∞), if it exists, is a critical point of the
function f, that is, a point where Df(x, y) = (0, 0).
3. (10 pts) Suppose that f : R2 −→ R2 is differentiable at (−2, 1) with
derivative
� �
Df(−2, 1) = −2 3 , −1 1
and that f(−2, 1) = (1, 3). Let g : R2 −→ R be the function g(y1, y2) = 2 2 y1 − y2.
(a) Show that the composite function g ◦ f : R2 −→ R is differentiable
at (−2, 1).
(b) Find the derivative of g f at (−2, 1).
4. (10 pts) Let F : R3 −→ R2
◦
be a C1 function. Suppose that F(4, −1, 2) =
(0, 0) and that
DF(4, −1, 2) = 1
0
−
1
1
−
4
1 .
(a) Show that there is an open subset U ⊂ R containing 4 and a C1
function f : U −→ R2 such that f(4) = (−1, 2) and such that
F(x, f(x)) = 0,
for every x ∈ U.
(b) Find the derivative Df(4).
5. (10 pts) (a) Show that, in a neighbourhood of (2, −1, 1), the subset
S = { (x, y, z) ∈ R3 | x3
y3 + y3
z3 + z3
x3 = −1 },
is the graph of a C1 function f : R2 −→ R.
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(b) Determine
∂f ∂f
∂x
(2, −1) and ∂y
(2, −1).
6. (10 pts) (2.5.9).
7. (10 pts) (2.5.11).
8. (10 pts) (2.5.22).
9. (10 pts) (2.6.21)
10. (10 pts) (2.6.24).
Just for fun: For any � > 0, find an example of a smooth function
f : R −→ R such that for f(x) = 0, if x ≤ 0 or x ≥ 1 and f(x) = 1 if
� < x < 1 − �. (Hint: consider the function g(x) = e−1/x2
.)
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