Description
4. Use implicit di erentiation to nd an equation for the tangent line to the graph of
sin(x + y) = y
2 cos(x) at point (0, 0). (5)
5. Compute the following integrals:
(a) Z 1
0
ln x dx
(b) Z
x
2 + 1
x
2 − 1
dx
(5+10)
6. Is the following improper integral convergent? There is no need to compute the
answer, but you should give detailed reasoning.
Z∞
0
ln x + e
−x
1 + x
2
dx
(10)
7. Consider the di erential equation
dy
dt
= t
3 y
3
.
(a) Solve the initial value problem with y(0) = 2.
(b) Does this equation have equilibrium points? Are they stable or unstable?
(10+5)
2
8. Show that, for u, v 2 R
3
,
kuk
2
kvk
2 = (u v)
2 + ku vk
2
.
(5)
9. Find the general solution to the system of linear equations Ax = b with
A =
0
BBBB@
2 0 2 4
0 1 0 1
2 −1 2 3
1 1 1 3
1
CCCCA
, b =
0
BBBB@
−2
−2
0
−3
1
CCCCA
.
(10)
10. Let L: R
4 → R
4 be the \shift mapping” de ned as follows:
L
0
BBBBBBB@
x1
x2
x3
x4
x5
1
CCCCCCCA
=
0
BBBBBBB@
0
x1
x2
x3
x4
1
CCCCCCCA
.
(a) Show that L is a linear transformation on R
4
.
(b) Write out the matrix S which represents L with respect to the standard basis.
(c) Find a basis for Range S and Ker S.
(d) State the \rank-nullity theorem” and verify explicitly that the result obtained
in part (c) matches the statement of the theorem.
(5+5+5+5)
3
