4. Use implicit di erentiation to nd an equation for the tangent line to the graph of sin(x + y) = y2 cos(x) at point (0,0). (5)
5. Compute the following integrals:
Z
1
(a)lnxdx
0 x2 + 1
(b) x 2 − 1 dx
(5+10)
6. Is the following improper integral convergent? There is no need to compute theanswer, but you should give detailed reasoning. ∞ dx lnx + e−x 1 + x2
0
(10) 7. Consider the di erential equation
dy 3 3
= t y .
dt
(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 R3,
kuk2 kvk2 = (u v)2 + ku vk2 .
(5) 9. Find the general solution to the system of linear equations Ax = b with
02
A = BBBB@102
0
1
−1
1
2
0
2 1
41
1CCCCA ,
3 3
0−21 b = BBBB@−−023CCCCA .
(10) 10. Let L: R4 R4 be the \shift mapping" de ned as follows:
→
0x11 0 0 1
BBx C
LBBBBBxxx4325CCCACCC = BBBB@BBBxxxx3412CCCCACCC .
@
(a) Show that L is a linear transformation on R4.
(b) Write out the matrix S which represents L with respect to the standard basis.
(c) Find a basis for RangeS and KerS.
(d) State the \rank-nullity theorem" and verify explicitly that the result obtainedin part (c) matches the statement of the theorem.
