- 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)
- Compute the following integrals:
Z |
1
(a)lnxdx
0 x2 + 1
(b) x2 − 1 dx
(5+10)
- 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
- Solve the initial value problem with y(0) = 2.
- Does this equation have equilibrium points? Are they stable or unstable?
(10+5)
2
- 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
02A = [email protected] | 01−11 | 202 1 | 411CCCCA ,3 3 | 0−21 b = [email protected]−−023CCCCA . |
(10) 10. Let L: R4 R4 be the shift mapping” de ned as follows:
→
0x11 0 0 1
BBx C
LBBBBBxxx4325CCCACCC = [email protected] .
@
- Show that L is a linear transformation on R4.
- Write out the matrix S which represents L with respect to the standard basis.
- Find a basis for RangeS and KerS.
- State the rank-nullity theorem” and verify explicitly that the result obtainedin part (c) matches the statement of the theorem.
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