[Solved] Calculus-Final

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) x2 − 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

• Solve the initial value problem with y(0) = 2.
• 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

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.