PRACTICE PROBLEMS FOR FINAL EXAM WITH SOLUTIONS. MATH 121A ...

PRACTICE PROBLEMS FOR FINAL EXAM WITH SOLUTIONS. MATH 121A

1. Find the maximum of the function f (x, y) = x2 - y2 at the region x2 + 2y2 1.

Solution. First, we look for critical points inside

f x

=

f y

=

0,

x

=

y

=

0,

f

(0, 0)

=

0.

Now we look for critical points on the boundary x2 + 2y2 = 1. One can use Lagrange multiplies, but is easier to use x2 = 1-2y2, f (y) = 1-3y2. One can see immediately,

that f (y) 1 hence the maximum is 1 and that happens when y = 0, x = ?1.

2. Find the Maclaurin series for the function: f (x) =

x2 0

e-t2 dt.

Solution.

e-t2

=

1

-

t2

+

t4 2!

-

t6 3!

+

t8 4!

-

...

e-t2 dt

=

t

-

t3 3

+

t5 5 ? 2!

-

t7 7 ? 3!

+

t9 9 ? 4!

-

.

.

.

Now use substitution t = x2

f

(x)

=

x2

-

x6 3

+

x10 5 ? 2!

-

x14 7 ? 3!

+

x18 9 ? 4!

-

...

3. Find all singular points for the function

z-1

z3 + 1

and evaluate the residue at each singular point including infinity. Solution. The singular points are

-1,

i

e3,

e-

i 3

.

They are all simple poles. Thus, the residue can be evaluated by the formula

Res (z0)

=

z-1 3z2

z=z0

,

Res (-1) =

-32 ,

Use that

Res

i

e3

=

e i 3 3e

-

2i 3

1

,

Res

e-

i 3

=

e-

i 3

-1

3e-

2i 3

.

e?

i 3

=

1 2

?

3 2

i,

e?

i 3

-

1

=

-

1 2

?

3 2

i

=

e , ?

2i 3

1

2

PRACTICE PROBLEMS FOR FINAL EXAM WITH SOLUTIONS. MATH 121A

to obtain Finally,

i

Res e 3

= Res

e-

i 3

= 1.

3

i

Res () = - Res (1) - Res e 3

- Res

e-

i 3

= 0.

4. Evaluate the integral

dx 0 (x2 + 1)2 .

Solution. Use large semi-circle contour C. The integral equals

1 2

dz

C (z2 + 1)2 = i Res (i) = 4 ,

Res (i)

=

d dz

(z

1 +

i)2 z=i

=

-2

1 (2i)3

=

1 4i

.

5. Evaluate the integral

sin z dz.

|z+1|=2 z (z + 1)

Solution. The integral is 2i (sum of residues inside the contour). There are two

singular points z = 0 and z = -1, they are both inside the contour |z + 1| = 2. But

z = 0 is a removable singularity,

Res (-1) =

sin -1 -1

=

sin

1,

sin z dz

|z+1|=2 z (z + 1)

= 2i sin

1.

6. Find the image of the first quadrant

Re z > 0, Im z > 0

under the conformal mapping w = ez. Solution. If z = x + iy, then w = ex (cos y + i sin y), or |w| = ex, the angle of w

is y. Therefore the image of x > 0, y > 0 is |w| > 1, the complex plane with removed unit disk.

7. Find the inverse Laplace transform of

1 (p + 2) (p2 + 4) .

Solution. The easiest way in this case is to use simple fractions

(p

+

1 2) (p2

+

4)

=

1 8 (p +

2)

-

p 8 (p2 + 4)

+

1 4 (p2 + 4) ,

and then the table

f

(t)

=

e-2t 8

-

cos 2t 8

+

sin 2t 8

.

PRACTICE PROBLEMS FOR FINAL EXAM WITH SOLUTIONS.

MATH 121A

3

8. Using Laplace transform solve the differential equation

y + 4y + y = e-t, y0 = 0, y0 = 1.

Solution.

p2Y

- 1 + 4pY

+Y

=

1 p+1

p2 + 4p + 1

Y

=

p p

+ +

2 1

Y

=

(p

+

p+2 1) (p2 + 4p

+ 1)

=

(p

+ 1)

p +2 p+2- 3

. p+2+ 3

To find y use the Bromwich integral formula for F (z) ezt.

Res (-1) =

-e-t 2

,

Res

3-2

=

e(3-2)t , 3-1 2

Res - 3 - 2 =

e-(3+2)t ,

- 3-1 2

y

=

-e-t 2

+

e(3-2)t

3

4

+

1

e-(3+2)t

- 3+1

+

4

.

9. Find the Fourier transform of the function

f (x) =

ex 0,

,|x|x|>|11.

Solution.

g

()

=

1 2

-

f

(x)

e-ixdx

=

1 2

1

exe-ixdx =

-1

1 2

1 -1

ex(1-i)dx

=

1 2 (1 - i)

e(1-i) - e(i-1)

=

sin ( + i) ( + i)

.

10. Let f (x) be as in Problem 9. Find the convolution of f (x) and ex.

Solution.

1

1

et- e d = et d = 2et.

-1

-1

11. Let f (x) have period 2

f (x) =

3x+1, 0 ................
................

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