math 55 - exercises for exam1

7/29/2019 Math 55 - Exercises for Exam1

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Math 55 Review Questions

1. Evaluate the directional derivative of f(x,y,z) =

xyz at (3, 2, 6) in

the direction ofv = 1,2, 2.

2. Determine the maximum rate of change of f(x, y) = sin xy at (1, )

and the direction in which it occurs.

3. Give an equation of the tangent plane to x + y + z = 3exyz at the point

(0, 1, 2).

4. Find the relative maximum and minimum values and saddle point(s)

of f(x, y) = xy 2x 2y x2 y2.

5. Find the absolute maximum and minimum values off(x, y) = x+yxyon the closed triangular region with vertices (0, 0), (0, 2), and (4, 0).

6. Use Lagrange multipliers to find the maximum and minimum values of

f(x,y,z) = x2 + y2 + z2 subject to the constraint x + y + z = 12.

7. Determine the dimensions of the rectangular box with largest volume

if the total surface area is given as 96 cm3.

8. Determine the volume of the solid that lies under the hyperbolic paraboloid

z = 3y2 x2 + 2 and above the rectangle R = [1, 1] [1, 2].

9. Evaluate

4

0 2

x

1

y3 + 1dy dx by reversing the order of integration.

10. Evaluate

R

y2

x2 + y2dA where R is the region that lies between the

circles centered at the origin of radii 1 and 2.

11. Set-up an iterated integral equal to the area of the region bounded by

x = y2 1 and x + y = 1.


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12. A lamina occupies the part of the disk x2+y2 1 in the first quadrant.Find its mass if the density at any point is twice its distance from the

x-axis.

13. Find the area of the surface z = 1+3x +2y2 that lies above the triangle

with vertices (0, 0), (0, 1), and (2, 1).

14. Find parametric equations for the surface obtained by rotating the part

of the curve y = ex where x 0 about the y-axis.

15. Find an equation of the tangent plane to the surface area of the helicoid

r(u, v) =

u cos v, u sin v, v

at the point where u = 12

and v = 3

.

16. Set-up an iterated integral equal to the surface area of the helicoid (in

item 15) for 0 u 1 and 0 v .


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Answers

1.

1

2. change of

1 + 2 in the direction of ,1

3. 5x + y + z 3 = 0

4. relative max at (2,2)

5. absolute max at (4, 0), absolute min at (0, 0)

6. no max, absolute min at (4,4,4)

7. 4 cm 4 cm 4cm

8.52

3

9.ln 9

3

10.3

2

11.

1

2

1y

y21dxdy

12.2

3

13.36

3

2 10 3224

14. r(u, v) =

ln u cos v,u, ln u sin v

, 1

u, 0

v

2

15.

3

2

x 1

4

1

2

y

3

4

+

1

2

z

3

= 0

16.

0

1

0

1 + u2 dudv

math 55 - exercises for exam1

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