Math 224 Fall 00

Final Exam Dec 12

Name_________________________

 

Good luck!  Please show all work since this will be graded with partial credit.

 

Problem

Points

Score

Prob

Pts

Score

 

 

1

30

 

9

15

 

Test 1

 

2

10

 

10

10

 

Test 2

 

3

10

 

11

10

 

Test 3

 

4

10

 

12

15

 

Test 4

 

5

15

 

13

15

 

HW/Q

 

6

10

 

14

10

 

Final

 

7

10

 

15

10

 

Total

 

8

10

 

16

10

 

%

 

 

 

 

Total

200

 

Grade

 

 

Note: Homework scores were only used if they helped the final grade.  They were not used if they lowered the grade.

 

 

Cylindrical Co-ord’s                                                                           Polar Co-ords:

                                      

Spherical Co-ords’s

 

                 

 

                                   

 

1.(3pts each) Circle T for True, F for False.

 

a)  T     F          Two vectors are orthogonal if their dot product equals zero.

 

b)                     The direction of a vector is defined to be a _______ vector

                        that points in the same direction as the vector.   

 

c)  T     F         

 

d)  T    F          A region is open if it consists of all interior points.

 

e)                     A point (x, y) is ________________ point of a region R if it is the center

                        of a disk that lies entirely in R.

 

f)                      The gradient of a differentiable function of two variables is always ___________          

                        to the function’s level curves.

 

g)  T     F          The absolute maximum of a bounded function may not occur at a boundary point.

 

h)  T     F          If the density of an object is constant, the center of mass is called the centroid

                        of the object.

 

i)  T      F          The area of a closed and bounded region R in the polar coordinate plane is

                       

 

j)                      A function has a saddle point when the ________________ is evaluated and

                        found to be negative.    

 

2.(10pts) Find the parametric equations for the line that passes thru the point (0, 1, 3) and

is parallel to the vector v = 2i – 3j + 2k.

 

 

 

 

 

 

3. (10pts) For the vectors A = 2i – 3j + 2k, and B = i – 1j + 2k find their cross product.

 

 

 

 

 

 

 

 

 

4.(10pts)  Sketch the figure:

 

 

 

 

 

 

 

 

 

 

5.(15pts) Show that the dot product of the position function r = sin(t)i + cos(t)j + 8k and the corresponding acceleration  has a constant value, and find this value.

 

 

 

 

 

 

 

 

 

 

6.(10pts) Find and describe the level curve for the function  at the point (2, 1).

 

 

 

 

 

 

 

 

 

 

7.(10pts) Given write the chain rule formula for

finding

 

 

 

 

 

 

 

8.(10pts) Solve the IVP, with initial conditions, when t = 0,

 

 

 

 

 

 

 

 

 

 

9. (15pts) To land on the green you must first clear a pine tree with your approach shot.  If you

hit the golf ball at a speed of 90 ft/sec and the ball leaves the ground at a 60 degree angle, will

it clear the top of the 60 foot tree that is 40 yards away?

 

 

 

 

 

 

 

 

 

 

 

10.(10pts) Find the limit: 

 

 

 

 

 

 

 

 

11.(10pts) The three dimensional Laplace equation is  appears in applied math.

Show that  satisfies a Laplace equation.

 

 

 

 

 

 

 

 

 

 

 

 

12. (15pts)  The dimensions L, H, W of a box are changing with respect to time.  At an instant in time

L = 1 m, H = 2 m, and W = 3 m, dL/dt = 1 m/s, dH/dt = 1 m/s, and dW/dt = -3m/s.  At what rate is the

volume of the box changing? (V = LHW).

 

 

 

 

 

 

 

 

 

 

 

13. (15pts)   Find the direction in which the function  is increasing most

rapidly at the point (-1, 1).

 

 

 

 

 

 

 

 

 

14.(10pts) Find all the local max, mins, and saddle points of the function, f(x, y), given that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15.(10pts) Integrate the function  over the disk

 

 

 

 

 

 

 

 

 

 

 

 

 

 

16.(10pts) Set up the double integral to find the volume of the region that lies under the paraboloid

, and above the triangle enclosed by the lines y = x, x = 0, and x + y = 2, in the xy-plane.