MAT 272Test 2 ReviewPLOPPWMBONT, NMTBAI**

Chapter 12–Vectors

1.Magnitude and direction2.Components

3.Dot Product

and are orthogonal if

4.Cross Product

The magnitude gives the area of the parallelogram bounded by and

Chapter 13–Derivatives of Functions of Many Variables

1.Partial Derivatives –rate of change of f in the x direction at the point

–rate of change of f in the y direction at the point

2.Local Linearity (Tangent Planes)

for near

3.Directional Derivative of f in the direction of at the point

4.Gradient Vector

a. Direction–that in which f is increasing at the greatest rate.

b. Magnitude–the rate of change in that direction

5.The differential of a function

6.The Chain Rule

a. If , ,

b. If , ,

**Possible List of Probable Problems Which Might Be On Next Test(Not Meant to be All Inclusive)

7.Second degree Taylor expansion for near

VECTOR PROBLEMS

1.Find the angle between and

2.For a unit cube, find the angle between a diagonal of the cube and the diagonal of an adjacent face.

3.A ship is moving N at 10 mph. A maggot runs SE across the deck at 5 mph. In what direction and how fast is the maggot moving relative to the earth’s surface?

4.Duke and Spike are pulling a wagon along a straight line. Duke is pulling with a force of 8 lbs. on a horizontal rope that makes an angle of 30 degrees with the direction of motion of the wagon. Spike is pulling on a horizontal rope that makes an angle of 45 degrees with the direction of motion. What is the magnitude of the force applied by Spike?

5.Using vectors, find the equation of the plane through the points , , and .

6.Find the area of the triangle defined by the three points in problem 5 and find the measures of the angles of the triangles.

Possible Answers: 1. 2. 3. E of N, 7.4 mph 4. 5.66 lbs.

5.

6.

derivative problems

1.Find an equation for the tangent plane to at the point .

2.Find the derivative of at in the direction toward .

3.For the function , find the direction of the steepest grade at .

4.If the temperature is given by and you are located at and want to get cool as soon as possible, in which direction should you head?

5. Find the rate of change of this function at the point in the direction

toward the point

6.A cylindrical piece of steel is initially 8 inches long and has a diameter of 8 inches. During heat treating the length and diameter each increase by 0.1 in. Use differentials to find the approximate increase in volume.

7.Find the rate of change with respect to time of the hypotenuse of a variable right triangle at the instant when the two sides are 7 and 10 inches if the first side is increasing at the rate of 6 in./min. and the second side is increasing at the rate of 4 in./min.

8. Find the local linearization and the second order Taylor polynomial for

near the point .

  1. Demonstrate the use of the definition of the directional derivative to find in the direction toward for the function .

10. The contour diagram below shows the contours of some function, . Use it to estimate the

following.

a) in the direction toward the point b) c)

d) if

Possible Answers

1. 2. 3. 4. 5. –1.28 6. 15.08 cu. in.

7. 6.72 in./min. 8.

9. –4

10. a) b) 0c) d)