MAT 272 Test 1 Review Chapter 11

MAT 272 Some Additional Practice Problems for Final Review

Topics

1. Functions of 2 variables, Graphs of functions of 2 variables 2. Contour diagrams 3. Critical points 4. Derivatives 5. Optimization

Sample Problems

1. f x , y  x2  y  2 . Sketch f 1, y and f x ,1

2. A cube is situated so that two of its corners have coordinates 0,0,0 and 3,3,3 . Find the coordinates of the other corners.

3. f x , y  2  cosx  y . Find the equation of a contour at 2.5 and sketch its graph.

4. For the function z  3 x2  y , sketch the intersection with the xy-plane, the xz-plane, and the yz-plane. Label the axes and the intercepts.

5. Find the equation of the plane which passes through the points (3,2,-1) , (1,-1,3), and (3,2,4).

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.

7. f x , y  x3  3x  y3  9y2  24y has a relative maximum, a relative minimum, and two saddle points on the intervals x 4,4 and y 2,6 . Find the x and y coordinates of these points.

8. f x , y  x 3  xy 2  y 2  3x 2 3  x  3 ,  3  y  3 Determine the coordinates of the relative maximum, relative minimum, and saddle points.

9. Find an equation for the tangent plane to f x , y  x2 y  xy3  2 at the point 3,2 .

10. Find the derivative of f x, y  xy 2 at 1,3 in the direction toward 4,5 .

11. For the function f x, y  8  4x2  2y2 , find the direction of the steepest grade at 1,1,2 .

12. If the temperature is given by Tx, y,z  3x2  5y2  2z2 and you are located at 0.3,0.2,0.5 and want to get cool as soon as possible, in which direction should you head? 13. f x, y,z  x3  y3  z . Find the rate of change of this function at the point 1,1,2 in the direction toward the point 2,3,5

14. 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.

15. 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.

16. The contour diagram below shows the contours of some function, f x, y. Use it to estimate the following.   a) f x 0,0.7 b) f y 0,0.7 c) Du f 1,0 if v  1,1

17. A trough is to be made from a rectangular piece of metal that is 12 inches wide. Equal lengths from each side along the length are to be bent up at the same angle to form slanted sides. Find the length that should folded up from each side and the angle that the sides should be bent so that the cross-sectional area of the trough is a maximum.

18. A soft drink manufacturer wants to design an aluminum can in the shape of a right circular cylinder to hold a given volume, V. If the objective is to minimize the amount of aluminum needed (top, sides, and bottom), what dimensions should be used? Possible Answers

1. f 1, y f x,1

2. 0,0,3 , 0,3,0 , 0,3,3 , 3,3,0 , 3,0,3 , 3,0,0

3. y  x   3

5. -15x + 10y = -25 6. 2 1 . 9 ° , 1 0 3 . 1 ° , 5 5 ° , 6 . 4 2

7. Max. 1,2 , Min. 1,4 , SP 1,2 , 1,4 8. 0,0 min, 1, 3 SD,  2,0 max

9. z  38  4x  27y 10. 3 1 3 11. 8,4 12. 1,1,1 13. –1.28 14. 15.08 cu. in. 1 3 V 8V 15. 6.72 in./min. 16. a) 0 b)  2.5 c)  2.12 17. 4 in, 60 18. r = 3 h= 3 0.4 2 2 2