(i) h(7,8,24) (ii) h(6,5,6) (iii) h( 7,8,9) (iv) h(10,9, 16) (iv) g 3,32 8

Transcription

1 M252 Practice Eam for Find and simplify the function values. f ( y, ) = 5 10y (i) f(0,0) (ii) f(0,1) (iii) f(3,9) (iv) f(1,y) (v) f(,0) (vi) f(t,1) 2. Find and simplify the function values. hyz (,, ) = y z (i) h(7,8,24) (ii) h(6,5,6) (iii) h( 7,8,9) (iv) h(10,9, 16) 3. Find and simplify the function values. y gy (, ) = (16t 6) dt (i) g(0,32) (ii) g(1,32) (iii) g 3,32 8 (iv) g 3 0, 8 4. Describe the domain and range of the function. f ( y, ) = 100 y Page 1

2 5. Describe the level curves of the function. Sketch the level curves for the given c-values. z = 6 2 3y, c = 0, 2, 4, 6 6. Find the limit and discusss the continuity of the function. ( y, ) ( 5,4) 2 ( y ) lim Find the limit and discusss the continuity of the function. ( y, ) (2, 10) ( + y ) lim Find the limit and discusss the continuity of the function. lim 7 6y ( y, ) (1,1) + 9. Find the limit and discusss the continuity of the function. lim 5e ( yz,, ) ( 1,0,8) y 6 z Page 2

3 10. Find the limit (if it eists). If the limit does not eist, eplain why. y 4 lim 3 + y ( y, ) (4,3) 11. Discusss the continuity of the function at the origin y, ( y, ) (0,0) 8 8 f(, y) = + y 2, ( y, ) = (0,0) 12. Use polar coordinates to find the limit. [Hint: Let = rcosθ and y = rsinθ, and note that ( y, ) (0,0) implies r 0.] lim y ( y, ) (0,0) + y 13. Discusss the continuity of the function. f(, y, z) = z + y 64 Page 3

4 14. Find each limit for the function 2 f ( y, ) 7 4y =. (i) lim 0 f ( +, y) f(, y) (ii) f y+ y f y lim y 0 (, ) (, ) y 15. Find both first partial derivatives. f(, y) = y Find both first partial derivatives. f y y (, ) = Find both first partial derivatives. z = 4 y 18. Find both first partial derivatives. 3 z = 2y Page 4

5 19. Find both first partial derivatives. z = e 6 10y 20. Find both first partial derivatives. 5 8 ( ) f ( y, ) = ln + y 21. Find both first partial derivatives. f ( y, ) = ln 7 y 22. Find both first partial derivatives. 5y z = + 9y 23. Find both first partial derivatives. f ( y, ) = 4 + y 10 4 Page 5

6 24. Evaluate f and fy at the given point. f(, y) = 3y 9 + 2y, (1,1) 25. For f(,y), find all values of and y such that f(,y) = 0 and fy(,y) = 0 simultaneously. f ( y, ) = 9 3 y+ 9y Find the first partial derivatives with respect to, y, and z. 2z w = 9 + 6y 27. Find the first partial derivatives with respect to, y, and z. H ( yz,, ) = cos(2+ 8y+ 7 z) 28. Evaluate f and fy at the given point. y f(, y, z) =, (6,8,3) + y + z Page 6

7 29. Find the four second partial derivatives. Observe that the second mied partials are equal. z = + 2y+ 8y 30. Find the four second partial derivatives. Observe that the second mied partials are equal. z = 11e y + 8ye 31. Find the total differential of the function z 10 9 = 5 y. 32. Find the total differential of the function 9 z =. y Find the total differential of the function z = y 34. For the function f(, y) = 5 4y: (i) Evaluate f(1,5) and f(1.09,5.09) and calculate (ii) Use the total differential dz to approimate z, and z. Page 7

8 35. For the function f(, y) = sin y: (i) Evaluate f(4,2) and f(4.08,2.03) and calculate (ii) Use the total differential dz to approimate z, and z. 36. The radius r and height h of a right circular cylinder are measured with possible errors of 8% and 3%, respectively. Approimate the maimum possible percent error in measuring the volume. 37. A triangle is measured and two adjacent sides are found to be 3 inches and 4 inches π 1 long, with an included angle of. The possible errors in measurement are inch for 4 10 the sides and 0.04 radian for the angle. Approimate the maimum possible error in the computation of the area. 38. Let w = y, where = 10sin t and y = 8cost. Find dw dt. 39. Let w= cos(2 4 y), where 9 = t and y = 7. Find dw dt. Page 8

9 40. Let w= ycos z, where 9 = t, y 2 = t, and z arccos = t. Find dw dt. 41. Let 3 3 w= + y, where = 4s+ t, y = 4s t. Find partial derivative at the point s = 2, t = 2. w ds and w dt and evaluate each 42. Let 7 6 w 7 y =, where s = e, derivative at the point s = 0, t = 1. y t = e. Find w ds and w dt and evaluate each partial 43. Let w y 3 = ( ), where r θ = + and y = r θ. Find w r and w θ. 44. Let yz w =, where y = r+ θ, and z = 9r 7θ. Find 2 = θ, 9 7 w r and w θ. 45. Differentiate implicitly to find dy d. y y y = 0 Page 9

10 46. Differentiate implicitly to find the first partial derivatives of z y + z = Differentiate implicitly to find the first partial derivatives of z. 3 y z + sin(6 + 7 ) = Differentiate implicitly to find the first partial derivatives of w y + z 7yw+ 9w = The radius of a right cylinder is increasing at a rate of 2 inches per minute, and the height is decreasing at a rate of 3 inches per minute. What is the rate of change of the volume and surface area (including both ends of the cylinder as well as the side) when the radius is 9 inches and height is 27 inches? 50. Find the directional derivative of the function at P in the direction of v. ( ) 1 f(, y) = 5 7y+ 10 y, P(1,4), v= ˆ i+ 3 ˆ j 2 Page 10

11 51. Find the directional derivative of the function at P in the direction of v. ( ) f(, y) = y, P(2,1), v = ˆ i+ ˆ j Find the directional derivative of the function at P in the direction of v. f (, y, z) = y + yz + z, P(1,1,1), v = 9ˆi+ 5ˆj-10 kˆ 53. Find the directional derivative of the function in the direction of u= cosθˆi+ sinθˆj. π f(, y) = sin(3 2 y), θ = Find the gradient of the function at the given point. f y y 2 (, ) = , (4,1) 55. Find the gradient of the function at the given point. gy (, ) = 9 e, (9,0) y Page 11

12 56. Find the gradient of the function at the given point. w= y yz+ z 6 10, (1,1, 2) 57. Use the gradient to find the directional derivative of the function at P in the direction of Q. g y y P Q (, ) = + + 1, (2, 4) (6,12) 58. Use the gradient to find the directional derivative of the function at P in the direction of Q. f(, y) = sin(7 )cos y, P(0,0), Q( π, π ) Find the gradient of the function and the maimum value of the directional derivative at the given point. 4 w= y z 7, (9,1,1) y 60. Find the directional derivative Du f (3,2) of the function f(, y ) = 10 in the 10 3 direction of u= cosθˆi+ sinθˆj. π (i) θ = ; (ii) θ = 4 2π 3 Page 12

13 y 61. Find the directional derivative Du f (3,2) of the function f(, y ) = 10 in the 10 5 direction of u = v. v (i) v = ˆi+ ˆ j; (ii) v = 3ˆi 4ˆj y 62. Find the directional derivative Du f (3,2) of the function f(, y ) = 10 in the 10 5 direction of u = v. v (i) v is the vector from (1,2) to ( 2,6); (ii) v is the vector from (3,2) to (4,5). y 63. Find f (, y) for function f(, y ) = y 64. For function f(, y ) = 6, find the maimum value of the directional derivative 6 9 at (3,2). Page 13

14 65. Use the gradient to find a normal vector to the graph of the equation at the given point. 2 9 y = 4, (10,896) 66. Find a unit normal vector to the surface + y+ z = 6 at the point (3,0,3). 67. Find a unit normal vector to the surface 2 + y + z = 18 at the point (4,1,1). 68. Find a unit normal vector to the surface y 4 2 z= 0 at the point ( 1, 6,36 ). 69. Find a unit normal vector to the surface + + = at the point ( 2, 1,2 ) 2 3 2y z Find an equation of the tangent plane to the surface ( 4,6, 20). gy (, ) y = at the point 71. Find an equation of the tangent plane to the surface ( 8, 1,1). 5 f ( y, ) = 10 yat the point 4 Page 14

15 2 72. Find an equation of the tangent plane to the surface + 9y + z = 504at the point ( 6, 6,12). 73. Find an equation of the tangent plane to the surface = y(7z 5) at the point ( 96, 6,3 ). 74. Find an equation of the tangent plane and find symmetric equations of the normal line to 2 the surface y z 201 1,10, = at the point ( ) 75. Find an equation of the tangent plane and find symmetric equations of the normal line to the surface y 4z 0 4, 10,10. = at the point ( ) 76. Find an equation of the tangent plane and find symmetric equations of the normal line to the surface 30 1,3,10. yz = at the point ( ) 77. Find the angle of inclination θ of the tangent plane to the surface point ( 1, 3, 8 ). y z + = 0 at the Page 15

16 78. Find the angle of inclination θ of the tangent plane to the surface 3 + 4y z = 0 at the point ( 6,6,252 ). 79. Find the distance from the point (0,0,0) to the plane 9+ 10y+ z = Find three positive numbers, y, and z whose sum is 3 and product is a maimum. 81. Find three positive numbers, y, and z whose sum is 24 and the sum of the squares is a maimum. 82. The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot eceed 36 inches. Find the dimensions of the rectangular package of largest volume that may be sent. 83. The material for constructing the base of an open bo costs 1.5 times as much per unit area as the material for constructing the sides. For a fied amount of money , find the dimensions of the bo of largest volume that can be made. Page 16

17 84. A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from 1 units of running shoes and y1 units of basketball shoes is: R = , where 1 and 2 are in thousands of units. Find 1 and 2 so as to maimize the revenue. 85. Find the least squares regression line for the points (1,0), (3,3), (10,6). 86. A store manager wants to know the demand y for an energy bar as a function of price. The daily sales for three different prices of the energy bar are shown in the table. Price, $ 1.06 $ 1.21 $ 1.45 Demand, y (i) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (ii) Use the model to estimate the demand when the price is Page 17