Math 214-5 Final Review Sheet The final exam is Tuesday, march 15 (9:10 am to 11:10 am). The exam is comprehensive and will cover chapters 12 through 14. The exam problems will be similar to your quiz and midterms problems. Thus, it is strongly recommended that you study your lecture notes, Mid-terms, and the quiz problems plus the problems given in this handout. The following sample problems are for the chapters which we have covered since Midterm 2 and they are meant to be as a guide, but there may be a topic or technique we discussed that is not included here.Please don't assume that means it will not appear on the exam. Remember, the final is COMPREHENSIVE! 1. Suppose that over a certain region of space the electrical potential V is given by 2 (, ,) 5 3 Vxyz x xy xyz = - + . Find the rate of change of the potential at P (3, 4, 5) in the direction of the vector v i j k = + - . In which direction does V change most rapidly at P? c. What is the maximum rate of change at P? 2. Find the equations of the tangent plane and the normal line to 2 2 2 x y z = + - at (-1, 1, 0). 3. Find the direction in which (, , ) xy f xyz ze = increases most rapidly at the point (0, 1, 2). What is the maximum rate of increase? 4. Find any extrema for 2 2 (, ) 3 fxy xy xy xy = - - and classify them. (Relative max/relative min. or saddle point) 5. Find the absolute maximum and minimum values of 2 2 2 3 (, ) 4 f xy xy xy xy = - - on the set D, where D is the closed triangular region in the xy-plane with vertices (0,0), (0,6), and (6,0). 6. Find the points on the surface 2 3 2 xy z = that are closest to the origin. 7. Using Lagrange multipliers method find the maximum and minimum values of (, ,) f xyz xyz = subject to. 2 2 2 3 x y z + + = 8. A closed rectangle box with a volume of 16 3 ft is made from two kinds of materials. The top and bottom are made of material costing 10 cents/ 2 ft and the sides from material costing 5 cents/ 2 ft . Find the dimensions of the box so that the cost of the materials is minimized. 9. Find the equation of the tangent plane to the surface ln z y x = at (1, 4, 0) 10. Find the linear approximation of the function 3 2 2 (, ,) f xyz x y z = + at the point (2, 3, 4) and use it to estimate the number 3 2 2 (1.98) (3.01) (3.97) + . 11. If cos( ) cos z xy y x = + , where 2 x u v = + and 2 y u v = - use the chain rule to find z u ∂ ∂ and z v ∂ ∂ 12. The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 rad/s. How fast is the area of the triangle changing when x = 40 in, y = 50 in, and 6 π θ = ?
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Math 214-5 Final Review Sheet The final exam is Tuesday, march 15 (9:10 am to 11:10 am). The exam is comprehensive and will cover
chapters 12 through 14. The exam problems will be similar to your quiz and midterms problems. Thus, it is
strongly recommended that you study your lecture notes, Mid-terms, and the quiz problems plus the problems given in this handout. The following sample problems are for the chapters which we have covered since Midterm 2 and they are meant to be as a guide, but there may be a topic or technique we discussed that is not included here.Please don't assume that means it will not appear on the exam. Remember, the final is COMPREHENSIVE!
1. Suppose that over a certain region of space the electrical potential V is given by2
( , , ) 5 3V x y z x x y x y z= − + .
Find the rate of change of the potential at P (3, 4, 5) in the direction of the vector v i j k= + −�
� ��
. In which direction does
V change most rapidly at P? c. What is the maximum rate of change at P?
2. Find the equations of the tangent plane and the normal line to2 2
2x y z= + − at (-1, 1, 0).
3. Find the direction in which ( , , )x y
f x y z z e= increases most rapidly at the point (0, 1, 2). What is the maximum rate of
increase?
4. Find any extrema for 2 2
( , ) 3f x y x y x y x y= − − and classify them. (Relative max/relative min. or saddle point)
5. Find the absolute maximum and minimum values of 2 2 2 3
( , ) 4f x y x y x y x y= − − on the set D, where D is the closed
triangular region in the xy-plane with vertices (0,0), (0,6), and (6,0).
6. Find the points on the surface 2 3
2x y z = that are closest to the origin.
7. Using Lagrange multipliers method find the maximum and minimum values of ( , , )f x y z x y z= subject to.
2 2 23x y z+ + =
8. A closed rectangle box with a volume of 163
ft is made from two kinds of materials. The top and bottom are made of
material costing 10 cents/2ft and the sides from material costing 5 cents/
2ft . Find the dimensions of the box so that the
cost of the materials is minimized.
9. Find the equation of the tangent plane to the surface lnz y x= at (1, 4, 0)
10. Find the linear approximation of the function3 2 2
( , , )f x y z x y z= + at the point (2, 3, 4) and use it to
estimate the number3 2 2
(1.98) (3.01) (3.97)+ .
11. If cos( ) cosz x y y x= + , where 2
x u v= + and 2y u v= − use the chain rule to find
z
u
∂
∂ and
z
v
∂
∂
12. The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate
of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 rad/s. How fast is the area of the triangle changing
when x = 40 in, y = 50 in, and 6
πθ = ?
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