Math 55 First Long Exam Exercises I.
Directional Derivative 1. Find the directional derivative of the function at point P in the direction of A. a. , = 2 − 3 , 5,5 , = 4 + 3 b. c. d. e.
, = − + √3 sec 2 , 1,1 , = 12 + 5 , , = 3 cos , 0,0,0 , = 2 + − 2# , , = cos + $ + ln , 1,0,1/2 , = + 2 + 2# ,ℎ , = ln + + e)* at 0,1 in the direction of + = . .
2. Find the directions in which the function increase and decrease most rapidly at point P. Then find the rate of change of the function in these directions. a. , , = / + + , 3, 6,2 b. , = + + ,
−1,1 c. , = + sin ,
1,0 d. ℎ , , = ln + − 1 + + 6 , 1,1,0 II.
Tangent Planes 1. Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point P. a. 2 â&#x2C6;&#x2019; 2 + â&#x2C6;&#x2019; 1 + â&#x2C6;&#x2019; 3 = 10, 3,3,5 b. â&#x2C6;&#x2019; 2 + = 2, 2,1, â&#x2C6;&#x2019;1 c. â&#x2C6;&#x2019; = 4 arctan , 1 + 5, 1,1 d. + 1 = cos , 1,1,0 2. Find the equation of the tangent plane to the surface with parametric equations = 7 , = 8 , = 7 + 28 at the point 1, 1, 3 . 3. Find the equation of the tangent plane to the surface with parametrization 9 7, 8 = â&#x152;Š;<= 7 4 + 2 ;<= 8 , = > 7 4 + 2 ;<= 8 , 2 = > 8â&#x152;Ş at the point 4,0,2 .
III.
Maximum and Minimum Values (Try to use Lagrange Multipliers if possible) 1. Find and classify the critical points (as local maxima, local minima, or saddle points) of the given function: (d) , = = > (a) , = 9 + + 2 â&#x2C6;&#x2019; + (e) , = B + B â&#x2C6;&#x2019; 2 (b) , = + + 1 (f) , = (c) , = 2 â&#x2C6;&#x2019; 4 â&#x20AC;&#x201C; â&#x2C6;&#x2019; 2 2. Find the absolute maxima and minima of the given function on the given set: C , = 1 + 2 + 3 on the triangle with vertices 0, 3 , 2, 1 , C>D 5, 3 E , = + 4 â&#x2C6;&#x2019; on the square â&#x2C6;&#x2019;1 â&#x2030;¤ â&#x2030;¤ 1, â&#x2C6;&#x2019;1 â&#x2030;¤ â&#x2030;¤ 1 ; , = G + 2 â&#x2C6;&#x2019; 2 on the disk + â&#x2030;¤ 4 3. Find the shortest distance from the point (3, 2, â&#x2C6;&#x2019;1) to the plane x â&#x2C6;&#x2019; y + 2z = 1. 4. Find the point(s) on the cone (z â&#x2C6;&#x2019; 2)2 = x2 + y2 that is closest to the point (1, 1, 2). 5. Among all the rectangular boxes of volume 125 H G , find the dimensions of that which has the least surface area. 6. Find the volume of the largest rectangular box with faces parallel to the coordinate planes that can be inscribed in the ellipsoid x2 + 4y2 + 4z2 = 16. 7. Find the maximum and minimum values of , = + subject to the constraint â&#x2C6;&#x2019; 2 + â&#x2C6;&#x2019; 4 = 0. 8. Find the maximum and minimum values of , = 3 + 4 on the circle + = 1.
IV.
Double Integrals
1. Evaluate the following: -
JKL
C. I I
M M
D D
E. I I 3 G M M
;. I I M
D D
D D
D. I I 2 sin D D M B
.I I M
M
D D 4−
O
. I I M
. I
√
M
N
B
1 D D +1
√ /B
I
ℎ. I I
√
1
/1 + +
D D
1 D D +
√ 1 + B /B
. I I M
M
+ D D +
2. SET-UP ONLY the iterated double integral using Cartesian Coordinates.
a. Find the volume of the region bounded by the paraboloid = + and below by the triangle enclosed by the lines = , = 0, and + = 2 in the xy-plane. b. Find the volume of the solid that is bounded above by the cylinder = and below by the region enclosed by the parabola = 2 − and the line = in the xy-plane. c. Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder + = 4, and the plane + = 3.
3. SET-UP ONLY : Use polar coordinates to find the volume of the given solid.
a. the solid in the first octant bounded by the coordinate planes, the plane = 3 and the parabolic cylinder = 4 − . b. the solid enclosed by the cone = + and the plane z = 3. c. the solid lying inside both the cylinder + = 4 and the sphere + + = 9. d. the solid enclosed by the hyperboloid − − = 1 and the plane = 2. e. A sphere of radius C. f. Inside both the cylinder + = 4 and the ellipsoid 4 + 4 + = 64. g. Find the volume of the solid lying inside all three cylinders + = 1, + = 1, and + = 1.
4. Find the surface area of
a. The part of the plane 2 + 5 + = 10 that lies inside the cylinder + = 9. b. The solid bounded by the parabolids = 3 + 3 and = 4 − − . c. The surface with parametrization 9 7, 8 = 〈;<= 7 4 + 2 ;<= 8 , = > 7 4 + 2 ;<= 8 , 2 = > 8〉 where 0 ≤ 7, 8 ≤ 25. d. The parametric surface given by Q 7, 8 = 〈78, 7 + 8, 7 − 8〉 where 7 + 8 ≤ 1 e. part of the surface = – that lies between the cylinders + = 1 and + = 4.
5. Find the mass and center of mass of the lamina that occupies the region R and has density given by the function R S, T . a. R is bounded by the curve = , the line = 1, and the coordinate axes; U , = b. R is the disk + ≤ 2 ; U , is twice the distance from (x, y) to the origin. c. R is bounded by the parabola = − and the line + = 0; U , = + . d. R is the smaller region cut from the ellipse + 4 = 12 by the parabola = 4 ; U , = 5 .