Math 280 Sample Final
WARNING
1)This sample final alone will NOT fully prepare you for the final. You must study the practice problems.
2)The partial solutions provided may be inaccurate. (why? I worked out the problems!!! I was a dummy even before my brain damage. Now ….. ) It is your responsibility to get the correct solutions. You may want to call some of your smart friends.
1)a) Compute the limit for if it exists. b) Compute the limit for if it exists.
2)Set up, but do not compute, an integral to compute each of the following: Assume the density is x.
a)M (the mass) for a solid bounded by , ,
b)M for a shell bounded by , ,
c)M for a thin plate bounded by , ,
d)M for a thin wirein the shape of ,
3)For ,
a)Find the directional derivative of F in the direction <3,4> at (1,2)
b)Find the rate and direction of maximum increase of f at (1,2).
c)Find the rate and direction of maximum decrease of f at (1,2)
d)Find the directions of no change of f at (1,2)
4)Find the extrema and saddle points of f for
5)Find the point on the sphere radius 3 centered at (0,0,0) that is closest to the point (2,3,4)
6)Verify divergence theorem for across the sphere
7)Verify circulation form of Green’s theorem over the region bounded by , oriented positively for .
8)Find the equations for the tangent plane and normal line to at P(2,-3,1)
9)Integrate
10)Compute , where R is bounded by the circle
11)Evaluate where R is a region bounded by (1,1) (2,2), (4,0), and (2,0).
12)Set up an integral to find the volume of the solid that is cut out of the sphere .
13)Find the curvature of the curve
14)Find T(t) and N(t) for i + j
15)Set up, but do not compute, an integral to find the volume of the region in the first octant bounded by and using a) dy dz dx b) dz dx dy
16)Use spherical coordinate to evaluate the integral
17)Determine whether the planeand the line are parallel, perpendicular, or neither.
18)Find an equation of the plane, each of whose points is equidistant fromand
19)A) Show that if a vector b is orthogonal to both vectors c and d, then b is orthogonal to 3c+5d
B) Does hold for any vectors a and b? Justify your answer. (This problem is not related to part A) C) Find the distance between and D)
1)Identify and sketch each of the following surfaces: A) B) c) d)
Math 280 sample final partial solutions
1)A) Use the squeeze theorem B) It does not exits: take x = 0 for Path 1 and y = 0 for Path 2
2)a) solid: use dv or
b) shell: use : you must first parametrize the surface: , . =
c) plate : use dA =
d) wire; use ds and First parametrize the curve as , , so
3)First find the gradient at (1,2)
a)The unit vector in the direction of <3,4> =
b)The maximum rate of increase occurs in the direction of the gradient: divide the gradient by its magnitude gives . The rate is
c)The maximum rate of decrease occurs in the direction opposite the gradient: multiply b) by -1 gives , The rate is
d)No change occurs in the direction perpendicular to the gradient: Take b), switch x and y, then multiply one of the components by -1 (so that the dot product with the gradient is 0). ,
4)Find the critical points by setting . Then test each critical point using Local min at (1/2, -1/4)
5)Lagrange with the optimization function and the constraint
6)A) for =3(volume of the unit sphere) = B) Using , you must parametrize the surface using , , . =. It is not all that obvious whether to use or . Here we use since the z-component must be positive in the first octant(when ) and if it is the right one in the first octant, then it is the one for the entire sphere .=
7) a) for the line integral, there are three pieces: , the x-axis, , the circular path, and the y-axis. Parametrize each piece , : : : to compute
b) For 2 dim’l integral, =(-1)area =
8) Find the gradient of f and evaluate it at the given point to find <a, b, c>: Tangent: 16(x-2)+6(y+3)+6(z-1)=0, normal x = 2+16t, y = -3+6t, z = 1+6t
9) Switch dx and dy
10) Observe that circle is a unit circle centered at (0,1) radius 1. It can be written in polar .
11) Let u = y – x , let v = y + x . The new boundary lines are , , ,
3(cos1 – 1)
12) First equate the equations to find the projection: we get
13)
14) i + j i + j
(To find a 2-dim’l N(t), switch the components of x and y .Then multiply one of the components by -1. Observe that the x-comp must be negative from the graph.
15) a) b)
16)
17) The plane is normal to the vector <3, 1, -1>. The line is parallel to <2, -1, 3>. If the line and plane are perpendicular, <3, 1, -1> and <2, -1,3> must be parallel. But they are not since one is not a multiple of the other. If the plane and the line are parallel, then <3, 1, -1> and <2, -1, 3> must be perpendicular. But they are not since their dot product is not zero. Neither
6)
The vector through and must be normal to the plane and their midpoint must be in the plane. and their midpoint is . Thus an equation is
18)
a) Recall that 1) two vectors are orthogonal iff their dot product is 0, and 2) the dot product is distributive and constants can be pulled out.
b) Yes, is orthogonal to , thus their dot product is 0
c) First make sure that they are parallel: this can be checked by seeing that their normal vectors is a multiple of each other. Use the formula . First pick any point in the first plane , say (0,0,-1) and any point in the second plane , say (0,0,-4) . Then compute the vector through (0,0,-1) and (0,0,-4) using terminal –initial =<0,0,-3>. This is your PS. Since the normal vector is <4,-1,-1>, .
19)
a) paraboloid b) cone c) cylinder d) plane