CS代考 MATH 241 CALCULUS 3- SANTILLI

EXAM #2 MATH 241 CALCULUS 3- SANTILLI
SHOW ALL WORK IN BLUEBOOK! INDIVIDUAL EFFORTS ONLY! GOOD LUCK.
1.)Find r , r and  when x  e2r cos and y  e3r sin and then check your results with another method. xy y
2.)An important integral in probability is e x2 2dx. Evaluate this integral by first letting I  e x2 2dx, then

I2  ex 2dx ey 2dy= e(x y ) 2dxdy.Usepolarcoordinates.  2  2   2 2
0 0  0 0
3.)Findtheabsoluteextremaforz3xyx2 y2 subjectedtotheconstraintx2 1y2.
a.)Find the absolute extrema by the use
b.)Verify your answer in a. by finding the absolute extrema by the method of incorporating the constraint. c.)Use the First Partial Test to determine the type of any critical points.
d.)Use the Second Partial Test to determine the type of any critical points.
    
4.)Evaluate the integral,     
34xy212xy z
dxdydzby transforming the integral into another 2 3
00 xy2 coordinate system of u, v, and w.
5.)Thesurfaceofamountainisdepictedbytheequation,h(x,y)500 2y2 x2 feet.Supposeaskierisatthe point (10,10) on the mountain and skis down the path of greatest decent.
a.)In what direction should the skier initially move in order to descend at the greatest rate? What is that rate?
b.)What would be the rate of change of altitude if the skier moved from his initial point in the direction of the point (10, 5, 350)?
c.)Find the path of greatest decent.
d.)Show that the path found in part c. and the contour curves of the mountain’s altitude are
orthogonal trajectories of each other.
6.)Consider a solid inside the portion of the sphere x2  y2  z2 1 that lies in the first octant. if the square root of the density at any point is proportional to the fourth power of the distance from the origin, find:
a.) the mass of the solid b.) the center of mass
c.) the moment of inertia about the origin d.) the center of gyration
7.)Show that the moment of inertia about the axis of a right circular cone having altitude h and a circular base of radius a is I  3 ma2. Assume that the cone has a constant density.
8.)Evaluate lim sin( x  y) Use polar coordinates.
(x,y)(0,0) xy
9.)Find the surface area of the part of the surface z  xy that lies within the cylinder x2  y2  1.
10.) Giventhesurfaces,x2 4y2 272z2 andx2 11y2 2z2,find:
a.) Symmetric equation for the tangent line to the curve of intersection at the point (3, -2, 1). b.) The tangent plane that contains the tangent line in part a. and the origin.
c.) The inclination of the tangent plane in part b.
d.) The equation of the normal line to the ellipsoid at the point (3, -2, 1).
11.) A triangle is measured and two adjacent sides are found to be 3 and 4 inches in length with the
included angle of π/4 radians. If the possible errors in measurements are 1/16 inch in the sides and 0.02 radians in the angle, use differentials to approximate the maximum possible error in computing the area of the triangle.

12.) Two objects are traveling in elliptical paths given by the following parametric equations: First object’spath: x1 4cost, y1 2sint;Secondobject’spath: x2 2sin2t, y2 3cos2t.Atwhatrateisthe distance between the two objects changing when t   ? Use the chain rule.
2222 13.) Ifw f(x.y),xrcos andyrsin,showthatw w w 1w .
14.) Find w , w and w for xyz  xzw  yzw  w2  5 by direct implicit differentiation and then by the use xy z
of the Implicit Function Theorem.
15.) Use a triple integral to find the volume of the solid in the first octant bounded by the surfaces y1x2 andz1x2.
      x  y  r  r 2  

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