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# Multivariable Calculus Engrs MATH 1920

Cornell

GPA 3.75

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This 2 page Class Notes was uploaded by Sandrine Wehner II on Saturday September 26, 2015. The Class Notes belongs to MATH 1920 at Cornell University taught by Staff in Fall. Since its upload, it has received 47 views. For similar materials see /class/214326/math-1920-cornell-university in Mathematics (M) at Cornell University.

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Date Created: 09/26/15

Math 1920 Prelim 2 October 30 2008 730 PM to 900 PM You are NOT allowed calculators the tedt or any other book or notes SHOW ALL WORK Write your name and LectureSection number on each booklet you use You may leave when you have nished but if you have not handed in your eacam booklet and left the room by 845 please remain in your seat so as not to disturb others who are still working 1 a 7 points Find the directions in which the function fdyz Qdy iyz increases and decreases most rapidly at the point 13017 71 1 Recall that a direction in three dimensional space is speci ed by a unit vector b 7 points Calculate the directional derivatives of f in these directions c 6 points Is there a direction at P0 in which the derivative of f is 73 Provide a reason for your answer 2 a 14 points Calculate the standard linear approximation of the function fdyz x2 7 3yz 2 at the point 112 b 6 points Use this to determine the approximate value of the function at the point 11 1121 3 Consider the region R inside the unit circle centered at the origin and outside the region de ned by 7d lt y lt x a 5 points Sketch R b 7 points Calculate the area of R c 8 points Find the centroid of R Note that the centroid y is given with the following equations HR 7 ffRydA 7 di d7 z an 177 ffRdA ffRdA 4 20 points Consider the points 1310707 1321707 P3ab in the plane7 a7 b constants Find the point Pdy such that the sum of the squares of the distances from P to each of 131132133 is minimized Be careful to justify your answer 5 20 points Find the maximum volume of a box determined by eight points ix7 iy i2 inside the ellipsoid 2 2y2 322 1 Math 1920 Prelim 1 October 2 2008 730 PM to 900 PM You are NOT allowed calculators the tedt or any other book or notes SHOW ALL WORK Write your name and LectureSection number on each booklet you use You may leave when you have nished but if you have not handed in your eacam booklet and left the room by 845 please remain in your seat so as not to disturb others who are still working 1 a 7 points Calculate the distance from 111 to the plane with the equation x 2y 32 0 b 7 points Find the equation of the plane through 000 110 and 123 2 14 points The position vector of a particle moving in space is given by the vector function I39t cos ti 2tj sin tk where t 2 0 For each t calculate the angle between the velocity vector and acceleration vector 3 14 points The equation 3ny yz zln2d 71 0 de nes x as a function of the two independent variables y and 2 Find the partial derivative dddz at the point 1 71 73 1 4 Let fdy i 2 271 a 5 points Determine the domain of fzy b 6 points Deterrnine and graph level curves in the x y plane for fdy 721 and 2 c 5 points Explain in complete sentences whether the level curve exists and what sets they form for fdy 71 712 and 0 5 Consider the curve I39t 4 sin ti 4 cos tj 3tk a 8 points Find the unit tangent vector of the curve b 8 points Find the point on the curve such that the length along the curve from t 0 is 107139 units in the direction of increasing t 6 14 points Let fyz x2 2y2 322 Integrate fdyz along the straight line segment from A to B where A 100 and B 011 7 12 points The covalent bonds of the molecule rnethane CH4 around the carbon atom can be described as three dirnensional unit vectors V1V2 V3 V4 where V1V2V3V407 and the angle 6 between any two of these vectors is same Use vector operations to derive an expression for 0 Please leave your answer in terms of inverse trig functions Hint Think about the vector dot product

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