18 02a Practice Exam 3 ESG Fall 2014 Remember to look over your notes and problem sets The practice give some example problems they are not intended to cover absolutely everything No books notes or calculators This should take about 80 minutes The actual test will be shorter designed to take 50 minutes Problem 1 The square with vertices A 1 1 B 1 1 C 1 1 and D 1 1 in the xy plane is the base of a pyramid The point P at the apex of the pyramid is on the z axis at a height 2 a Give the components of the vectors PA and PB b Find the angle on one of the faces at the apex c Find the area of any one of 1 0 3 2 Problem 2 Let A 1 1 1 a Find A the four faces touching the apex 1 1 2 b Use part a to solve x z 1 3x 2y z 0 x y 2z 4 c For what c will the system of equations x z 0 3x 2y z 0 x y cz 0 have a non zero solution d For the value of c found in part c find a non zero solution to the system Problem 3 D C A If A 1 2 3 B 2 2 4 C 5 4 5 D 2 3 5 compute the volume of the parallelpiped shown B Problem 4 a Write the equation of the plane containing the three points 1 1 1 1 2 1 2 2 3 b Find the distance from the point 0 0 3 to the plane in part a Problem 5 Find the intersection of the line x y z 2 3 0 t 1 3 5 and the plane 2x 3y z 7 Problem 6 a Write the curve y sin x in parametric form r t x t i y t j dT dr ds b For your answer in part a find T t Note some of the derivatives dt dt ds are messier than our typical problem work carefully Problem 7 Let P be a point halfway along a radius of a circle of radius a Use vector methods to write the parametric equations for the curve traced out by P as the circle rolls along the x axis Assume the circle starts with both its center and P on the y axis 18 02a Practice Exam 3 ESG Fall 2014 2 Problem 8 Let w f x y z where f x y z x3 y z 3 x exyz Compute all of the following f f f w w a x y z x y b fxy c fzyx f 1 2 0 x 2w 2w x2 y x f y 1 2 0
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