# Correction practice midterm Exercise 1

Correction practice midterm Exercise 1. Find a cartesian equation of the plane that passes through the points A(2, – 1,1) and containing the lines of parametric equations x = 1+4, y = -1 + 2t, z = 3 – t. Find the distance of the point B(5,2,1) to this plane. Exercise 2. Let Lį be the line through the points (1,2, -1) and (2, -1,3). Let L2 be the line through the points (2, – 1,1) and (3,1,2). Calculate the distance between L1 and L2. Exercise 3. Find the intersection between the sphere of center A(1,2, -1) and radius 2 with the line defined by the point B(3,3,2) and the vector ū= (2,1,1) Exercise 4. Find and sketch the domain of the following function f(x,y) 1 In(1 + y2 – 3) Exercise 5. Consider the function F(x,y) { I sin(my), (x,y) = (0,0) 0, (x,y) = (0,0). 1. Prove that F is continuous at (0,0). 2. Find Fr(0,0) and F,(0,0). 3. Check the differentiability of Fat (0,0). 4. Calculate F2(x,y) and Fy(x,y) for any (x,y) = (0,0). Exercise 6. Let consider the vector function r(t) = (cos(t)e”, sin(t)et,e’) defined for t € [0,1]. 1. Find the length of the curve of r. 2. Find the curvature at t= 0. 1 3. Find the TNB frame at t=0. Exercise 7. [Bonus] Consider a circle C of radius r > 0 and centered at a point O. Let A be a point outside the disc delimited by this circle. Given a line through the point A and intersecting the circle C at two different points M and N. Show the identity |AM||AN| = |OÀ12 – p2
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