MATH 237 Fall 2008 Practice Test 1: Geometry and Vector Calculus Problems - Prof. Carl Dro, Exams of Advanced Calculus

Download MATH 237 Fall 2008 Practice Test 1: Geometry and Vector Calculus Problems - Prof. Carl Dro and more Exams Advanced Calculus in PDF only on Docsity! MATH 237 Fall 2008 Practice Test 1 NOTES: – None of these problems is likely to be on the real test in exactly this form. – There will be other kinds of problems on the real test. – You will not be able to use your book when you take the real test! Maybe you should consider doing this one without your book. 1. Consider the three points (1, 2, 2), (−1, 0, 2) and (1,−1, 3). (a) Determine whether or not these three points are collinear. (b) If they are collinear, find parametric equations for the line containing them; if they’re not collinear, find an equation of the plane containing them. 2. Let ~u =~ı− 2~ + 3~k and let ~w =~ı +~−~k. Write ~u as the sum of one vector which is parallel to ~w and another which is perpendicular to ~w. 3. Let ℓ be the line with parametric equations    x = −1 + 2t y = 3t z = 1 − t and let P be the plane with equation 2x + y − 4z = 8. (a) Find the point of intersection of ℓ and P . (b) Find the sine of the angle of intersection between ℓ and P . 4. Let ℓ be the line ~r(t) = (1, 3, 2) + t(1,−1, 0), and let P be the point (3,−1, 0). (a) Find the distance from P to ℓ. (b) Find an equation for the plane that contains both ℓ and P . 5. Let ~r(t) = t2~ı − t~ + 2 3 t3~k, and let C be the curve parametrized bt ~r. Find parametric equations for the line which is tangent to C at the point where t = 2. 6. Let ~f(t) = t~ı + t2~ + t3~k and let ~g(u) = u~ı + (u + 6)~ − 2u2~k. (a) Find the point where the curves parametrized by ~f and ~g intersect. (b) Find the cosine of the angle at which these two curves intersect.