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# Mathematics 10A Multi-variable Calculus Final Exam Practice Math 010A

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This 3 page Test Prep (MCAT, SAT...) was uploaded by Avid Notetaker on Friday April 1, 2016. The Test Prep (MCAT, SAT...) belongs to Math 010A at University of California Riverside taught by Meng Zhu in Winter 2016. Since its upload, it has received 27 views. For similar materials see Calculus:Several Variables in Mathematics (M) at University of California Riverside.

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Date Created: 04/01/16

1 Practice for Final Exam - Winter 2016 Mathematics 010A - Calculus of Several Variables 3 1. Write down both the Cartesian and parametric equations of the line in R through the point (1;0;▯1) that is parallel to the line (x ▯ 4) = y = z + 2. 3 2 Answer: x = 1 + 3t, y = 2t, z = ▯1 + t x▯1 = y = z+1. 3 2 1 2. Write down both the Cartesian and parametric equations of the line in R through the point (2;1;2) and perpendicular to the plane 3x + 2y + z = 1 . Answer: x = 2 + 3t, y = 1 + 2t, z = 2 + t x▯2 y▯1 3 = 2 = z ▯ 2. 3. Find the plane through point (2;4;▯1) with normal vector n = (2;3;4) Answer: 2x + 3y + 4z = 12 4. Find an equation of the plane through points (1;3;2), (3;▯1;6) and (5;2;0). Answer: 6x + 10y + 7z = 50 x+1 y+2 z▯2 5. Find the plane through (1;0;2) that contains the line2 = ▯1 = ▯3. Answer: ▯(x ▯ 1) + y ▯ (z ▯ 2) = 0. 6. Find the line of intersection of planes x + y + z = 1 and x ▯ 2y + 3z = 1. x▯1 y z Answer: 5 = ▯2 = ▯3 7. Find the distance between the point (1;▯2;3) and the line l : x = 2t▯5; y = 3▯t; z = 4: p Answer: 285=5. 8. Determine the distance between 2 lines 1 : t(2;▯1;0) + (▯1;3;5) and l2: t(0;3;1) + (0;3;4) 2 p Answer: 7= 41. 9. Determine the distance between 2 planes given by x ▯ 3y + 2z = 1 and x ▯ 3y + 2z = 8 p14 Answer: 2. 10. Determine the distance from point (1;0;▯1) to the plane x + 3y ▯ 5z = 2 Answer: p4. 35 p 11. Find the tangent line of the curve r (t) = (2cost;2sint;4cos2t) at ( p p p Answer: x = 3 ▯ t, y = 1 +3t, z = 2 ▯ 4 3t. 12. Reparametrize the curve r (t) = (e sint;2;e cost) with respect to arc length parameter based at t = 0. p Answer: s(t) = 2(e ▯ 1), p p p p ▯ s + 2 s + 2 s + 2 s + 2 r (s) = (p sinln( p );2; p cosln( p )) 2 2 2 2 13. Find the curvature ▯ of r (t) = (e sint;2;e cost). 1 Answer: ▯(t) =p2e. ▯ 14. Find the torsion ▯ of r (t) = (t;sint;cost) Answer: ▯(t) =p . 2 15. Evaluate the limit or explain why the limit does not exist e ey a) lim (x;y)!(0x + y + 2 2x + y 2 b) lim (x;y)!(▯1;x + y2 2x + y2 c) lim (x;y)!(0x + y 2 3 2 2 d) lim x + 2xy + y (x;y)!(0;0) x + y 4 4 x ▯ y e) (x;y)!(0;x + y 2 2 (x + y) f)(x;y)!(0;0)2 2 x + y Answer: a) 1=2, b) 6=5, c) DNE, d) 0, e) 0, f) DNE ▯ 16. Compute the divergence and curl of the vector ▯eld F = (x ;xe ;2xyz). ▯ y ▯ y Answer: divF = 2x + xe + 2xy, curlF = (2xz;▯2yz;e ). 17. Find the equation of the tangent plane of f(x;y) = (x + y)exy at (0;1;1). Answer: 2x + (y ▯ 1) ▯ (z ▯ 1) = 0. 18. Find the local maximum and minimum of f(x;y) = xy(1 ▯ x ▯ y). Answer: critical points are (0;0), (0;1), (1;0) and (1=3;1=3). Local maximum 1=27 at (1=3;1=3), no local minimum. 2 2 2 19. Find the absolute maximum and minimum of f(x;y) = x + y ▯ x y + 4 in the region D = f(x;y)j jxj ▯ 1 and jyj ▯ 1g, Answer: Absolute maximum is 7 achieved at (1;▯1) and (▯1;▯1), absolute minimum is 4 achieved at (0;0). 20. Use Lagrange multiplier to ▯nd the maximum and minimum value of f(x;y) = y ▯ x 2 2 2 2 subject to the constraint x + y = 1. Answer: the maximum is 1 at (0;1) and (0;▯1) and the minimum is -1 at (1;0) and (▯1;0).

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