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Calculus III Practice Exam, Exams of Calculus

Multiple Choice Problems and Partial Credit Problems.

Typology: Exams

2018/2019

Uploaded on 02/11/2022

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MATH 20550: Calculus III
Practice Exam 1
Multiple Choice Problems
1. Find an equation for the line through the point (3,โˆ’1,2) and perpendic-
ular to the plane 2xโˆ’y+z+ 10 = 0.
(a) xโˆ’3
2=y+1
โˆ’1=zโˆ’2 (b) x+3
2=yโˆ’1
โˆ’1=zโˆ’2
(c) xโˆ’2
3=y+1
โˆ’1=zโˆ’2
2(d) 3xโˆ’y+ 2z+ 10 = 0
(e) 3xโˆ’2y+z+ 10 = 0
2. Find an equation of the plane that passes through the point (1,2,3) and
parallel to xโˆ’y+z= 100.
(a) xโˆ’y+zโˆ’2 = 0 (b) xโˆ’y+z+2 = 0 (c) x+2y+3z= 100
(d) xโˆ’1 = 2 โˆ’y=zโˆ’3 (e) xโˆ’1 = y+1
2=zโˆ’1
3.
3. Find the distance between the point (โˆ’1,โˆ’1,โˆ’1) and the plane x+ 2y+
2zโˆ’1 = 0.
(a) 2 (b) 0 (c) 6 (d) -2 (e) -6
4. Find values of bsuch that the vectors <โˆ’11, b, 2>and < b, b2, b > are
orthogonal
(a) 0, 3, -3 (b) 0, 11, 3 (c) 0, -11, 2
(d) 0, 2, -2 (a) 0, 11, 2
5. Find the area of the triangle with vertices at the points (0,0,0), (1,0,โˆ’1)
and (1,โˆ’1,2).
(a) โˆš11
2(b) โˆš11 (c) โˆš6 (d) โˆš6
2(e) 1
6. Which vector is always orthogonal to bโˆ’projab.
(a) a(b) b(c) aโˆ’b(d) |a|b(e) projba
7. Find the parametric equations of the intersection of the planes xโˆ’z= 0
and xโˆ’y+ 2z+ 3 = 0.
(a) The line given by x=โˆ’t,y= 3 โˆ’3tand z=โˆ’t.
(b) The line given by x=โˆ’2โˆ’t,y= 1 โˆ’3tand z=โˆ’t.
(c) The line given by x= 1 + t,y= 6 โˆ’tand z= 1 + 2t.
(d) The plane 3x+ 3yโˆ’3z+ 3 = 0.
(e) The line given by x= 1 + t,y= 6 and z= 1 โˆ’t.
8. Find an equation for the normal plane to the vector function
r(t) = hetโˆ’1, t2,cos(1 โˆ’t)i
when t= 1.
1
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MATH 20550: Calculus III Practice Exam 1

Multiple Choice Problems

  1. Find an equation for the line through the point (3, โˆ’ 1 , 2) and perpendic- ular to the plane 2x โˆ’ y + z + 10 = 0. (a) xโˆ’ 2 3 = y โˆ’+1 1 = z โˆ’ 2 (b) x+3 2 = y โˆ’โˆ’ 11 = z โˆ’ 2 (c) xโˆ’ 3 2 = y โˆ’+1 1 = zโˆ’ 2 2 (d) 3x โˆ’ y + 2z + 10 = 0 (e) 3x โˆ’ 2 y + z + 10 = 0
  2. Find an equation of the plane that passes through the point (1, 2 , 3) and parallel to x โˆ’ y + z = 100. (a) xโˆ’y+zโˆ’2 = 0 (b) xโˆ’y+z+2 = 0 (c) x+2y+3z = 100 (d) x โˆ’ 1 = 2 โˆ’ y = z โˆ’ 3 (e) x โˆ’ 1 = y+1 2 = zโˆ’ 3 1.
  3. Find the distance between the point (โˆ’ 1 , โˆ’ 1 , โˆ’1) and the plane x + 2y + 2 z โˆ’ 1 = 0. (a) 2 (b) 0 (c) 6 (d) -2 (e) -
  4. Find values of b such that the vectors < โˆ’ 11 , b, 2 > and < b, b^2 , b > are orthogonal (a) 0, 3, -3 (b) 0, 11, 3 (c) 0, -11, 2 (d) 0, 2, -2 (a) 0, 11, 2
  5. Find the area of the triangle with vertices at the points (0, 0 , 0), (1, 0 , โˆ’1) and (1, โˆ’ 1 , 2). (a)

โˆš 11 2 (b)^

11 (c)

6 (d)

โˆš 6 2 (e) 1

  1. Which vector is always orthogonal to b โˆ’ projab.

(a) a (b) b (c) a โˆ’ b (d) |a|b (e) projba

  1. Find the parametric equations of the intersection of the planes x โˆ’ z = 0 and x โˆ’ y + 2z + 3 = 0.

(a) The line given by x = โˆ’t, y = 3 โˆ’ 3 t and z = โˆ’t. (b) The line given by x = โˆ’ 2 โˆ’ t, y = 1 โˆ’ 3 t and z = โˆ’t. (c) The line given by x = 1 + t, y = 6 โˆ’ t and z = 1 + 2t. (d) The plane 3x + 3y โˆ’ 3 z + 3 = 0. (e) The line given by x = 1 + t, y = 6 and z = 1 โˆ’ t.

  1. Find an equation for the normal plane to the vector function

r(t) = ใ€ˆetโˆ’^1 , t^2 , cos(1 โˆ’ t)ใ€‰

when t = 1.

(a) x + 2y โˆ’ 3 = 0. (b) x + y + z โˆ’ 3 = 0. (c) x = 1 + t, y = 1 + 2t, z = 1. (d) Does not exist since rโ€ฒ(0) = 0. (e) x + 2y โˆ’ 3 = 0 and z = 0.

  1. The equation of the sphere with center (4, โˆ’ 1 , 3) and radius

5 is (a) (xโˆ’4)^2 +(y +1)^2 +(z โˆ’3)^2 = 5 (b) (xโˆ’4)^2 +(y +1)^2 +(z โˆ’3)^2 = 25 (c) (xโˆ’4)^2 +(y+1)^2 +(zโˆ’3)^2 =

5 (d) (x+4)^2 +(yโˆ’1)^2 +(z+3)^2 = 5 (e) (x โˆ’ 4)^2 + (y โˆ’ 1)^2 + (z โˆ’ 3)^2 = 5

  1. Suppose a particleโ€™s position at time t > 0 is described by

r(t) =< t^2 , โˆ’ 2 t, ln t >.

Find the tangential component aT and normal component aN of the ac- celeration at t = 1.

(a) aT = 1, aN = 2 (b) aT = 2, aN = 1 (c) aT = 1, aN = โˆ’ 2 (d) aT = 1, aN =

(e) aT =

5 , aN = 1

  1. Find the volume of the parallelepiped determined by the vectors < 1 , 2 , 7 >, < 0 , โˆ’ 3 , 4 >, and < 0 , 0 , 6 >. (a) 18 (b) 12 (c) 0 (d) 16 (e) 20
  2. Let f (x, y) = esin^ x^ + x^5 y + ln(1 + y^2 ).

Find โˆ‚

(^2) f โˆ‚xโˆ‚y. (a) 5x^4 (b) (^) 1+^2 yy 2 (c) 20x^3 y (d) esin^ x^ cos x (e) esin^ x^ cos x + x^5 + (^) 1+^2 yy 2

Partial Credit Problems

  1. Find the length of the curve

r(t) =< โˆ’ ln t, t^2 , 2 t >, 1 โ‰ค t โ‰ค e.

  1. Find the unit normal vector as a function of t, N(t), where r(t) = ใ€ˆ 3 t, 4 cos(t), 4 sin(t)ใ€‰.