MATH 230, CALCULUS 3 - Spring 2018 MIDTERM EXAMINATION I … · Math 230 - Exam 1 Spring 2018 2. (8...

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MATH 230, CALCULUS 3 - Spring 2018MIDTERM EXAMINATION I

FEBRUARY 6, 2018

INSTRUCTIONS

• PLEASE, WRITE YOUR SOLUTIONS CLEARLY AND COMPLETELY.ANSWERS WITHOUT WORK WILL BE GIVEN NO CREDIT.Please, BOX your final answer.

• Unless otherwise stated, you CAN use the results discussed in class or in thetextbook.

• You may not use CALCULATORS, BOOKS, or PERSONAL NOTES.

• The use of mobile phones and other electronic devices is not permitted andall such devices must be turned off before entering the examination room andremain off for the duration of the examination. Remove any smartwatches.

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Math 230 - Exam 1 Spring 2018

1. (a) (4 points) Find the vector pointing in the same direction as the vector 〈4, 3,−5〉with length 3.

(b) (4 points) Find the equation of the sphere centered at (1,−2, 3) containing thepoint (−1, 1, 4).

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2. (8 points) For the following two parts, consider the vectors a = i − 4 j + 5k andb = 2 i + 2 j− k.

(a) (5 points) Compute the vector projection of a onto b.

(b) (3 points) Calculate the angle between a and b.

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3. (8 points) For the following two parts, consider the points P (5, 1, 0), Q(6, 2,−1),and R(4, 0, 0).

(a) (5 points) Compute the area of the triangle with vertices P,Q, and R.

(b) (3 points) Is the vector 〈1, 1, 1〉 parallel to the plane containing P, Q, and R?

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4. (12 points) The planes y − z = 1 and x + 2z = 3 intersect in a line L.

(a) (7 points) Find parametric and symmetric equations of L.

(b) (5 points) Find the equation of a third plane that contains L and is perpen-dicular to the plane x + y − 2z = 3.

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5. (8 points) Find the distance from the point P (−1, 1, 2) to the plane 2x− y + z = 5.

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6. (10 points) Consider the points P (1, 3, 2), Q(2, 4, 3), and R(0, 3, 0).

(a) Find a vector equation of the line containing the points P and Q.

(b) Find the equation of the plane containing the points P , Q, and R.

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7. (12 points) Match the following equations with their picture, and give the name ofeach surface.

A B C

Equation Picture Name of Surface

2x2 + y2 − z2 = 4

x2 + 9y2 − z2 = 0

x2 + 9y2 + 9z2 = 1

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8. (8 points) Find parametric equations for the tangent line to the curve r (t) =⟨2 sin t, 2 cos t, 4 sin2 t

⟩at the point

(1,√

3, 1).

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9. (8 points) Find the arc length of the part of r (t) = 〈1, t2, t3〉 between the points(1, 0, 0) and (1, 4, 8).

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10. (8 points) Find the curvature of r (t) =

⟨t, t2,

t3

4

⟩at (2, 4, 2) .

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11. (10 points) A particle travels with acceleration a (t) = 〈− cos t, e2t,−5〉 with aninitial velocity of v (0) = 〈0, 3, 1〉 and an initial position of r (0) = 〈−1, 3, 2〉. Findthe position and velocity function of this particle.

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MATH 230, CALCULUS 3 - Spring 2018 MIDTERM EXAMINATION I … · Math 230 - Exam 1 Spring 2018 2. (8...

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