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Practice exams, Exams of Calculus

Minnesota State University Moorhead (MSUM)Calculus

Practice Final, Spring 2013, Cheat Sheet.

Typology: Exams

2018/2019

Uploaded on 02/11/2022

ekansh
ekansh 🇺🇸

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Calculus III: Practice Final
Name:
Circle one: Section 6 Section 7 .
•Read the problems carefully.
•Show your work unless asked otherwise.
•Partial credit will be given for incomplete work.
•The exam contains 10 problems.
•The last page is the formula sheet, which you may detach.
•Good luck!
Question: 1 2 3 4 5 6 7 8 9 10 Total
Points: 10 10 10 10 10 10 10 10 10 10 100
Score:
1
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Practice exams and more Exams Calculus in PDF only on Docsity!

Calculus III: Practice Final

Name:

Circle one: Section 6 Section 7.

  • Read the problems carefully.
  • Show your work unless asked otherwise.
  • Partial credit will be given for incomplete work.
  • The exam contains 10 problems.
  • The last page is the formula sheet, which you may detach.
  • Good luck!

Question: 1 2 3 4 5 6 7 8 9 10 Total Points: 10 10 10 10 10 10 10 10 10 10 100 Score:

  1. (10 points) Circle True or False. No justifation is needed.

(a) The curve traced by 〈cos^2 (t), sin^2 (t)〉 is a circle. True False

(b) The plane 3x + 2y − z = 0 is perpendicular to the line x = 3t, y = 2t, z = −t. True False

(c) The function f (x, y, z) =

{sin(x+y+z) x+y+z if^ x^ +^ y^ +^ z^6 = 0 1 if x + y + z = 0 is continuous at (0, 0 , 0). True False

(d) If the acceleration is constant, then the trajectory must be a straight line. True False

(e) The complex number e2+3i^ has magnitude 2. True False

  1. Use the contour plot of f (x, y) to answer the questions. No justification is needed.

(a) (3 points) Mark any three critical points of f. Label them A, B, and C. Identify whether they are local minima, local maxima or saddle points.

(b) (2 points) Draw a vector at (1, 1) indicating the direction of ∇f at (1, 1). (c) (3 points) Determine the sign of

  1. ∂f∂x (3, 4):
  2. ∂f∂y (2, 3):
  3. Duf (5, 3) where u is the South–East direction: (d) (2 points) Give a (admittedly rough) numerical estimate of ∂f∂x (1, 1).
  1. (a) (5 points) Write parametric equations for the tangent line at 〈 1 , 0 , 1 〉 to the curve traced by 〈t^2 , ln t, t^3 〉.

(b) (5 points) Write an equation of the normal plane to the curve at the same point.

  1. A ball of unit mass is thrown with the initial velocity of i + j. It experiences the force of gravity of magnitude 10 units in the −j direction and a force due to the wind of magnitude 1 unit in the i direction. Suppose the ball is initially at (0, 4). (a) (7 points) Find the position of the ball at time t.

(b) (3 points) Where is the ball when it hits the ground?

  1. Suppose u = exy^ where x = st + s + t and y = st − s − t.

(a) (2 points) Find the value of u when s = 2 and t = 2.

(b) (8 points) Find an approximate numerical value of u when s = 2.01 and t = 1.98.

  1. (10 points) A race track is in the shape of an ellipse with minor radius 1 km and ma- jor radius 2 km as shown. A car is going along this track at a constant speed of 100 km/h. Find the tangent and normal com- ponent of its accelaration when it as at P.

Race track

P

  1. (10 points) You want to design a cylindrical cup that can hold 100π ml coffee. To minimize the material to be used, you decide to minimize the surface area. What is the radius and height of the optimal cup? (Ignore the thickness of the walls.)