Unformatted text preview: CME 100
Vector Calculus for Engineers
Prof. Eric Darve Midterm Exam – Wednesday, October 28th, 2009
This is a 1 hour and 15 minute exam. There are 3 pages. Total number of points in the exam: 100.
You can bring a single sheet of paper to the exam but no other written material. You may use
your own scratch paper. Calculators are not allowed.
Show all work. Problem 1
5 points. Find the magnitude of the torque about P if a 50 N force is applied as shown. 50 N
30 deg 40 cm
P
50 cm Problem 2
5 points. Find the equation of the plane through (2,1,0) and parallel to x + 4y − 3z = 1. Problem 3
10 points. Find a parameterization of the curve of intersection of the cylinder x2 + y 2 = 16 and the
plane x + z = 5. Problem 4
10 points. Find the length of the curve:
r(t) = (2t3/2 , cos 2t, sin 2t), 0≤t≤1 Problem 5
We consider a trajectory r(t), the velocity vector v (t), and the acceleration vector a(t).
1. 5 points. Show that the tangential component of the acceleration is equal to:
aT = v (t) · a(t)
v (t) CME 100 2/3 2. 5 points. Show that the normal component of the acceleration is equal to:
aN = v (t) × a(t)
v (t) 3. 5 points. Find the tangential and normal components of the acceleration vector for the
trajectory r(t) = (t, 2t, t2 ).
4. 5 points. Find the curvature at time t for the same trajectory. Problem 6
10 points. Find the local maxima, minima, and saddle points of the function:
f (x, y ) = 3xy − x2 y − xy 2 Problem 7
10 points. A particle of mass m = 5 is moving on the surface given by the equation sin(xyz ) =
x + y + z . At time t, the x coordinate is 1 + t and the y coordinate is t2 . At t = 0, z (0) = −1. Give
1
an expression for the kinetic energy 2 mv 2 of the particle at time t = 0. Problem 8
10 points. The region R in the xy plane is shown below in gray. Evaluate:
x2 dxdy
R y
1
R (1,1) −1 1 x −1 Problem 9
1. 4 points. Write Matlab code to calculate:
1
100 99
i=0 1
i
1 + 100 2. 2 points. What is this sum approximately equal to?
3. 4 points. Write Matlab code to plot the graph of z = x2 + y 2 , for −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1. CME 100 3/3 Problem 10
1. 5 points. Show that for x < 1:
∞ xn = 1 (1 − x)
n=0 2. 5 points. Use the result of Question 1 to prove that:
1
0 1
0 ∞ 1
dxdy =
1 − xy Note: this result can be used to prove that n≥1 1/n n=1 2 1
n2 = π 2 /6. ...
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This note was uploaded on 08/24/2011 for the course CME 100 at Stanford.
 '08
 DARVE,KHAYMS

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