School

University of WaterlooDepartment

Mechanical EngineeringCourse Code

ME201Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**3 pages of the document.** ME 201 ADVANCED CALCULUS

8 February 2006

Midterm Examination R. Culham

•This is a two-hour, closed-book examination, where you are permitted to use:

–Schaum’s - Mathematical Handbook of Formulas and Tables

–one 8.5 in.×11 in. crib sheet. (one side)

•There are 6 questions to be answered. Read the questions very carefully.

•Clearly state all assumptions.

•It is your responsibility to write clearly and legibly.

•Reduce all equations to their simplest possible form.

Question 1 (11 marks)

For the curve given by

~

R(t) = ˆ

i(2 ln t)−ˆ

j 1

t+t!−ˆ

k

ﬁnd:

i) velocity and acceleration vectors

ii) the angle (radians) between the velocity and acceleration vectors at t= 1

iii) the unit tangent vector

iv) the principal unit normal vector

v) the length of the curve over the interval 1≤t≤2

Question 2 (7 marks)

Find the curvature of the path

~

R(t) = ˆ

i(etcos t) + ˆ

j(etsin t) + ˆ

k(et)

when t= 0.

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