DEPARTMENT OF MATHEMATICS UNIVERSITY OF LIBERIA
FENDALL CAMPUS, MONROVIA, LIBERIA CALCULUS III (MATH 301) FINAL EXAMINATION
FIRST SEMESTER ACADEMIC YEAR 2016/2017
INSTRUCTION: THIS PAPER CONTAINS TEN (10) QUESTIONS. YOU ARE ALLOWED TO ANSWER ALL QUESTIONS. EACH QUESTION CARRIES TWELVE (10) MARK. THE TIME ALLOCATED FOR THIS PAPER IS TWO (2) HOURS. SOLVE EACH PROBLEM IN A STEP – STEP AND ACCURATE MANNER FOR FULL CREDIT. CALCULATORS ARE ALLOWED BUT UNNECESSARY.
1. Find the following limit if it exists.
(a)
(b)
2. Proof the following Mathematical Theorems.
For two nonzero vectors , in , if is the angle between , . Show that
3. Find the velocity and position of an object at any time , given its acceleration
, its initial velocity is and its initial position is
4. Evaluate the following vector integrals
5. Find the direction ratio and cosine of a vector that is perpendicular
to both and .
6. Define what is meant by the term “a vector valued –function”?
8. Show that the curve of a path traced out by the vector – value function is given by
where, is the curvature of the curve and are the first and second derivatives of the vector - value function.
9. Let be the angle between two nonzero vectors . Show that
10.If are continuous on the interval , then the arc length to the curve
on that interval is given by
Consider a curve defined parametrically by , and , where
are all continuous for . If the curve is traversed exactly once as increases from . Write down the formula for the arc length if the curve in 3D is traced out by the endpoint of the vector valued – function
