

Midterm Solutions
 (20 pts) Find the length of the portion of the helix
r(t) = (4 cos t)i + (4 sin t)j + 3tk
corresponding to the interval 0 t /2.
Solution
Arc length = ^{}
Arc length = ^{}
 (20 pts) Can anything be said about the speed of a particle whose acceleration is always orthogonal to its velocity? Give reasons for your answer.
Solution
The particle has velocity V and acceleration A given by
^{} and ^{}.
This implies ds/dt = 0 or d^{2}s/dt^{2} = 0. Setting aside the case ds/dt = 0 as trivial (the particle is not moving), we conclude that d^{2}s/dt^{2} = 0, i.e. ds/dt = const (the speed is constant).
 (20 pts) Show that the function
has no limit as (x, y) approaches (0, 0).
Solution
As the point (x, y) approaches (0, 0) along y=kx, then
This limit depends on k and therefore the function has no limit at the origin.
 (20 pts) Find _{} for
 f = e^{x} sin (x + y)
 f = 1 + xy^{2} + ln y
Solution

 (20 pts) Find the directions in which the function
 increases most rapidly at the point (1, 1);
 decreases most rapidly at the point (1, 1).
What is the direction of the level curve
f(x, y) = const = f(1, 1)
at this point?
Solution
 The most rapid increase occurs along the vector v
 The most rapid decrease occurs along the vector v
The level curve has the direction
perpendicular to v.
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Created by Henry Fink
Last updated: Sunday, September 28, 1997

