Midterm Solutions

1. (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 =

2. (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 d2s/dt2 = 0. Setting aside the case ds/dt = 0 as trivial (the particle is not moving), we conclude that d2s/dt2 = 0, i.e. ds/dt = const (the speed is constant).

3. (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.

4. (20 pts) Find for

1. f = e-x sin (x + y)

2. f = 1 + xy2 + ln y

Solution

5. (20 pts) Find the directions in which the function

1. increases most rapidly at the point (1, 1);

2. 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

1. The most rapid increase occurs along the vector v

2. 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