Question 1. Linear independence/dependence

5 pts

(a) Show that the following vectors are linearly dependent:

v1  =       , v2  =      , v3  =       .

2 subpts

(b) Hence   nd a non-trivial solution to the equation c1 v1 + c2v2 + c3v3  = 0.     3 subpts

Question 2. Linear transformations

10 pts

1. Find the standard matrix of the transformation T : R2  → R2  that corresponds to re ection through the line x2  = 2x1  followed by re ection through the line x2  = 13x1 . Identify the transformation T (it has a simple geometric description).

You are given that the standard matrix for a re ection Ta  through x2  = ax1  is:

   .

3 subpts

2. Find the standard matrix of the transformation T : R3  → R3 that corresponds to the anti-clockwise rotation by an angle  about the x1-axis.

Check that your matrix has the intended eect of leaving all points on the x1-axis invariant under the rotation.                                                                       3 subpts

3. Let T : R5  → R4  be the linear transformation

T(x1 ,x2 ,x3 ,x4 ,x5 ) =

(x1 + 4x4 + 5x5 ,   x1   x3 ,x2 + 2x3 ,x1       2x2       2x3 ).

Write down the standard matrix of this transformation.  Determine whether T is

one-to-one and whether T is onto.                                                              4 subpts

Question 3. Related Rates

4 pts When air expands adiabatically, its pressure P and volume V are related by the equation PV1.4  = c, where c is a constant. Suppose that at a certain instant the volume is 350 cm3 and the pressure is 50 kPa, and the pressure is increasing at a rate of 10 kPa/min.  At what rate is the volume decreasing at this instant?

Question 4. Dierentials

3 pts

The electrical resistance R of a wire is given by

R = kr2            k=constant, r=radius of wire.

Use dierentials to estimate the percentage error in the measured value of r if we want the percentage error in R to be within 1%.

Question 5. Derivatives and Graphs

8 pts

Consider the following function:

g(x) = 2x  3x2/3           for    1  x  8.

(a) Find the open intervals on which the function is increasing  and on which  it is

decreasing;

(b) Find limx→0  g ′ (x), and limx→0+  g ′ (x).

What is the behaviour of the function at the origin?

(c) Find all local and absolute maximum and minimum points;

2 subpts

1 subpts

2 subpts