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First Semester 2022

Math1013, Mathematics and Applications 1

Assignment 2

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 = 1 3x1 . 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 e ect 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. Di erentials

3 pts

The electrical resistance R of a wire is given by

R = k r2 k=constant, r=radius of wire.

Use di erentials 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

(d) Find any in ection points, and open intervals where the function is concave up, and open intervals where it is concave down. 2 subpts

(e) Sketch the graph of the function (hand-drawn is preferred but a computer generated graph is okay). 1 subpts