1.      i)  Evaluate each limit, giving brief reasons for your answer.

2ex + cosx

x→∞ 6ex − sin x

x→0 x2  − 3x

ii)  Evaluate each of the following integrals:

a)  Z cos x sin5 xdx,

b)  Z  dx.

iii)  The   oor of x is denoted ⌊x⌋ and gives the greatest integer less than or equal to x. For example, ⌊3.1⌋ = 3.

Let f : R → R be de  ned by

f(x) = ⌊3x − 1⌋ .

a)  Write down the value of f(12).

b)  By considering left and right limits, state whether or not f is con- tinuous at x = 13 .

√2     √2 .

a)  Write z in polar form.

b)  Find the smallest positive integer n such that zn  = 1.

c)  Find all possible positive integers m such that zm  = 1.

v)  Consider the following Maple output >     with(LinearAlgebra):

>      A := <<607,207,75>|<-2286,-783,-288>|<1414,483,176>>;

 607   − 2286   1414

A  :=     207    − 783    483

>      A^2;

  1297   −4896    3024  

− 207     783      −483

>      A^3;

 607   − 2286   1414

 207    − 783    483  

a)  Find A100 .

b)  Is A invertible? Give reasons for your answer.

vi)   a)  Express sin  in terms of ei  and e −i .

b)  Find constants a, b and c such that

sin4   = acos4 + bcos2 + c.

c)  Hence or otherwise evaluate

Z

2.      i)  Find the gradient dydx of the curve de  ned by xy + y2  = 4 at the point where y = 1.

ii)   Let f : R → R be given by f(x) = 3x − cos x.

a)  Show that f has a zero in the interval   0, 2 .

b)  What is the minimum value of f′ ?

c)  Explain why f has an inverse on R.

d)  Find the value of g′ ( − 1), where g is the inverse of f .

iii)  Sketch the graph of the polar curve r = 2 − cos2 for 0 ≤  ≤  .

iv)  For some values of the real parameters a, b, c and d, the curve ax2 + by2 + cx + dy = 1

passes through the points A(1, 1),B(2, 3) and C(0, 1).

a)  Explain why the following equations can be used to determine the values of a, b, c and d for which the curve passes through the points.

a    +   b    +   c    +   d    =   1

4a   +   9b   +   2c   +   3d   =   1

b                +   d    =   1.

b)  Use Gaussian Elimination to solve the system of linear equations in part (a).

c)  Are there zero, one, or in  ntely many curves of the form

ax2 + by2 + cx + dy = 1 which pass through the points A, B and C? d)  Using your answer from part (b),   nd the parabola of the form

y = x2 +   x +   which passes through A, B and C .

v)   a)  Write    as a linear combination of  6   and  2 .

3.      i)   a)  Clearly state the Mean Value Theorem.

b)  Apply the Mean Value Theorem to f : R → R given by f(t) = e2t − t

x,

e2x  ≥ 2x + 1.

ii)   a)  De  ne the hyperbolic function cosh x in terms of exponentials.

b)  Use the de  nition to prove that

cosh2x = 2 cosh2 x − 1.

iii)  Evaluate the integral  xlnxdx.

iv)  The point (x(t),y(t)) is moving clockwise on the circle x2 + y2  = 100.

dy        x dx

dt         y dt .

b)  If the velocity in the x direction at x = 6 is 2 units/sec,   nd the

velocity in the y direction.

v)  Consider the function f de  ned for all real x by f(x) = 0 x  dt.

a)  Find ddx  f(x3 )  .

b)  Explain why f is an even function.

c)  By considering the improper integral,  1 ∞  dt,  explain why lim f(x) exists.

x→∞

vi)  The Maple output below shows a calculation and then the upper Rie- mann sum for the function f : (0, ∞ ) → R, given by f(x) = lnx, on the interval [1, 6] using the partition shown. The value of the upper Riemann sum is approximately 6.181297752.

Find, to two decimal places, the value of the corresponding lower Rie- mann sum.

> [seq(i*0.5, i = 2..12)];

[1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0] > map(ln,%);

[0., .4054651081, .6931471806, .9162907319, 1.098612289, 1.252762968,

1.386294361, 1.504077397, 1.609437912, 1.704748092, 1.791759469]

> with(Student[Calculus1]):

> RiemannSum(ln(x), x = 1..6, method = upper, output = plot);

4.      i)  Let a =  4(1)   and b =  5(5)  be two vectors in R2 .

a)  Calculate projb (a).

b)  Sketch the vectors a, b and projb (a) in the plane.

ii)  Calculate the inverse of the matrix A =   

0   0   1

a)  Find DC .

b)  What is the size of CD?

iv)  Let u =     and v =     .

a)  Calculate u × v .

b)  Hence or otherwise   nd          60(0)0

v)  Let P1  and P2  be the planes in three dimensional space with Cartesian equations 12x + 9y + 3z = 75 and z = 51 − 4x − 3y respectively.

a)  Explain why P1  and P2  are parallel.

b)  Show that (5, 1, 2) is a point on P1 .

c)  It is given that the line L with parametric vector equation

    =     +         ;     ∈ R,

passes through the point (5, 1, 2) and is perpendicular to both P1 and P2 . Determine where the line L meets the plane P2 .

d)  Hence, or otherwise,   nd the shortest distance between P1  and P2 .

e)  Find a point Q with the property that the set of all points equidistant from (5, 1, 2) and Q is the plane P2 .

vi)   Suppose  that  A  and  B  are  two  n  × n  matrices  and  that  AB − A  is

invertible.  Prove that BA − A is also invertible.