1.      i)  Calculate each of the following limits, or explain why it doesn’t exist.

3x2 + sinx3

x→∞ x2 + 2x + 1

ii)  Find the following integrals.

a)  I1  = Z x cos xdx

b)  I2  = Z  dx

iii)  Consider the curve de  ned implicitly by

x2 y − xy3 − x2  = −2.

a)  Calculate the value of dydx at the point (2, 1).

b)  Hence write down a Cartesian equation of the tangent line at the point (2, 1).

c)  Express your tangent line from part (b) in parametric vector form.

iv)  Let α be a positive real number. Write the complex number −α + αi in

v)   a)  Solve z6 + 1 = 0.

b)  Factorise z6 + 1 into linear factors with complex coecients.

c)  Factorise z6 + 1 into linear or irreducible quadratic factors with real coecients.

d)  Factorise z6 + 1 into irreducible factors with rational coecients.

2.      i)  Let p : [0, ∞) → R be the function de  ned by     p(x) = ln  x + 19 + e −x .

a)  Show that p has at least one root in the interval [0, 1].

b)  Show that p has exactly one root in the interval [0, 1].

c)  Explain how your arguments in parts (a) and (b) can be extended to determine the number of roots of p in the interval [0, ∞).

ii)   a)  Write down the formula for sinh(x) and cosh(x) in terms of exponen- tials.

b)  Hence, prove that for n ∈ Z and x ∈ R

cosh(x) + sinh(x)  n  = cosh(nx) + sinh(nx)

iii)  Sketch the graph of the polar curve

r = 1 + cos(3θ)

for 0 ≤ θ < 2π .

iv)  A function f : [1, 5] → R has the following properties

•  f has a global maximum at 3,

•  f is continuous everywhere except 4,

•  f has no global minimum.

Draw a sketch of the graph of a possible f .

(You do not need to give a formula for your function.)

v)  Let

w =   1     1(1)

√6   1  .

You are given that the following vectors form an orthonormal set. (You do not need to prove this.)

v1  =  1 ,    v2  =      v3  =   .

a)  Calculate v1  · w .

b)  Express w as a linear combination of v1 , v2  and v3 .

vi)  Consider the lines ℓ 1  and ℓ2  in R3  de  ned below.

ℓ 1  :                   x =   + λ  1 ,    λ ∈ R

ℓ2  :                   x1  = 4,                =

a)  Show that the lines ℓ 1  and ℓ2  intersect.

b)  Explain the Maple code below and how it is used to   nd the shortest distance from the point P(1, 2, 3) to the plane containing the lines ℓ 1 and ℓ2 .

>     with(LinearAlgebra):

>      AP := <2,0,1> - <1,2,3>;

   1  

−2

>      v1 := <1,2,-1>;

   1   2

−1

 0  2     

>     n := CrossProduct(v1,v2);

   8  

−3

>      l := abs(n.AP)/sqrt(n.n);

l  :=

v)  Using the Maple code below, or otherwise,   nd the determinant of

 0(1)   2(2)   1(1)   1

>     with(LinearAlgebra):

>      B := <<1,0,2,3>|

 1 0

 2

<2,2,4,0>|<1,1,2,1>|<4,-1,-2,0>>;

>      RowOperation(B,[3,1],-2);

 1   2   1     4   

0   2   1    − 1

 0   0   0   − 10

− 1

− 10

− 12

 1   2   1     4    0   2   1    − 1

 0   0   0   − 10

4.      i)  Let f : R → R be the function de  ned by

f(x) = 0 x4  sin(t3 + 1)dt.

(Do NOT try to evaluate this integral.)

a)  Explain why f is an even function.

b)  Find f′ (x).

ii)  Find all real values of a and b such that the function de  ned by, f(x) =

is dierentiable for all R.

iii)   a)  State the Mean Value Theorem.

b)  Prove that, for all x ≥ 0,

1 − x ≤ e −x .

iv)  The function f is de  ned by

f : (0, 2) → R  where  f(x) = e−x (1 − x).

a)  Explain why f has an inverse  .

b)  Find the domain and range of g .

c)  Evaluate g′ (0).

v)  Do the following improper integrals converge or diverge?  Give reasons for your answer.

a)  0 ∞  dx

b)  1 ∞ 4xlnx dx

vi)  Use the ǫ-de  nition of the limit to show that

lim (2 + e−x(sin(x)+2) sin(x)cos(x)) = 2.

x→∞