Section A

Attempt all questions in this section. Each question is worth 10 marks.

A1. The functions f and g are defined by f (x) = and g(x) = x2 + 1.

(a) Calculate f\ (x) and g\ (x).

d

dx

(c) Calculate (i) f\ (g(x)), and then (ii) g\ (x) . f\ (g(x)).

(d) Why should the results from parts (b)(ii) and (c)(ii) be equal?

A2. (a) The function h is defined by h(t) = . Prove that h is injective, and find h-1 .

← dx

(b) Calculate (i)   0 θ sin θ dθ,    (ii)      x2  _ 2x + 2 , using the substitution x _ 1 = tan u.

A3. (a) Give the limit definition of the derivative f\ (x), for a function f (x).

Use it to find f\ (x) when f (x) = 1

(b) Evaluate (i)π(l) ╱ 、 ,  (ii) π(l) ╱ 、 ,  (iii)π(l) ╱ 、 .

A4. (a) Use the quotient rule to show that tan θ = sec  θ, and then determine\2  tan\\ θ .

Hence determine the Taylor polynomial P2 (θ) of degree 2, centred at 0, for tan θ .

(b) Verify the Mean Value Theorem for f (x) = x + on the interval [1/2, 2].

A5. The function u is defined by u(x, y) = x + 3xy2 + x2y3 .

(a) Show that u has a stationary point at (_2, 1), and determine the nature of this point.

(b) The function u has a second stationary point. By considering simultaneously the conditions ∂u/∂x = ∂u/∂y = 0, show that this stationary point necessarily occurs when xy = _2. Hence use the condition ∂u/∂x = 0 to find the location of this second stationary point.

A6. The function g is defined by g(x) = x3 _ 3x4

(a) Find the stationary points of g, and determine their nature.

(b) Sketch y = g(x).

(c) Give the range of g, using appropriate set notation.

(d) Is g odd, even, or neither odd nor even? Justify your assertion.

Section B

Attempt all questions in this section. Each question is worth 20 marks.

B1. The function h is defined by h(x) = 1

(a) Give the domain of h, using appropriate set notation.

(b) Sketch y = h(x), marking the location of the single stationary point.

(c) Prove that it is impossible to write h(x) in the form C + D

x     x2         x3         x + 1 ,

and find the corresponding values of A, B , C and D .

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(e) By treating as an improper integral, evaluate        h(x) dx.

1

Why does this imply that log 2 > 1/2?

B2. The exponential function exp can be defined by the Taylor series (centred at 0)

exp x =

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