CALCULUS

1. Let f(x) = 2x – 1

3x – 1

.

(i) Write down the domain of f. [1]

(ii) Find the composite functions f ? f and f ? f ? f, and their domains. [4]

2. For each of the following, evaluate the limit. (L’Hˆopital’s rule is not allowed.)

(a) lim

x?- 1

2

4x

3 – 3x – 1

4x

4 + 4x

3 + 5x

2 + 4x + 1

. [4]

(b) limx?2

v

x

2 + 1 –

v

2x + 1

v

x

3 – x

2 –

v

x + 2

. [4]

(c) limx?0

(b-xc + b2x + pc + b3xc). [4]

(d) limx?0

sin x + sin 3x + sin 5x

sin 2x + sin 4x + sin 6x

. [4]

3. Prove the following limits using only the precise definition of limits.

(a) limx?-2

x

3 = -8. [4]

(b) limx?-1

1

v

x

2 + 3

=

1

2

. [4]

(c) limx?8

3x + 2

2x + 3

=

3

2

. [4]

4. Let P 6= O be a point on the parabola y = x

2 and Q be the point where the perpendicular

bisector of OP intersects the y-axis. As P approaches the origin O along the

parabola, does Q have a limiting position? If so, find it. [4]

†

5-mark

5. Prove that the equation [3]

x

5 – 1102x

4 – 2015x = 0

has at least three real roots.

6. Using the definition of derivative, prove that [4]

d

dx

1

v

x

2 + 2

= –

x

(x

2 + 2)3/2

.

7. Let [6]

f(x) = (

ax3

cos(1/x) + bx + b, if x < 0,
v
a + bx, if x = 0,
where a, b are positive real constants. Determine the values of a and b, if any, that
make f differentiable everywhere.
8. Find dy
dx and d
2
y
dx2
at (x, y) = (1, -2) if [4]
2x
2 + 2xy + xy2 - 3x + 3y + 7 = 0.
(You may assume that the derivatives exist at (1, -2) without proof.)
9. Find the absolute maximum and minimum values of [4]
f(x) = (x
2 - x - 5)x
2/5
, -1 = x = 2,
and indicate the values of x where they occur.
10. Suppose that limx?a
f(x) = -8 and limx?a
g(x) = c, where c is a constant. Prove that [4]
limx?a
(f(x) + g(x)) = -8.
11. Prove that if the curve y = x
3 + px + q is tangent to the x-axis, then [4]
4p
3 + 27q
2 = 0.
12. Let f be a nonconstant function that is continuous on [a, b], where a < b. Prove that
the range of f is a finite closed interval [c, d], where c < d. [4]
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