a) exists and equals

b) exists and equals

c) does not exist because x – 1 → 0

d) does not exist because the left hand limit is not equal to the right hand limit.

 is

a) 2

b) – 2

c)

d)

For x ε R,  is equal to

a) e

b)

c)

d)

 equals

a) – π

b) π

c)

d) 1

The left hand derivative of f (x) = [x] sinπx at k, k an integer is

a) (k – 1)π

b) (k – 1)π

c) kπ

d)  kπ

Evaluate if sin |x| - |x| is differentiable at x = 0.

a) Yes

b) No

Let f : ℝ → ℝ be such that f (1) = 3 and  then

 equals

a) 1

b)

c)

d)

If  where n is a non–zero real number, then a is equal to

a) 0

b)

c) n

d)

given that  and

a) does not exist

b) is equal to

c) is equal to

d) is equal to 3

Multiple choices

If x + |y| = 2y, then y as a function of x is

a) Defined for all real x

b) Continuous at x = 0

c) Differentiable for all x

d) Such that  for x < 0

Multiple choices

The function

a) continuous at x = 1

b) differentiable at x = 1

c) continuous at x = 3

d) differentiable at x = 3

Let f (x) =   then for all x

a)

b) f is differentiable

c)  is continuous

d) f is continuous

Multiple choices

If f (x) = then

a) f (x) is continuous ∀ x  ℝ

b) f (x) > 0 ∀ x > 1

c) f (x) is continuous but not differentiable ∀ x  ℝ

d) f (x) is not differentiable at two points

Evaluate

a) 0

b)

c)

d) 1

Evaluate

a) 0

b)

c)

d) 1

Evaluate

a) 0

b)

c) 1

d) 2

Find  if f(x) =

a) 0

b)

c)

d)

Evaluate

a) 2asina

b) a²cosa

c) 2asina + a²cosa

d) 2a

If f (x + y) = f (x) + f (y) for all x and y. If the function f is continuous at x = 0 then f is continuous for all x.

a) True

b) False

Determine the values of a, b, c for which the function

is continuous at x = 0

a)

b)

c)

d)

Let f be a twice differentiable function such that  and , . Find h (10) if

h (5) = 1.

a) 0

b) 1

c) 2

d) 4

Let

Determine the function g (x) = f (f(x)) and hence find the points of discontinuity of g if any.

a) g(x) is continuous for all x except x = 1 and x = 2

b) g(x) is continuous for all x except x = 1

c) g(x) is continuous for all x except x = 2

d) g(x) is continuous for all x

Let f(x) =

Discuss the continuity of  on [0, 2]

a)  is continuous for all x  ℝ

b)  is continuous for all x  ℝ except at x = 1

c)  is continuous for all x  ℝ except at x = 1 and x = 2

d)  is continuous for all x  ℝ except at x = 0, x = 1 and x = 2