exercise module 1 derivatives

EXERCISE: THE DERIVATIVES

1.

2.

3.

Find the following limits: 4x 2 − x a) lim x →− 2 x 3 − 5

b)

lim

1 x→ 2

6x − 3 x(1 − 2x )

2   x + 2 ; x  −2    Given a piecewise function: f ( x ) =  x3 − 4 ; − 2  x  2   4   x −1 ; x  2  a) Find the limits of f(x) at x = –2, x = 2 and x approaches ∞. b) Determine the continuity of f(x) at x = 2.

Determine the continuity of g(x) at x = –5 and x = –3 given  x + 3 , − 5  x  −3  g ( x ) = 0 , x = −3  2 9 − x , x  −3

4.

By using the first principles, find the differentiation of the function: f ( x ) = x − 

5.

Find dy/dx: a)

y = 2x + e − 32x + tan−1 2x

b)

y = ( 6 − 3x ) (1 + x )

c)

3x 2 y = sin   +  x  7x + 5

d)

y=

e)

y = ln (ln2x )

f)

2

2

x2 − x x2 + 1

3y3 x + ( x + y ) = 3 − 2x 3

ALL RIGHTS RESERVED MOOC MAT099

1

EXERCISE: THE DERIVATIVES

ANSWERS:

1.

2.

a)

0

b)

–6

a) b)

3.

lim f ( x ) does not exist ; lim f ( x ) = 4 ; lim f ( x ) = 0

x →−2

x→2

x →

f ( x ) continuous at x = 2

g ( x ) discontinuous at x = −5 g ( x ) continuous at x = −3

4.

f ' ( x) =

5.

a) b) c)

1

2 x− dy 1 2 = − 32x ( 2ln3 ) + dx 2x + e 1 + 4x2 dy = 2 6 + 3x − 3x 2 ( 3 − 6x ) dx dy 2 15 2 = − 2 cos   + dx x x   ( 7x + 5 )2

(

)

d)

dy x 2 + 2x − 1 = 2 dx x2 + 1

e)

dy 1 = dx x ln 2x

f)

3 dy −2 − 3y − 3 ( x + y ) = 2 dx 9xy 2 + 3 ( x + y )

(

)

2

ALL RIGHTS RESERVED MOOC MAT099

2