1) a) f(x) = 1 – x3

f ׳(x) = - 3x2

=2

3

2 3

3 1

xdx

x− −

−∫

= ( )1

3 24

13

x C− − +

b) dV

dt=10

A = 4πr2

dA

dr= 8πr

3

2

4

3

4

V r

dAr

dr

π

π

=

=

2 11

10 2.54

dA dV dA

dt dt dV

m s−

= ×

= × =

c)

Area= xdy∫

=

4 2

2

4

2 4

y ydy

−

+−

∫

=

4 42 3

2 2

1 14

2 2 4 3

y yy

− −

+ −

= 9 unit2

2. y = x2 + 2 ln xy

2

2dy dy

x x ydx xy dx

= + +

2 2

1 2dy

xy dx x

− = +

12

21

xdy x

dx

y

+

=

−

when x = y =1

4dy

dx= −

Further differentiation

2

2

21 2ln 2 2ln

d y dy dyy x

y dx dxdx

− + − = +

when x = y = 1

2

232

d y

dx=

3. 6x + 16 = A(2x + 4 ) + B

A= 3, B = 4

6x + 16 = 3( 2x + 4 ) + 4

( )

1

2

2

1

2

2

1 1

2 2

2 2

1

2 1 1

2

2

6 16

4 13

3 2 4 4

4 13

3(2 4) 4

4 13 4 13

1 23[ln( 4 13)] 4 tan

3 3

43ln 2

3 4

3ln 23

xdx

x x

xdx

x x

xdx dx

x x x x

xx x

π

π

−

−

− −

−

−

−

+=

+ +

+ +=

+ +

+= +

+ + + +

+ = + + +

= +

= +

∫

∫

∫ ∫

4. ( )2

tan secy x x= +

= ( ) ( )22 tan sec sec tan secdy

x x x x xdx

= + +

= 2 22sec 2 tan sec tan secx x x x x + +

= ( )2

2sec tan secx x x+

2

cos

dy y

dx x=

cos 2dy

x ydx

= (shown) *

From *

( )2

2cos sin 2

d y dy dyx x

dx dxdx+ − =

( ) ( ) ( )3 2 2 2

3 2 2 2cos sin sin cos 2

d y d y d y dy d yx x x x

dxdx dx dx dx

+ − − − + − =

3 2

3 2cos cos 2

d y dy d yx x

dxdx dx− =

y(0) = 1

y2 = (0)׳

y4 = (0)׳׳

y10 = (0)׳׳׳

f(x) = 1 + 2x + 2x2 +

5

3x

3 + …

4

2

-2

-4

-5 5 10

( )2

18

4

1

4

dA dr dA

dV dV dr

rr

ππ

= ×

=

=

5. The vertical asymptote, x = -1

2 1lim

1

12

lim1

1

2

x

x

x

x

x

x

→±∞

→±∞

+

+

+

=

+

=

The horizontal asymptote, y = 2.

when x = 0, y = 1

When y = 0, x = - ½

(0, 1), and ( - ½ , 0 )

Let y = 2 1

1

x

x

+

+

x = 2 1

1

y

y

+

+

y = 1

2

x

x

−

−

1 1: , 2

2

xf x x

x

− −∴ → ≠

−

}{

}{

1

1

: , 2

: , 1

f

f

D x x R x

R y y R y

−

−

= ∈ ≠

= ∈ ≠ −

6.

y = ex

xdye

dx=

when x = a, adye

dx=

when x = a, y = ea

The equation of the tangent at the point where x = a is,

y – ea = e

a ( x – a )

y = ea( 1+x – a )

At (0,0),

ea ( 1 – a ) = 0

a = 1

The equation of tangent which passes through origin is

y = e ( 1 + x – 1 )

y = ex

For y = mx to intersect y = ex at two distinct points, m > e.

7.

( )

( )

0

0

0

( )

ln

ln 27

dxk x x

dt

dxkdt

x x

x x kt c

x kt c

= − −

= −−

− = − +

− = − +

∫ ∫

When t = 0, x = 63

c = ln36

when t = 6ln 2, x = 45,

k = 1

6

( )

1

6

1

6

1ln 27 ln 36

6

27 36

27 36 ( )

t

t

x t

x e

x proven

−

−

− = − +

− =

= +

(a) x – 27

=

7

636−

= 11.2°C

The fall of temperature after being left in the room for 7

minutes is 11.2°C

(b) 636

t

e−

= 2

1

6t

e = 18

t = 6 ln 18

t = 17.3 minutes

The time that elapses is 17.3 minutes.

8. y = 225 x−

x 225 x−

0 5

0.5 4.9749

1.0 24

1.5 91

4

2.0 21

2.5 75

4

3.0 4

3.5 51

4

4.0 3

4.5 19

4

5.0 0

6

4

2

-2

-4

-5 5

x = - 1

y = - 1

h = 5 0 1

10 2

−=

Area = ( )1 1

5 2 36.306442 2

+

= 19.40322 unit2

= 19.4

b) underestimated

c) Area =

5

2

0

25 x dx−∫

= ( )2

2

0

5 1 sin 5cos d

π

θ θ θ−∫

= 2

2

0

25 cos d

π

θ θ∫

=

2

0

25(cos 2 1)

2d

π

θ θ+∫

= 2

0

25 sin 2

2 2

π

θθ

+

= 6.25π unit2

(d)

( )21

5 19.40324

3.105

3.11

π

π

π

≈

≈

≈