math help

profileMsBiologist
assignment_222222.docx
Home>Mathematics homework help>Algebra homework help>math help

Question 1 [3 + 3 + 3 + 10 + 4 + 4 = 27 marks]

a) Differentiate the following with respect to x:

i)

df/dx = -15x^2

ii)

df/dx= (x^0.5+4)/2x

b) Functions are not differentiable at: x=0

(1)

Y=x/|x| (function “jumps”)

(2)

Y=|x| (V – form)

(3)

Y=x^-1 (function is unbounded and goes to infinity)

(4)

Y=x^-3 (derivative is infinite)

(c) Construct the graph of the derivative for the following function.

y = f(x)

(d) If possible, differentiate with respect to x:

(i)

Df/dx = (2x+1)*exp(2x)

(ii)

-(sinx-xcosx)/x^2

(iii)

8 x^3*sec^2(x^4)*tan(x^4)

(iv)

Tan(x)

(e) Use implicit differentiation to find if .

dy/dx = 3x^2/(2*(-2y^3+cosy))

(f) Use logarithmic differentiation to find if .

dy/dx = 3x^2

Question 2 [5 + 5 + 5 = 15 marks]

a) Use to determine whether f(x) = |x| is

differentiable at x = 0.

b) Find the derivative of y = sin x from first principles.

You may assume .

c) Show that . You may assume the differentiation result for the sine function.

Question 3 [3 + 4 = 7 marks]

a) Find in terms ofand if .

b) Find the value of the constants a and b to ensure the following function is differentiable for all real values of x:

Question 4 (8 + 5 + 3 + 4 + 3 = 23 marks)

a) Find for the following (give reasons if not possible):

i)

ii)

iii)

b) Oil spilled from a ruptured tanker spreads in a circle whose radius increases at a constant rate of 6 km2/hour. How fast is the radius of the spill increasing when the area is 9 km2.

c) Use the formula to approximate .

d) Let . Find and if x = 1 and .

e) Find all of the values of x for which is concave down.

Question 5 (6 + 4 + 3 + 4 = 17 marks)

a) If possible, find the absolute maximum and minimum values of the function on the interval [0, 4].

b) Consider which is continuous on the interval [0, 2] and differentiable on the interval (0, 2). If possible, find all the values of c which are satisfied by the Mean Value Theorem on [0, 2].

.

c) Determine the sign of the first and second derivatives at each of the points A, B, and C in the following graph of

Point A

Point B

Point C

Sign of

Sign of

d) The graph below shows the derivative of Determine the values of x, if any, where f has:

Values of x

Relative minima

Relative maxima

Inflection points

Question 6 (3 + 5 + 3 = 11 marks)

a) Suppose that and for all values of x. What is the largest possible value for

b) For what values of the constants a and b is (1,3 ) a point of inflection of the curve ? Give reasons.

c) i) Draw a sketch of a function that is concave down.

ii) Explain why the second derivative is negative for a function that is concave down.

Page 3 of 4

(

)

x

ln

x

x

x

f

2

+

=

x

xe

y

2

=

x

x

sin

y

=

(

)

4

2

x

tan

y

=

(

)

x

cos

ln

y

1

-

=

dy

dx

y

sin

y

x

2

4

3

=

+

x

x

y

2

=

(

)

0

()()

'lim

h

fxhfx

fx

h

®

+-

=

(

)

(

)

00

sincos1

lim1,lim0

hh

hh

hh

®®

-

==

1

2

1

sin

1

d

x

dx

x

-

=

-

'

h

'

f

'

g

(

)

(

)

(

)

(

)

cos2

hxfgx

=

2

,2

()

1,2

axbx

fx

xx

+£

=

+>

ì

í

î

dy

dx

ln2

yx

=+

(

)

ln1)ln(1

yxx

=-+-

2

x

yx

=

()()()'()

fxfaxafa

»+-

1

9

.

2

4

yx

=

dy

y

D

0.1

dxx

=D=

32

32

yxxx

=-+

432

31618

yxxx

=-+

3

()

fxxx

=-

().

yfx

=

'

y

"

y

().

yfx

=

(

)

3

75

fxx

=-

(1)2

f

-=

'()6

fx

£

(1)?

f

32

yaxbx

=+

()

yfx

=