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Evaluate limx→14x+1x-1 Unit Objective 5 Does not exist 0 4 Determine the limx→4x2-2 2 6 14 Does not exist What is the value of the limx→2x2-4x-2 4 Does not exist 6 0 limx→33x2+2x-8 is equal to 6 5 4 3 Evaluate limx→∞x2-5x+15x2-3x+7 1/5 ∞ 0 5 limx→02x2-3xx is equal to Unit Objective 3 0 -3 Does not exist Use the following function to answer questions 7 and 8 y=3x-x2x-22 Determine the horizontal asymptote of the function above y=1 y=3 y=2 y=-1 Determine the vertical asymptote of the function above x=0 x=4 x=3 x=2 Find the horizontal asymptote of fx=x-1x+1 y=-1 y=1 y=2 there is no horizontal asymptote Find the vertical asymptotes of fx=x2-2x2-4 x=2 x=-2 x=-2 and x=2 x=4 A function is continuous at a point a if what condition(s) is/are satisfied f(a) is defined limx→af(x) exists limx→af(x)=f(a) All of the above Determine the value(s) of x for which fx=2x2-x-3x2-x-2 is not continuous Unit Objective x=2 and x=-1 x=-2 and x=-1 x=-2 and x=1 x=2 and x=1 Let f be defined by the following: fx=x3 x≤2ax2 x≥2 For what value of a is the function continuous for all real values of x ? 1 2 3 All of the above What are the set of values for x , for which the function fx=x+1x2-3x-10 is discontinuous? x=5 and x=2 x=-5 and x=-2 x=5 and x=-2 x=-5 and x=2 Find the value(s) of x for which the function fx=x x<1x2 x>1 is discontinuous? 1 2 3 Continuous for all values of x When differentiating from first principle the derivative of f(x) is given by: limh→0fx+h-f(x)h limh→0fx+h+f(x)h limh→0fx+h-f(x-h)h limh→0fx-h-f(x)h Let fx=3x3+4x2-2x+10

Math May 03, 2023

Evaluate limx→14x+1x-1

Unit

Objective

5
Does not exist
0
4

Determine the limx→4x2-2

2
6
14
Does not exist

What is the value of the limx→2x2-4x-2

4
Does not exist
6
0

limx→33x2+2x-8

is equal to

Evaluate limx→∞x2-5x+15x2-3x+7

1/5
∞
0
5

limx→02x2-3xx

is equal to

Unit

Objective

3
0
-3
Does not exist

Use the following function to answer questions 7 and 8

y=3x-x2x-22

Determine the horizontal asymptote of the function above

1. y=1
2. y=3
3. y=2
4. y=-1

Determine the vertical asymptote of the function above

x=0
x=4
x=3
x=2

Find the horizontal asymptote of fx=x-1x+1

1. y=-1
2. y=1
3. y=2
4. there is no horizontal asymptote

Find the vertical asymptotes of fx=x2-2x2-4

x=2
x=-2
x=-2 and x=2
x=4

A function is continuous at a point a

if what condition(s) is/are satisfied

f(a) is defined
limx→af(x)

exists
limx→af(x)=f(a)
All of the above

Determine the value(s) of x

for which fx=2x2-x-3x2-x-2

is not continuous

Unit

Objective

x=2 and x=-1
x=-2 and x=-1
x=-2 and x=1
x=2 and x=1

Let f

be defined by the following:

fx=x3 x≤2ax2 x≥2

For what value of a

is the function continuous for all real values of x

1
2
3
All of the above

What are the set of values for x

, for which the function fx=x+1x2-3x-10

is discontinuous?

x=5 and x=2
x=-5 and x=-2
x=5 and x=-2
x=-5 and x=2

Find the value(s) of x

for which the function fx=x x<1x2 x>1

is discontinuous?

1
2
3
Continuous for all values of x

When differentiating from first principle the derivative of f(x) is given by:

limh→0fx+h-f(x)h
limh→0fx+h+f(x)h
limh→0fx+h-f(x-h)h
limh→0fx-h-f(x)h

Let fx=3x3+4x2-2x+10.

What is f'x

?

1. 9x2+8x+2
2. 3x2+4x-2
3. 9x2+8x-2
4. 9x2-8x+2

Let fx=3x

. What is f'(x)

?

Unit

Objective

-32x3
32x32
3(x)3
-3(x)3

Differentiate y=2x3

with respect to x

.

1. 3x3
2. 2x2
3. 2x
4. 3x

Differentiate y=3x-63

with respect to x

2
1
-2
0

SECTION B

Instructions: Answer any TWO(2) questions from this section.

Unit

Objective

a) Determine the limit of the following functions:

i. limx→5x-5x2-25

[5 marks]

ii. limh→0x+h-xh

[5 marks]

iii. limx→∞x2-5x+15x2-3x+7

[5 marks]

b) Find the vertical and horizontal asymptotes of the following functions:

i. fx=3xx2-x-6

[5 marks]

ii. fx=1-4x-2x2

[4 marks]

c) Find the constants a

and b

so that the function is continuous on the entire real line

fx=2 x≤-1 ax+b -1-2 x≥3

[6 marks]

a) Differentiate fx=3x2-4x

from first principles [6 marks]

b) Differentiate the following with respect to x

i. fx=4x+62x+3

[6 marks]

ii. fx=3x-x

[4 marks]

c) Find the equation of the tangent to the curve x3-3xy2+y3=1

at the point (2, -1)

[8 marks]

d) Find dydx

in terms of t

for the curve parametrically defined by x=3t+4

and

y=6t3-3

. Hence find the point on curve at which the gradient is zero. [6 marks]

a) Find the following:

i. 3x-x dx

[3 marks]

ii. 4x+12 dx

[3 marks]

iii. -3-11x2-1x3 dx

[7 marks]

b) By using the substitution u=1-x

Evaluate 01x1-x dx

[8 marks]

c) A region is bounded by the curve y=x2,

the lines x=1

and x=2

and y=3x.

Find the area enclosed by this region. [9 marks]

a) Air is being pumped into a spherical balloon at a rate of 8 cm3s-1

. Find the rate of

change of the radius of the balloon when the radius of the balloon is 10 cm.

Volume of a sphere is 43πr3

[10 marks]

b) Find the equations of the tangent and the normal to the curve y=2x2-2x-42

at the

point (3, 2). [10 marks]

c) Determine the constants a

and b

so that the function fx=x3+ax2+bx

have

stationary points when x=-1

and x=3.

Determine also the nature of these stationary

points.

Instructions: On the computerized answer sheet provided, shade the letter that corresponds with the most appropriate response for each of the following.

Evaluate limx→14x+1x-1

Unit

Objective

5
Does not exist
0
4

Determine the limx→4x2-2

2
6
14
Does not exist

What is the value of the limx→2x2-4x-2

4
Does not exist
6
0

limx→33x2+2x-8

is equal to

Evaluate limx→∞x2-5x+15x2-3x+7

1/5
∞
0
5

limx→02x2-3xx

is equal to

Unit

Objective

3
0
-3
Does not exist

Use the following function to answer questions 7 and 8

y=3x-x2x-22

Determine the horizontal asymptote of the function above

1. y=1
2. y=3
3. y=2
4. y=-1

Determine the vertical asymptote of the function above

x=0
x=4
x=3
x=2

Find the horizontal asymptote of fx=x-1x+1

1. y=-1
2. y=1
3. y=2
4. there is no horizontal asymptote

Find the vertical asymptotes of fx=x2-2x2-4

x=2
x=-2
x=-2 and x=2
x=4

A function is continuous at a point a

if what condition(s) is/are satisfied

f(a) is defined
limx→af(x)

exists
limx→af(x)=f(a)
All of the above

Determine the value(s) of x

for which fx=2x2-x-3x2-x-2

is not continuous

Unit

Objective

x=2 and x=-1
x=-2 and x=-1
x=-2 and x=1
x=2 and x=1

Let f

be defined by the following:

fx=x3 x≤2ax2 x≥2

For what value of a

is the function continuous for all real values of x

1
2
3
All of the above

What are the set of values for x

, for which the function fx=x+1x2-3x-10

is discontinuous?

x=5 and x=2
x=-5 and x=-2
x=5 and x=-2
x=-5 and x=2

Find the value(s) of x

for which the function fx=x x<1x2 x>1

is discontinuous?

1
2
3
Continuous for all values of x

When differentiating from first principle the derivative of f(x) is given by:

limh→0fx+h-f(x)h
limh→0fx+h+f(x)h
limh→0fx+h-f(x-h)h
limh→0fx-h-f(x)h

Let fx=3x3+4x2-2x+10.

What is f'x

?

1. 9x2+8x+2
2. 3x2+4x-2
3. 9x2+8x-2
4. 9x2-8x+2

Let fx=3x

. What is f'(x)

?

Unit

Objective

-32x3
32x32
3(x)3
-3(x)3

Differentiate y=2x3

with respect to x

.

1. 3x3
2. 2x2
3. 2x
4. 3x

Differentiate y=3x-63

with respect to x

2
1
-2
0

SECTION B

Instructions: Answer any TWO(2) questions from this section.

Unit

Objective

a) Determine the limit of the following functions:

i. limx→5x-5x2-25

[5 marks]

ii. limh→0x+h-xh

[5 marks]

iii. limx→∞x2-5x+15x2-3x+7

[5 marks]

b) Find the vertical and horizontal asymptotes of the following functions:

i. fx=3xx2-x-6

[5 marks]

ii. fx=1-4x-2x2

[4 marks]

c) Find the constants a

and b

so that the function is continuous on the entire real line

fx=2 x≤-1 ax+b -1-2 x≥3

[6 marks]

a) Differentiate fx=3x2-4x

from first principles [6 marks]

b) Differentiate the following with respect to x

i. fx=4x+62x+3

[6 marks]

ii. fx=3x-x

[4 marks]

c) Find the equation of the tangent to the curve x3-3xy2+y3=1

at the point (2, -1)

[8 marks]

d) Find dydx

in terms of t

for the curve parametrically defined by x=3t+4

and

y=6t3-3

. Hence find the point on curve at which the gradient is zero. [6 marks]

a) Find the following:

i. 3x-x dx

[3 marks]

ii. 4x+12 dx

[3 marks]

iii. -3-11x2-1x3 dx

[7 marks]

b) By using the substitution u=1-x

Evaluate 01x1-x dx

[8 marks]

c) A region is bounded by the curve y=x2,

the lines x=1

and x=2

and y=3x.

Find the area enclosed by this region. [9 marks]

a) Air is being pumped into a spherical balloon at a rate of 8 cm3s-1

. Find the rate of

change of the radius of the balloon when the radius of the balloon is 10 cm.

Volume of a sphere is 43πr3

[10 marks]

b) Find the equations of the tangent and the normal to the curve y=2x2-2x-42

at the

point (3, 2). [10 marks]

c) Determine the constants a

and b

so that the function fx=x3+ax2+bx

have

stationary points when x=-1

and x=3.

Determine also the nature of these stationary

points. [10 marks]

[10 marks]

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