Homework answers / question archive / 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

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

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Evaluate limx→14x+1x-1

 

Unit

Objective

 

 

1. 5
2. Does not exist                                                                                                                                      
3. 0                                                                                                                                 
4. 4

 

2. Determine the limx→4x2-2

   

 

1. 2
2. 6
3. 14
4. Does not exist

 

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

   

 

1. 4
2. Does not exist
3. 6
4. 0

 

4. limx→33x2+2x-8

   

     is equal to

 

1. 6
2. 5
3. 4
4. 3

 

5. Evaluate limx→∞x2-5x+15x2-3x+7

   

 

1. 1/5

   

2. ∞
3. 0
4. 5

 

 

6. limx→02x2-3xx

   

     is equal to

 

Unit

Objective

 

 

1. 3
2. 0
3. -3

   

4. Does not exist

 

Use the following function to answer questions 7 and 8

y=3x-x2x-22

 

 

7. Determine the horizontal asymptote of the function above

 

1. 1. y=1

      

   2. y=3

      

   3. y=2

      

   4. y=-1

      

 

8. Determine the vertical asymptote of the function above

 

1. x=0

   

2. x=4

   

3. x=3

   

4. x=2

   

 

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

   

 

1. 1. y=-1

      

   2. y=1

      

   3. y=2

      

   4. there is no horizontal asymptote

 

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

    

 

1. x=2

   

2. x=-2

   

3. x=-2  and x=2

   

4. x=4

   

 

 

 

11. A function is continuous at a point a

    

     if what condition(s) is/are satisfied

 

1. f(a) is defined
2. limx→af(x)

   

    exists
3.    limx→af(x)=f(a)

   

4. All of the above

 

12. Determine the value(s) of x

    

     for which fx=2x2-x-3x2-x-2

    

     is not continuous

 

Unit

Objective

 

 

1. x=2 and x=-1

   

2. x=-2 and x=-1

   

3. x=-2 and x=1

   

4. x=2 and x=1

   

 

13. 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. 1
2. 2
3. 3
4. All of the above

 

14. What are the set of values for x

    

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

    

     is discontinuous?

 

1. x=5 and x=2

   

2. x=-5 and x=-2

   

3. x=5 and x=-2

   

4. x=-5 and x=2

   

 

15. Find the value(s) of x

    

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

    

     is discontinuous?

 

1. 1
2. 2
3. 3
4. Continuous for all values of x

   

 

 

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

 

1.     limh→0fx+h-f(x)h

   

2.     limh→0fx+h+f(x)h

   

3.     limh→0fx+h-f(x-h)h

   

4.     limh→0fx-h-f(x)h

   

 

17. Let fx=3x3+4x2-2x+10. 

    

     What is f'x

    

    ?

 

1. 1. 9x2+8x+2

      

   2. 3x2+4x-2

      

   3. 9x2+8x-2

      

   4. 9x2-8x+2

      

 

18. Let fx=3x

    

    .  What is f'(x)

    

    ?

 

Unit

Objective

 

 

1. -32x3

   

2. 32x32

   

3. 3(x)3

   

4. -3(x)3

   

 

19. Differentiate y=2x3

    

      with respect to x

    

    .

 

1. 1. 3x3

      

   2. 2x2

      

   3. 2x

      

   4. 3x

      

 

20. Differentiate y=3x-63

    

     with respect to x

    

 

1. 2
2. 1
3. -2
4. 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.

                                                           

1. Evaluate limx→14x+1x-1

   

 

Unit

Objective

 

 

1. 5
2. Does not exist                                                                                                                                      
3. 0                                                                                                                                 
4. 4

 

2. Determine the limx→4x2-2

   

 

1. 2
2. 6
3. 14
4. Does not exist

 

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

   

 

1. 4
2. Does not exist
3. 6
4. 0

 

4. limx→33x2+2x-8

   

     is equal to

 

1. 6
2. 5
3. 4
4. 3

 

5. Evaluate limx→∞x2-5x+15x2-3x+7

   

 

1. 1/5

   

2. ∞
3. 0
4. 5

 

 

6. limx→02x2-3xx

   

     is equal to

 

Unit

Objective

 

 

1. 3
2. 0
3. -3

   

4. Does not exist

 

Use the following function to answer questions 7 and 8

y=3x-x2x-22

 

 

7. Determine the horizontal asymptote of the function above

 

1. 1. y=1

      

   2. y=3

      

   3. y=2

      

   4. y=-1

      

 

8. Determine the vertical asymptote of the function above

 

1. x=0

   

2. x=4

   

3. x=3

   

4. x=2

   

 

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

   

 

1. 1. y=-1

      

   2. y=1

      

   3. y=2

      

   4. there is no horizontal asymptote

 

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

    

 

1. x=2

   

2. x=-2

   

3. x=-2  and x=2

   

4. x=4

   

 

 

 

11. A function is continuous at a point a

    

     if what condition(s) is/are satisfied

 

1. f(a) is defined
2. limx→af(x)

   

    exists
3.    limx→af(x)=f(a)

   

4. All of the above

 

12. Determine the value(s) of x

    

     for which fx=2x2-x-3x2-x-2

    

     is not continuous

 

Unit

Objective

 

 

1. x=2 and x=-1

   

2. x=-2 and x=-1

   

3. x=-2 and x=1

   

4. x=2 and x=1

   

 

13. 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. 1
2. 2
3. 3
4. All of the above

 

14. What are the set of values for x

    

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

    

     is discontinuous?

 

1. x=5 and x=2

   

2. x=-5 and x=-2

   

3. x=5 and x=-2

   

4. x=-5 and x=2

   

 

15. Find the value(s) of x

    

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

    

     is discontinuous?

 

1. 1
2. 2
3. 3
4. Continuous for all values of x

   

 

 

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

 

1.     limh→0fx+h-f(x)h

   

2.     limh→0fx+h+f(x)h

   

3.     limh→0fx+h-f(x-h)h

   

4.     limh→0fx-h-f(x)h

   

 

17. Let fx=3x3+4x2-2x+10. 

    

     What is f'x

    

    ?

 

1. 1. 9x2+8x+2

      

   2. 3x2+4x-2

      

   3. 9x2+8x-2

      

   4. 9x2-8x+2

      

 

18. Let fx=3x

    

    .  What is f'(x)

    

    ?

 

Unit

Objective

 

 

1. -32x3

   

2. 32x32

   

3. 3(x)3

   

4. -3(x)3

   

 

19. Differentiate y=2x3

    

      with respect to x

    

    .

 

1. 1. 3x3

      

   2. 2x2

      

   3. 2x

      

   4. 3x

      

 

20. Differentiate y=3x-63

    

     with respect to x

    

 

1. 2
2. 1
3. -2
4. 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]