MAT1581

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Supplementary/aegrotat/special examination candidates May 2011 Do not hand in any assignments to UNISA. To pass as supplementary candidate you need to obtain 50% in the May/June examination. If you are writing an aegrotat or special examination and it is your first exam in the module your yearmark gained in 2010 will count towards your final mark. You will be writing the same examination as the first semester 2011 candidates. To help you prepare for the examination do the assignments for the first semester. The solutions will be posted on MyUNISA at the end of April. You can then mark for yourself. You are always welcome to ask questions of the lecturer. 1. LECTURER Your lecturer for this year is Miss LE Greyling, Florida Campus, Block C, Room 535. If you experience any problems with the mathematical content you are welcome to contact her: 1) by telephone (011-471-2350), if she is not available leave a message on the

answering machine. The message must contain your name, the subject and a telephone number where you can be reached. You may also ask her to call you if she is in office and you do not have sufficient funds on your phone.

2) by e-mail ( lgreylin@unisa.ac.za ) 3) by fax (011-471-2142) 4) or personally. For a personal visit you must make an appointment by

telephone or e-mail. You must be prepared to come to the Florida Campus in Roodepoort for a personal visit.

Assignments 2011

2. ASSIGNMENT 01 SEMESTER 1 TO PLAN YOUR REVISION FINISH THIS BY 7 MARCH

DO NOT SEND TO UNISA FOR MARKING.

SOLUTIONS WILL BE PLACED ON MYUNISA

This assignment is a written assignment based on Study Guide 1 and 2 QUESTION 1 Solve for x : 1.1 2(2 4 ) 5 2n x x n x+ = + (4)

1.2 3 14 (16)x − −= (3)

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QUESTION 2

V-belt tensions driving a pulley are related by ( )sin / 22

1

T eT

µβα

= .

For 2 112 , 0,3,T T= µ = β = π what is the belt angle α in degrees? (5) QUESTION 3

Find the middle term in the binomial expansion of 6

2

8 2x

−

(4)

QUESTION 4 The percentages of three chemical substances X, Y and Z in a chemical process is reflected in the following system of equations:

1002 0

4 0

x y zx y

x z

+ + =− =

− + =

Solve the equations for y only using Cramèr's rule. (6) QUESTION 5

Resolve into partial fractions: ( )( )2

21 1

xx x− −

(7)

QUESTION 6 Solve for x : 2 2cos sin 5 sin , 0 360x x x x− = ≤ ≤ ° and sin 0.x > (5) QUESTION 7 Identify the following curve and sketch: 2 2( 4) ( 2) 4x y+ + − = (3) QUESTION 8 8.1 Solve for x and y if 2 3 4 3x yj j xj y+ − + = − (5)

8.2 Express in the form a bj+ : ( )( )−− + − 11 3 2j j (3) 8.3. Find the cube roots of 64 48° (3) 8.4 Write the principal cube root of 64 48° in exponential form. (2)

MAXIMUM: 50

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3. ASSIGNMENT 02 SEMESTER 1 TO PLAN YOUR REVISION FINISH THIS BY 7 MARCH

DO NOT SEND TO UNISA FOR MARKING.

SOLUTIONS WILL BE PLACED ON MYUNISA

QUESTION 1

Given that 22

4xf x

x+=−

( ) determine the following:

1.1 ( )2f (1)

1.2 ( )2

limx

f x→

(1)

1.3 ( )limx

f x→∞

(3)

QUESTION 2 Differentiate the following with regard to x and simplify if possible:

2.1 2 1

2 1xyx

−=+

(4)

2.2 33 cosec y x x= (3)

2.3 2 1 siny n x x = − . (4)

QUESTION 3 Use the second derivative test to find the values of x where the function

3 23 2y x x= − + reaches a maximum or a minimum. Sketch the function showing all important points. (8) QUESTION 4

Given the equation ( )32 2y x x= − find the slope of the tangent line at the point

( ) ( ), 1 , 1x y = − . (4) QUESTION 5

The displacement s meters of a particle from a fixed point after time t seconds is

given by 3

25st

= − .

Determine expressions for: 5.1 The velocity after time t (2)

5.2 The acceleration after time t (2)

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QUESTION 6 Determine the following integrals:

6.1 4

4

2y dyy

⌠⌡

− (3)

6.2 28

4 1x dx

x⌠⌡ +

(2)

6.3 2 1xe dx+∫ (3)

6.4 2(1 )x dx

x

⌠⌡

+ (5)

QUESTION 7 Consider the function 34)( 2 +−= xxxg . Calculate the actual total area Between this function and the x-axis between x = 0 and x = 3. (6) [Hint: Start by sketching the function]

MAXIMUM: 50

4. ASSIGNMENT 01 SEMESTER 2 DO NOT SEND TO UNISA FOR MARKING

SOLUTIONS WILL NOT BE GIVEN AS THE DUE DATE FOR THIS ASSIGNMENT IS AFTER YOUR EXAMINATION DATE. This assignment is a written assignment based on Study Guide 1 and 2 QUESTION 1 Solve for x : 1.1 ( )log log 2 log15x x+ − = (4)

1.2 42 2 17x x−+ = (4) 1.3 3 2. 59x xe e = (3) QUESTION 2 Find the first four terms of the binomial expansion of ( ) 51 x −− (4)

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QUESTION 3 Solve the following system of equations for z only using Cramèr's rule:

3 2 73 4 7 8

2 2

x y zx y z

x y z

− + − =+ − = −+ + =

(8)

QUESTION 4

Resolve into partial fractions:

2 3 53

x xx+ +

+ (7)

QUESTION 5 Solve for x: 22 sin cos 1 0x x− − = for 0 2x≤ < π. Give your answer in terms of π. (5) QUESTION 6 Identify the following curve and give the equation in standard form: 2 2 6 4 3x y x y+ − + = (3) QUESTION 7 A circle has a radius of 15 cm. Calculate the length of arc which subtends an angle of 217°. (3) QUESTION 8 8.1 Find the cube roots of z if 5(cos225 sin 225 )z j= ° + ° (5)

8.2 Express in the form a bj+ :(3 5 )(2 6 )

2 4j j

j− −

+ (3)

8.3 Express the complex number 2 j− + in exponential form. (4)

MAXIMUM: 50

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5. ASSIGNMENT 02 SEMESTER 2 DO NOT SEND TO UNISA FOR MARKING

SOLUTIONS WILL NOT BE GIVEN AS THE DUE DATE FOR THIS ASSIGNMENT IS AFTER YOUR EXAMINATION DATE. This assignment is a written assignment based on Study Guide 3. QUESTION 1 Determine the following limits:

1.1 9

lim 4 11x

x→

− (2)

1.2 0

sinlimx

x xx→

− (3)

QUESTION 2 Differentiate the following with regard to x : 2.1

3xy e−= (1)

2.2 12 sin cosy x x= (3)

2.3 ( )tan secy n x x= + (2)

2.4 ( )( )2 3y t t= − − (1)

2.5 5 11

xy nx

−= + (4)

QUESTION 3

3.1 Find the equation of the normal line to the curve 4 1

5xy

x+=

at the point (1;1). (6)

3.2 The strength of a person’s reaction to a certain drug is given by

12

( )3xR x x C = −

, where x represents the quantity of the drug taken

by the person and C is a contant. The sensitivity to the drug is given by '( )R x . Find the sensitivity if x = 87 and C = 59. (4)

QUESTION 4 Use the second derivative test to find the local maximum and minimum points

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of the curve 3

25 9 43xy x x= − + − . (6)

QUESTION 5 Determine:

5.1 4 42

3( ) 7x xx n e dxx

⌠⌡

+ + +

(5)

5.2 ⌡⌠

−− dx

xx

21 (3)

5.3 ∫ dxxe x 3sec23tan (3)

5.4 ⌡⌠

+− dx

xx

21 (4)

QUESTION 6 The total maintenance of a machine is given by the area under the curve

2( ) 10 2M t t t= + + , where ( )M t is the rate of maintenance in thousands of rands and t is the number of years the machine is being used. How much will a company spend on maintenance if the machine is used for 5 years? (5)

MAXIMUM:50