MAT1512
EXAM PACK
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1. Educational Aid: These study notes are designed to serve as educational aids and should not be considered as a
substitute for individual research, critical thinking, or professional guidance. Students are encouraged to
conduct their own extensive research and consult with their instructors or academic advisors for specific
assignment requirements.
2. Personal Responsibility: While every effort has been made to ensure the accuracy and reliability of the
information provided in these study notes, the seller cannot guarantee the completeness or correctness of all
the content. It is the responsibility of the buyer to verify the accuracy of the information and use their own
judgment when applying it to their assignments.
3. Academic Integrity: It is crucial for students to uphold academic integrity and adhere to their institution's
policies and guidelines regarding plagiarism, citation, and referencing. These study notes should be used as a
tool for learning and inspiration, but any direct reproduction of the content without proper acknowledgment and
citation may constitute academic misconduct.
4. Limited Liability: The seller of these study notes shall not be held liable for any direct or indirect damages,
losses, or consequences arising from the use of the notes. This includes, but is not limited to, poor grades,
academic penalties, or any other negative outcomes resulting from the application or misuse of the information
provided.
, 1 MAT1512
January /February 2023
UNIVERSITY EXAMINATIONS
January/February 2023
MAT1512
Calculus A
Examiners:
First: DR Z.I. ALI
Second: MR S. BLOSE
100 Marks
2 Hours
Closed book and online examination, which you have to write within 2 hours
and submit online through the link: https://myexams.unisa.ac.za/portal
Use of a non-programmable pocket calculator is NOT allowed
This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.
This examination allows attachment documents only as part of your
submission.
Declaration: I have neither given nor received aid on this examination.
Answer All Questions and Submit within the stipulated timeframe.
Late submission will not be accepted.
This paper consists of 4 pages.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
, 2 MAT1512
January /February 2023
QUESTION 1
(a) Determine the following limits (if they exist):
x 1 2x 1
(i) lim (3)
x 0 3x 4 2 x 4
(ii) lim
x
x 2
xx (3)
1 cos 3 x
(iii) lim (3)
x sin 2 x
sin 2 x
(iv) lim (2)
x 0 sin 3x
3x 2 6
(v) lim (2)
x 5 2x
(vi) lim
x
25x 2
x 5x (3)
(b) Use the Squeeze Theorem to determine the following limit:
x 2 x sin x
lim (3)
x x 2 cos x
(c) Consider the function f given below:
ax if x 1
f x x 2 a b if 1 x 1
bx if 1 x
(i) Determine the one-sided limits lim f x and lim f x . (2)
x 1 x 1
(ii) Find the one-sided limits lim f x and lim f x . (2)
x 1 x 1
(iii) Hence or otherwise determine the numerical values of a and b . (2)
[25]
, 3 MAT1512
January /February 2023
QUESTION 2
(a) Using the first principles of differentiation, find the first derivative of f x 3x 2
2
x
at x 1 . (5)
(b) Find the derivatives of the following functions by using the appropriate rules of
differentiation:
sin x cos x
(i) f x (3)
sin x cos x
(ii) g x e 4 x sin 4 x (3)
2 4
x x
(iii) F x tan t dt and G x t dt (5)
x x2
(c) The curve C has the equation
cos 2 x cos 3 y 1 , x , 0 y
4 4 6
dy
(i) Find in terms of x and y . (4)
dx
(ii) The point P lies on C where x .
6
Find the equation of the tangent to C at P , giving your answer in the form
ax by c 0
where a , b and c are integers. (5)
[25]
QUESTION 3
(a) Determine the following integrals:
x2 4
(i) x 2 dx (2)
sin x
(ii) 2 5 cos xdx (2)
1 cos x
(iii) x sin xdx (2)
EXAM PACK
Recent exam questions and answers
Summarised study notes
Exam tips and guidelines
DISCLAIMER & TERMS OF USE
1. Educational Aid: These study notes are designed to serve as educational aids and should not be considered as a
substitute for individual research, critical thinking, or professional guidance. Students are encouraged to
conduct their own extensive research and consult with their instructors or academic advisors for specific
assignment requirements.
2. Personal Responsibility: While every effort has been made to ensure the accuracy and reliability of the
information provided in these study notes, the seller cannot guarantee the completeness or correctness of all
the content. It is the responsibility of the buyer to verify the accuracy of the information and use their own
judgment when applying it to their assignments.
3. Academic Integrity: It is crucial for students to uphold academic integrity and adhere to their institution's
policies and guidelines regarding plagiarism, citation, and referencing. These study notes should be used as a
tool for learning and inspiration, but any direct reproduction of the content without proper acknowledgment and
citation may constitute academic misconduct.
4. Limited Liability: The seller of these study notes shall not be held liable for any direct or indirect damages,
losses, or consequences arising from the use of the notes. This includes, but is not limited to, poor grades,
academic penalties, or any other negative outcomes resulting from the application or misuse of the information
provided.
, 1 MAT1512
January /February 2023
UNIVERSITY EXAMINATIONS
January/February 2023
MAT1512
Calculus A
Examiners:
First: DR Z.I. ALI
Second: MR S. BLOSE
100 Marks
2 Hours
Closed book and online examination, which you have to write within 2 hours
and submit online through the link: https://myexams.unisa.ac.za/portal
Use of a non-programmable pocket calculator is NOT allowed
This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.
This examination allows attachment documents only as part of your
submission.
Declaration: I have neither given nor received aid on this examination.
Answer All Questions and Submit within the stipulated timeframe.
Late submission will not be accepted.
This paper consists of 4 pages.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
, 2 MAT1512
January /February 2023
QUESTION 1
(a) Determine the following limits (if they exist):
x 1 2x 1
(i) lim (3)
x 0 3x 4 2 x 4
(ii) lim
x
x 2
xx (3)
1 cos 3 x
(iii) lim (3)
x sin 2 x
sin 2 x
(iv) lim (2)
x 0 sin 3x
3x 2 6
(v) lim (2)
x 5 2x
(vi) lim
x
25x 2
x 5x (3)
(b) Use the Squeeze Theorem to determine the following limit:
x 2 x sin x
lim (3)
x x 2 cos x
(c) Consider the function f given below:
ax if x 1
f x x 2 a b if 1 x 1
bx if 1 x
(i) Determine the one-sided limits lim f x and lim f x . (2)
x 1 x 1
(ii) Find the one-sided limits lim f x and lim f x . (2)
x 1 x 1
(iii) Hence or otherwise determine the numerical values of a and b . (2)
[25]
, 3 MAT1512
January /February 2023
QUESTION 2
(a) Using the first principles of differentiation, find the first derivative of f x 3x 2
2
x
at x 1 . (5)
(b) Find the derivatives of the following functions by using the appropriate rules of
differentiation:
sin x cos x
(i) f x (3)
sin x cos x
(ii) g x e 4 x sin 4 x (3)
2 4
x x
(iii) F x tan t dt and G x t dt (5)
x x2
(c) The curve C has the equation
cos 2 x cos 3 y 1 , x , 0 y
4 4 6
dy
(i) Find in terms of x and y . (4)
dx
(ii) The point P lies on C where x .
6
Find the equation of the tangent to C at P , giving your answer in the form
ax by c 0
where a , b and c are integers. (5)
[25]
QUESTION 3
(a) Determine the following integrals:
x2 4
(i) x 2 dx (2)
sin x
(ii) 2 5 cos xdx (2)
1 cos x
(iii) x sin xdx (2)
