MAT1512 EXAM PACK 2023

MAT1512
EXAM PACK

Solutions, Explanations, workings, and references




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, 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
xx  (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)