MAT1512 EXAM PACK 2023

MAT1512 EXAM PACK
2023




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, 1 MAT1512
January /February 2022

UNIVERSITY EXAMINATIONS




January/February 2022

MAT1512

Calculus A
Examiners:
First: DR S.B. MUGISHA
Second: DR Z. ALI


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 2022

QUESTION 1

(a) Determine the following limits (if they exist):


6 x 2
(i) lim (3)
x2 3  x 1
x3
(ii) lim (3)
x  3  x2  9
x 2  4x  2x
(iii) lim (3)
x   2x
1 x
(iv) lim (3)
x  1 1 x

2x
(v) lim (3)
x 0 3  x9
sin t  tan 2t
(vi) lim (3)
t 0 t
(b) Use the Squeeze Theorem to determine the following limit:

5k 2  cos 3k
lim . (3)
k  k 2  10

 x  2 if x2
(c) Let G ( x)  
x  2 if x2
(i) Draw the graph of Gx  . (1)
(ii) Determine lim G( x) . (2)
x 2

(iii) Is Gx  continuous at x  2 ? Give a reason for your answer. (1)
[25]



QUESTION 2

(a) By the first principles of differentiation, find the derivative of f x   2 x 2  3x  4
at x  2 . (5)
(b) Find the derivatives of the following functions by using the appropriate rules of
differentiation:
x2  x
(i) f x   (3)
sin x cos x
(ii) g x   cos5 x 
 
sin x 2
(3)

, 3 MAT1512
January /February 2022

z3
(iii) F z    sin 3xdx (5)
z

(c) Given: sinx  y   2 x , find the following:
dy
(i) by using implicit differentiation. (4)
dx
(ii) the equations of the tangent line and normal line to the curvesinx  y   2 x at
the point 0,  . (5)
[25]



QUESTION 3

(a) Determine the following integrals:
 2  2
(i)   x  x2 

x  2 dx
x 
(3)

5x  e 3 
2x
(ii)   7 e3x dx
e   (3)

1
(iii)  4  3x 3 dx
 
(4)


4

 tan x   sec x  dx
3 3
(iv) (5)
0

(b) Let f  x   x 2  2 and g x    x , then
(i) Sketch the graphs of f and g on the same axes. (4)
(ii) Find the area enclosed by f x   x 2  2 and g x    x . (6)
[25]




QUESTION 4

(a) Solve the following Initial Value Problem:
dx 3t 2  sec2 t
 ; x0  5 . (6)
dt 3x 2