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MAT1512 EXAM PACK 2023 A+ ALL QUESTIONS
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  • MAT1512 - Calculus A
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  • MAT1512 - Calculus A

MAT1512 EXAM PACK 2023 A+ ALL QUESTIONS

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  • October 9, 2023
  • 82
  • 2023/2024
  • Exam (elaborations)
  • Only questions

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  • 2023
  • mat1512 exam pack 2023
  • mat1512 exam
  • mat1512
  • mat1512 calculus a mat1512
  • university of south africa

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  • MAT1512 - Calculus A
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MAT1512 Exam Papers Bundle




MAT1512
Exam Papers Bundle

2010-2023




Past Question Papers
Contains Questions ONLY

, 2 MAT1512
January /February 2023

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

lim x2 + 4x − 2x
(iii) (3)
x→ − 2x
(iv) 1− x (3)
lim
x→1 1− x
(v) lim 2x (3)
x→ 0 3 − x+9
(vi) sin t − tan 2t (3)
lim
t→ 0 t
(b) Use the Squeeze Theorem to determine the following limit:

5k 2 − cos 3k
lim 2 . (3)
k→  k +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) = 2x2 + 3x + 4
at x = −2 . (5)
(b) Find the derivatives of the following functions by using the appropriate rules of
differentiation:
() 2
(i) f x = x + x (3)
sin x cos x
( )
g (x ) = (cos 5x )
sin x 2
(ii) (3)

, 3 MAT1512
January /February 2023

z3

(iii) F (z) =  sin 3xdx (5)
z

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



QUESTION 3

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

5x  e 3
2x

(ii)   7 e3x dx
e  + (3)

1
(iii) (iii)  (4 − 3x )
3
dx (4)


4
(iv)(iv)
 (tan x ) (sec x ) dx (5)
3 3

0

(b) Let f (x) = x2 − 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) = x2 − 2 and g(x) = − x . (6)
[25]




QUESTION 4

(a) Solve the following Initial Value Problem:
dx
=
3t 2 + sec2 t ( )=
; x0 5. (6)
dt 3x2

, 4 MAT1512
January /February 2023

(b) If z = cos(xy)+ y cos x , where x = u2 + v2 and y = uv .
z
Use the Chain Rule for partial derivatives to find u . (5)

( )
(c) Let F(x, y) = 2y − 3xy + cot xy 2 .
(i) Find the first partial derivatives Fx and Fy . (4)
dy
(ii) Using (c)(i) above, find . (4)
dx
dy
(iii) If F(x, y) = 0 , then find using implicit differentiation to confirm your
dx
answer in part (c)(ii) above. (6)
[25]


TOTAL: [100]



UNISA 2022

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