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MAT1512 EXAM PACK 2023 A+ ALL QUESTIONS
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MAT1512 EXAM PACK 2023 A+ ALL QUESTIONS
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MAT1512 Exam Papers BundleMAT1512
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 x2
(c) Let G(x) =
x − 2 if x2
(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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