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MAT1512 EXAM PACK 2023
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MAT1512 EXAM PACK 2023
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MAT1512EXAM
PACK 2023
, 2 MAT1512
January /February 2022
QUESTION 1
(a) Determine the following limits (if they exist):
6 x 2
(i) lim (3)
x2 3 x 1
x3
(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 x9
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 x2
(c) Let G ( x)
x 2 if x2
(i) Draw the graph of Gx . (1)
(ii) Determine lim G( x) . (2)
x 2
(iii) Is Gx 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: sinx 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 curvesinx 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
; x0 5 . (6)
dt 3x 2
, 4 MAT1512
January /February 2022
(b) If z cosxy y cos x , where x u 2 v 2 and y uv .
z
Use the Chain Rule for partial derivatives to find . (5)
u
(c) Let F x, y 2 y 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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