Well-structured MAT1613 EXAM PREPARATION PACK - DISTINCTION GUARANTEED. Contains recent exam questions and answers, and Summarised study notes. All you need to pass the OCTOBER NOVEMBER 2023 EXAMS
MAT1613
EXAM PACK
Recent exam questions and answers
Summarised study notes
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, MAT1613
Oct/Nov 2021
Memorandum
QUESTION 1
If the derivative f ′ (x) of a function f is given by
x2
f ′ (x) = 1 −
(3x − 2)2
(a) Write f ′ (x) in quotient form and use the sign pattern to determine the interval(s) (6)
over which f increases and over which it decreases.
(b) Determine f ′′ (x) and use the sign pattern of f ′′ (x) to determine:
(i) the intervals where the graph of f is concave up and where it is (7)
concave down
(3)
(ii) the x − coordinate(s) of the local extreme point(s).
[16]
Solution
(a) ′ (x)
x2 (6)
f =1−
(3x − 2)2
(3x − 2)2 − x 2
f ′ (x) =
(3x − 2)2
′ (x)
9x 2 − 12x + 4 − x 2
f =
(3x − 2)2
8x 2 − 12x + 4
f ′ (x) =
(3x − 2)2
′ (x)
4(2x 2 − 3x + 1)
f =
(3x − 2)2
4[(2x − 1)(x − 1)]
f ′ (x) =
(3x − 2)2
(b) 1 2 (5)
Critical points for f ′ (x) are: , and 1. These critical points split the number
2 3
line into four distinct intervals:
1 1 2 2
x< , <x< , < x < 1, x>1
2 2 3 3
Note:
Critical points (stationary points) are points on a graph where f ′ (x) = 0
or f ′ (x) does not exist.
2
f ′ (x) is undefined at and (3x − 2)2 is always positive, i. e. , it
3
doesn′ t affect the sign pattern, hence we do not include it on the sign
table.
f rises where they are positive signs on the sign table.
f falls where they are negative signs on the sign table.
1 1 2 2
f rises over (−∞, ) ∪ (1, ∞) and f falls over ( , ) ∪ ( , 1)
2 2 3 3
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