Exam (elaborations)
MAT1613 EXAM PACK 2023
- Course
- Institution
- Book
MAT1613 Latest exam pack questions and answers, perfect for OCTOBER NOVEMBER 2023 exam preparation
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
[Calc 2] 11.3: Polar Coordinates
Seller
Follow
VarsityC
Reviews received
Content preview
MAT1613EXAM PACK
Solutions, Explanations, workings, and references
+27 81 278 3372
, 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.
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
, -2- MAT1613
Oct/Nov 2021
Memorandum
Sign pattern:
undefined
1 1 2 2 x>1
x< <x< <x<1
2 2 3 3
(2x − 1) − + + +
(x − 1) − − − +
f ′ (x) + − − +
1 2
1
2 3
Note:
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
(b) 2x(3x − 2)2 − 6x 2 (3x − 2) (7)
f ′′ (x) = 0 − [ ]
(3x − 2)4
′′ (x)
(3x − 2)[2x(3x − 2) − 6x 2 ]
f = −[ ]
(3x − 2)4
6x 2 − 4x − 6x 2
f ′′ (x) = − [ ]
(3x − 2)3
−4x
f ′′ (x) = − [ ]
(3x − 2)3
4x
f ′′ (x) =
(3x − 2)3
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
, -3- MAT1613
Oct/Nov 2021
Memorandum
(i) 2
Critical points for f ′′ (x) are: 0 and
. These critical points split the
3
number line into three distinct intervals:
2 2
x < 0, 0<x< , x>
3 3
Sign pattern:
undefined
x<0 2 2
0<x< x>
3 3
4x − + +
(3x − 2) − − +
f ′ ′(x) + − +
2
0
3
Note:
2
f ′ (x) is undefined at and (3x − 2)3 is not always positive,
3
i. e. , it affects the sign pattern, hence it must be investigated.
f is concave up where they are positive signs on the sign
table.
f is concave down where they are negative signs on the sign
table.
2
f is concave up on (−∞, 0) ∪ ( , ∞) and f is concave down
3
2
down on (0 < x < ).
3
Local extreme point(s) occur whenf ′ (x) = 0
(ii)
4[(2x − 1)(x − 1)] 1
2
= 0 ⇒ 4[(2x − 1)(x − 1)] = 0 ⇒ x = or x = 1
(3x − 2) 2
1
∴ the x − coordinate(s) of the local extreme poit(s) are x = and x = 1.
2
[16]
© 2021
JTS Maths Tutoring
Keeping you mathematically informed
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller VarsityC. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.75. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
50064 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now