Exam (elaborations)
MAT1512 EXAM PACK 2023
- Course
- MAT1512 EXAM PACK 2023
- Institution
- University Of South Africa
MAT1512 EXAM PACK 2023
[Show more]
Content preview
MAT1512EXAM
PACK 2023
, 1 MAT1512
January /February 2022
UNIVERSITY EXAMINATIONS
January/February 2022
MAT1512
Calculus A
Examiners:
First: DR S.B. MUGISHA
Second: DR Z. ALI
100 Marks
2 Hours
Closed book and online examination, which you have to write within 2 hours
and submit online through the link: https://myexams.unisa.ac.za/portal
Use of a non-programmable pocket calculator is NOT allowed
This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.
This examination allows attachment documents only as part of your
submission.
Declaration: I have neither given nor received aid on this examination.
Answer All Questions and Submit within the stipulated timeframe.
Late submission will not be accepted.
This paper consists of 4 pages.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
, 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
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 EFT, 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 this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller examexpert01. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R46,58. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
66579 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now