1. How fast is the surface area of a spherical balloon increasing when the radius is 10 cm and
the volume is increasing at 15 cm3
/sec? [7]
2. Let f be the function defined by
f (x) = xe−x
.
(a) Determine the y–intercept.
(b) Determine the horizontal and vertical asymptotes.
(c) Use the...
IMPORTANT INFORMATION:
This tutorial letter contains important
information about your module.
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,1 INTRODUCTION AND WELCOME
Welcome to module MAT1613 on Calculus. I hope you will find it both interesting and rewarding.
This module is offered as a semester module. You will be well on your way to success if you start
studying early in the year and are committed to do the assignments properly.
I hope you will enjoy this module, and wish you success with your studies.
Tutorial matter
You will find all the study material online. The topics and outcomes for this module are specified
in your study guide and the topics must be studied in both the textbook and study guide.
A list of topics with references to the page numbers in the textbook will be posted on myUnisa.
Tutorial Letter 101 and the study guide which will be available at the time of registration all other
tutorial letters and some extra material will be uploaded later in the semester. The solutions to
the assignments will also be uploaded online about a week after the closing date.
Please access the myUnisa website at http://my.unisa.ac.za
Tutorial Letter 101 contains important information about the scheme of work, resources and assign-
ments for this module. I urge you to read it carefully and to keep it at hand when working through
the study material, preparing the assignments, preparing for the examination and addressing ques-
tions to your lecturers.
In this tutorial letter you will find the assignments as well as instructions on the preparation and
submission of the assignments. This tutorial letter also provides information with regard to other
resources and where to obtain them. Please study this information carefully.
Certain general and administrative information about this module has also been included. Please
study this section of the tutorial letter carefully.
You must read all the tutorial letters carefully, as they always contain important and, sometimes,
urgent information.
2 PURPOSE AND OUTCOMES OF THE MODULE
2.1 Purpose
This module will be useful to students interested in developing those skills in integral and differential
calculus which can be used in the natural economic, social and mathematical sciences. Students
credited with this module will have the knowledge of those basic techniques in differential and
integral calculus which are used in related rates problems, graph sketching, evaluating integrals,
calculating volumes and areas, and in maximum and minimum problems.
2
, MAT1613/001
2.2 Outcomes
2.2.1 Calculate and use the derivatives of a function to sketch a graph of the function.
2.2.2 The first derivative is used to determine the relationship between the rates of change of
various quantities in the rates-of-change word problem.
2.2.3 The student is able to solve maximum or minimum word problems using the theory of deriva-
tives.
2.2.4 Ability to use L’Hôpital’s rule to determine limits of indeterminate forms.
2.2.5 Calculation of the volumes of solids of revolution.
2.2.6 An improper integral is tested for convergence or divergence and evaluated if convergent.
2.2.7 The student is able to use various integration techniques to evaluate integrals.
2.2.8 The student is able to calculate the Taylor polynomial of any order at a given point.
3 LECTURER(S) AND CONTACT DETAILS
3.1 Lecturer
The lecturers responsible for this module will be announced in a TL early in 2021. All queries
that are not of a purely administrative nature but are about the content of this module should be
directed to this person or these persons. Email is the preferred form of communication to use.
If you phone me please have your study material with you when you contact the lecturers. If you
cannot get hold of them, leave a message with the Departmental Secretary. Please clearly state
your name, time of call and how the lectureres can get back to you (preferable by email.) You
are always welcome to come and discuss your work with them, but please make an appointment
before coming to see them (if possible). Please come to these appointments well prepared with
specific questions that indicate your own efforts to have understood the basic concepts involved.
You are also free to write to them about any of the difficulties you encounter with your work for this
module. If these difficulties concern exercises which you are unable to solve, you must send your
attempts so they can see where you are going wrong, or what concepts you do not understand. .
3
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