MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 3 (12 pts)
Please print names and IDs in ink :
NSID:
Family Name:
First Name:
INSTRUCTIONS:
1. Time Limit: 30 minutes
2. No cheating.
5. Show all your work.
Student ID:
3. Closed book.
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 2 (12 pts)
1. (3 pts) Find the roots of the polynomial x6 10x4 + 9x2 . If a root is repeated, give its
multiplicity. Also, write the polynomial as a product of linear factors.
x6 10x4 +
2.2. Rough definitions of limits
Whether tangent line problems or instantaneous
velocity problems, we are dealing with functions.
So it is necessary to give the definition of a limit
for general functions.
Topics:
I. Two-sided limits
II. One-sided limits
2.8. Derivative functions
Definition :
If we replace a by x in the definition of
derivative in the last section, we obtain
f ( x h) f ( x )
f ( x) lim
h 0
h
where x is a variable, we call this the
derivative function of f , denoted by f .
1
Other Notation
2.7. Derivatives
From section 1, we see that either tangent
problem or velocity problem, we consider :
f ( x) f (a )
lim
.
x a
x a
This special type of limit plays a very important
role in mathematics and applications.
It is necessary to give a name for t
Chapter 2. Limits and Derivatives
Introduction
The invention of the calculus was accomplished by
the mathematician and physicist Newton (16421727) and the mathematician, logician, and
philosopher Leibniz (1646-1716).
The entire calculus is based on the
1.4 Exponential Functions
An exponentia l function is a function of
the form f ( x) a x where a 0.
The a is called the base and x the exponent.
D f R and R f (0, ) if a 1.
f is called exponential function because
the variable x is the exponent.
Comment
MATH 102.3
Pre-Calculus Mathematics
Term 2
Course Section: W02
Delivery: Online
Start Date: Jan 5, 2016
End Date: April 7, 2016
Class Syllabus
Your Instructor
Stavros Stavrou, M.Sc, M.Ed
Department of Mathematics & Statistics
Class Syllabus
Table of Conte
MATH123 CALCULUS I for Engineers
Lab Section:
Solution Key for Midterm 1 (50 pts)
Part 1. Handwritten Answers 4 questions
Write down the solution to the following problems, giving all necessary details, in the space
provided.
1. (a) (4 pts) If 2 + cos x f
MATH123 CALCULUS I for Engineers
Lab Section:
Solution Key for Midterm 2 (50 pts)
Part 1. Handwritten Answers 4 questions
Write down the solution to the following problems, giving all necessary details, in the space
provided.
1. (5 pts) Let r > 1 be a fix
University of Saskatchewan, Department of Mathematics and Statistics
MATH 123.3, Calculus I for Engineers ALL SECTIONS, 2016-2017 Term 1
Lectures Section 01: MWF 11:30 am - 12:20 pm BIOL 106
Instructor: Derek L.Postnikoff
Office: McLean Hall 238
e-mail: d
1.3 New Functions From Old Functions
We only have a few types of basic functions :
polynomial
rational
exponentia l
logarithmi c
trigono metric
inverse trigonome tric
root functions
They are building blocks. In this section, we learn
how to create
2.6. Limits at infinity and infinite
limits at infinity
To better understand the behavior of the given
function, sometimes, we need to compute the
limits at infinity : lim f ( x).
x
This is another application of limits.
Topics :
I. Finite limits at infi
Chapter 3
Differentiation Rules and formulas
We see from the last chapter, finding a
derivative function using the definition
involves a long computation. Sometimes,
the process is very long and painful.
We could only find derivative functions
for simple
MATH123 CALCULUS I for Engineers
Quiz 1
(12 pts)
Please print names and IDs in ink :
September 14, 2016
NSID:
Family Name:
INSTRUCTIONS:
Lab Section:
First Name:
1. Time Limit: 30 minutes
2. No cheating.
5. Show all your work.
Student ID:
3. Closed book.
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 3 (12 pts)
1. Evaluate the following limits or explain why they do not exist:
(x 2)(x + 2)
x2 4
x+2
= lim
= lim
2
2
x2
x2 x 4x + 4
x2 x 2
(x 2)
(a) (2 pts) lim
We must take the right an
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 9 (12 pts)
1. (6 pts) A rectangle with sides parallel to the coordinate axes is inscribed in the ellipse
x2 + 4y 2 = 4. Find the largest possible area for this rectangle.
A = 2x 2y = 4x
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 8 (12 pts)
1. (4 pts) Locate and classify all local extreme values of f (x) =
Determine if any of these are also absolute extreme values.
f (x) =
x
, where 2 x 2.
1 + x2
1 + x2 2x x
1 x
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 7 (12 pts)
1. (2 pts) Simplify and nd the exact value of this expression:
loga
logb b
a3
3
a
= loga logb b
3
a
loga a3 = loga
2. (2 pts) Solve log3 (x + 5) log3 (1 +
1
8
3
a3= 3=
3
3
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 6 (12 pts)
1. (3 pts) A snowball is melting. By approximately what percentage will the radius of
4
the snowball decrease when the volume (V = r3 ) decreases by 9%?
3
If V is the volume
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 5 (12 pts)
1. (4 pts) Does the graph of f (x) = (1 x)5/3 have a tangent line at x = 1? If yes, what
is the tangent line? You may use may the Dierentiation Rules to nd the derivative
of
1.5 Inverse and log functions
In this section, we look at logarithmic
functions, which are another very important
type of functions. We define logarithmic
functions as inverse of exponential functions.
Topics:
A. Inverse functions
B. Logarithmic functions
Chapter 1: Functions
Our world is full of changing quantities. Almost all
quantities are functions of time. For instance, the motion
of planets, sounds, etc.
Functions are fundamental objects in calculus.
They are used to describe relationship between
q
2.3. Calculating limits using the limit laws
Main Idea: Instead of using definitions, we establish
some limit laws and theorems and use these to
compute limits
Topics:
I. Laws
II. Limits of polynomials and rational functions
III. Limits of quotient functi
2.5. Continuity
In this section, we use limits to
formulate the notion of continuity.
This can be treated as one
application of limits.
I. Continuity at one point
II. Continuity on an interval
III. Intermediate value theorem
1
I. Continuity at one point
D
MATH123 CALCULUS I for Engineers
Lab Section: L
Solution Key for Quiz 4 (12 pts)
2
x 2a if x < 2
2
if x = 2 is a continuous function.
1. (4 pts) Find a and b so that f (x) =
b 3x if x > 2
For f (x) to be continuous we need lim f (x) = lim + f (x) = f (2
Mathematics 123.3: Solutions to Lab Assignment #3
1.1:2 The point P (1, 3) lies on the curve y = 1 + x + x 2 .
(a) If Q is the point (x, 1 + x + x 2 ), nd the slope of the secant line P Q for the following values of x:
(i) 2 (ii) 1.5 (iii) 1.1 (iv) 1.01 (
Appendix A: Numbers, Inequalities,
and Absolute values
I. Numbers
There are many number systems
Real number system (1-dimension system)
Complex number system (2-dimension system)
Hamilton Number system (3-dimension system)
Calculus is based on real number
MATH123 CALCULUS I for Engineers
Solution Key for Midterm 2 (50 pts)
1. (3 pts) Evaluate f 000 (x) for f (x) = x2 +
1
x2
f 0 (x) = 2x + (2)x3
f 00 (x) = 2 + (3)(2)x4
f 000 (x) = 0 + (4)(3)(2)x5 = 24x5 , or
24
x5
2. (6 pts) Compute the velocity v(t) and t