Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Limits (Algebraically)
Limits Infinities and Zeros
It is useful to have the following symbolic fractions when dealing with limits. Note that infinity
means positive or negative infinity.
A.
infinity
=
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
The Product and Quotient Rules
The Product Rule
Theorem (The Product Rule)
Let f and g be differentiable functions. Then
[f(x) g(x)] ' = f(x) g '(x) + f '(x) g(x)
Proof:
We have
Example
Find
d
(2  x2
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Infinite Limits
Vertical Asymptotes
Consider the function
f(x) = x/(x  1)
From the graph we see that there is a vertical asymptote at x = 1. We say
and
Formal Definition
Definition
We say
if for any
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Limits a Geometric and Numeric Approach
Limits Using Tables
Consider the function
x2  1
f(x) =
x+1
Notice that this function is undefined at x = 1. In calculus undefined is not as precise as
possibl
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
The Chain Rule
The Chain Rule
Our goal is to differentiate functions such as
y = (3x + 1)10
The last thing that we would want to do is FOIL this out ten times. We now look for a better
way.
The Chain
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
A Preview of Calculus
Local graph experiment
Try this experiment: Write down an arbitrary function and an arbitrary value of x. Enter the
function into a graphing calculator and zoom in on the point.
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Continuity
Continuity
If a graph has no holes asymptotes, or breaks then the function is continuous. Or if you can draw
the function without lifting your pencil then it is continuous. Below is a forma
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Implicit Differentiation
Implicit and Explicit Functions
An explicit function is an function expressed as y = f(x) such as
y = sinx
y is defined implicitly if both x and y occur on the same side of th
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
EpsilonDelta
The Formal Definition of the Limit
 Definition of a Limit
Let f(x) be a function and L be a number we say that
if for any choice of , the team can respond with a positive
number so that
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Derivatives The Easy Way
Constant Rule and Power Rule
We have seen the following derivatives:
1.
If f(x) = c, then f '(x) = 0
2.
If f(x) = x, then f '(x) = 1
3.
If f(x) = x2, then f '(x) = 2x
4.
If f(
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
MATH 105
CALCULUS and ANALYTIC GEOMETRY
Monday, Wednesday, Friday 8:00 to 9:40 AM
Room E 106
5 UNITS
Instructor Larry Green
Phone Number Office: 5414660 Extension 341
email: [email protected]
C
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Derivatives and Interpretations
Velocity
Recall that the difference between speed and velocity is that velocity has direction and speed
does not. In other words, the speed is the absolute value of vel
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Name
MATH 105 MIDTERM II KEY
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 (7 Points each) Please answ
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Name
MATH 105 PRACTICE MIDTERM 1 KEY
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the f
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
MATH 105 PRACTICE MIDTERM III KEY
Please work out each of the given problems. Credit will be based on the steps that you show
towards the final answer. Show your work.
PROBLEM 1 Please answer the foll
Differential Calculus for Engineering and other Hard Sciences
MATH 105

Fall 2011
Name
MATH 105 FINAL
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or