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
= infinity
finite
0
B.
= 0
finite nonzero
C.
finite
= 0
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)(x4  5)
dx
Solution:
Here
f(x) = 2  x2
and
g(x) = x4
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 choice of N there exists a such that whenever
x  c <
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
possible. Instead one asks, what does the y value "look like"
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 Rule
If
y = y(u)
is a function of u, and
u = u(x)
is a
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. You will notice that your calculator
shows a line. Now,
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 formal definition.
Definition of Continuity
A function is co
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 the equation such as
x2 + y2 = 4
we can think of y as fun
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 with a "perfect calculator" the team will
win. That is
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(x) = x3, then f '(x) = 3x2
5.
If f(x) = x4, then f '(x)
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]
Class Grades
Web Page: http:/www.ltcconline.net/greenl/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 velocity. We have seen that the
secant line can be used to
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 answer the following true or false. If false, explain why
o
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 following true or false. If false, explain why or provid
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 following true or false. If false, explain why or provide a
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 false. If false, explain why or provide a
counter examp