Lecture 2
Definition of a limit:
lim x a f ( x )=L if for every error bound , there is a choice of error for the
input, , so that 0<|xa|< guarantees |f ( x )L|<
The idea is that no matter how close you need the answer to get to get L, that can be
achieve

Eric 1
1. Plot volts vs cm-H2O steadily increasing and then decreasing the height
2. F = kx (static method), a = w2 x (dynamic method), w = (k/m)1/2
3. Because of the cos value (90o = 0), and its linear with the measurements for accelerometer. (start with

Robertus Eric - Prelab
1.Because it measure the temperature differences, same metal will makes delta equal to 0, so
they need to be made out of two different metals
2.As basic of known temperature, which is 0 C, for calculation purposes.
3. q= hA (T hot T

Lecture 6
Continuity Part 1:
What does it mean for a function to be continuous?
o A function f is continuous at c if lim x c f ( x) both exist
o Intuitively, this is meant to convey the idea that the graph of f near c
leads to f (c)
If f is not continuo

Lecture 7
Continuity Part 2:
Other functions
Trigonometric functions:
o We first consider f ( x )=sin ( x )
o Note that sin(x )< x for all x> 0
o
o Since 0<sin ( x)< x
+
for 0<x <
2
the squeeze theorem tells us that
x 0 sin ( x )=0
lim
x 0 sin ( x )=0

Implicit Differentiation: Chain rule means we can take the derivative of an equation, even if it
isnt a function
Example: Find the slope of the curve defined by x 2+ y 2 =4
Obviously one option is to solve for y
Alternatively, we can take the derivative

Product Rule:
d
d
d
fg=f
g+ g
f
dx
dx
dx
Example: Find
d
x ex
dx
Product Rule says that it is
e
( x) ( 1 )=x e x +e x
d
d
x e x +e x
x=x e x +
dx
dx
Proof of the Product Rule
g ( x+ h )g ( x )
f ( x )f ( x )
g (x) d
d
lim h 0 f ( x+ h )
+g( x)
=f ( x ) g

Lecture 13
Our goal is to find rules for the derivative of as many functions as possible
d
cc
c=l h 0
=l h 0 0=0
First we deal with the simplest functions
dx
h
Thus, the derivative of any constant is zero
( x +h )x
h
=l x 0 =1
Next we do the identity f

Lecture 10
The Derivative of a Function
lim f ( x +h ) f (x )
Given a function f(x), we define the derivative h 0
h
'
f ( x )=
This gives us a new function, since the defining limit depends only on x
d
f
Another notation for f ' ( x ) is
dx
o This is call

Slope of the tangent line
The slope of the tangent line should be the limit of the slope of the secant lines
Therefore, the slope of the tangent line is:
f ( x )f ( a)
M= lim x z xa
where (a, f(a) is the point of tangency
The expression inside the limi

Lecture 5
The Precise Definition of the Limit:
Recall, lim x c f ( x )=L if for every choice of output varience , there is a choice
of input varience so that 0< xc implies that f ( x )
Example: Consider f (x)=5 x2
o Look at lim x 1 f ( x)
o
F(1)=52=3

Lecture 4
The Limit Laws:
Suppose lim x c f ( x ) and lim x c g ( x ) both exist. We then get the following:
lim x c g ( x )
lim x c f ( x ) +
o
lim x c [ f ( x )+ g ( x ) ] =
lim x c g ( x )
lim x c f ( x )
o
lim x c [ f ( x )g ( x ) ] =
lim x c g ( x )

Lecture 3
Notation One Sided Limit
+
xc
We write lim to indicate the limit from the right side, also called the limit as x
decreases to c
xc
We write lim to indicate the limit from the left side, also called the limit as x
increases to c
Example: f ( x

Lecture 1
What is a function?
Input
function
output
How do we represent functions?
Verbally
o Example: The temperature of water coming out a faucet in terms of time is a function
o Example: A persons phone number over time may not be a function, since

Robertus Eric
1. Ratio of relative change in electrical resistance R to the mechanical strain or
= R/R
Poissons ratio negative transverse strain over longitudinal strain dA/A / -dL/L
Yung modulus = kL3/3I where L = length beam, I = moment of inertia)
Ela