Transcript Math Review and Examples

Math Review
• Units, Scientific Notation, Significant Figures, and Dimensional analysis
• Algebra –
–
–
–
–
Per Cent Change
Solving simultaneous equations
Cramers Rule
Quadratic equation
Coversion to radians
• Vectors
–
–
–
–
–
Unit vectors
Adding, subtracting, finding components
Dot product
Cross product
Examples
• Derivatives
– Rules
– Examples
• Integrals
– Examples
The system of units we will use is the
Standard International (SI) system;
the units of the fundamental quantities are:
• Length – meter
• Mass – kilogram
• Time – second
• Charge - Coulomb
Fundamental Physical Quantities and
Their Units
Unit prefixes for powers of 10, used in the SI
system:
Scientific Notation
Scientific notation: use powers of 10 for
numbers that are not between 1 and 10 (or,
often, between 0.1 and 100); exponents add if
multiplying and subtract if dividing:
Accuracy and Significant Figures
If numbers are written in scientific notation, it
is clear how many significant figures there are:
6 × 1024 has one
6.1 × 1024 has two
6.14 × 1024 has three
…and so on.
Calculators typically show many more digits
than are significant. It is important to know
which are accurate and which are meaningless.
Other systems of units:
cgs, which uses the centimeter, gram, and
second as basic units
British, which uses the foot for length, the
second for time, and the pound for force or
weight – all of these units are now defined
relative to the SI system.
Accuracy and Significant Figures
The number of significant figures represents the
accuracy with which a number is known.
Terminal zeroes after a decimal point are
significant figures:
2.0 has 2 significant figures
2.00 has 3 significant figures
The number of significant figures represents
the accuracy with which a number is known.
Trailing zeroes with no decimal point are not
significant. A number like 1200 has only 2
significant figures whereas 1200. has 4 significant
figures.
Dimensional Analysis
The dimension of a quantity is the particular
combination that characterizes it (the brackets
indicate that we are talking about
dimensions):
[v] = [L]/[T]
Note that we are not specifying units here –
velocity could be measured in meters per
second, miles per hour, inches per year, or
whatever.
Problems
Involving
Percent
Change
A cart is traveling along a track. As it passes through a photogate its speed is
measured to be 3.40 m/s. Later, at a second photogate, the speed of the cart is
measured to be 3.52 m/s. Find the percent change in the speed of the cart.
new  original
%Change 
100%
original
%Change 
3.52
m
m
 3.40
s
s 100%
m
3.40
s
%Change  3.5%
Simultaneous Equations
2x  5y  11
x  4 y  14
FIND X AND Y
x  14  4y
2(14  4y)  5y  11
28  8y  5y  11
13y  39
y  3
x  14  4(3)  2
Cramer’s Rule
c1
c2
x
a1
a2

b1
b2
c1b2  c2b1

b1 a1b2  a2b1
b2
(11)(4)  (14)(5) 44  70 26


2
(2)(4)  (1)(5)
8  5
13
a1
a2
y
a1
a2

a1 x  b1 y  c1 2x  5y  11
a2 x  b2 y  c2 x  4 y  14
c1
c2
a c  a2 c1
 1 2
b1 a1b2  a2b1
b2
(2)(14)  (1)(11) 28  11 39


 3
(2)(4)  (1)(5)
8  5 13
Quadratic Formula
EQUATION:
ax  bx  c  0
2
SOLVE FOR X:
b  b  4ac
x
2a
2
SEE EXAMPLE NEXT PAGE
Example
2x 2  x  1  0
a2
b 1
c  1
1  12  4(2)(1)
x
2(2)
1  9 1  3
x

4
4
1  3
x
 1
4
1  3 1
x

4
2
Derivation
ax 2  bx  c  0
b
c
x  ( )x  ( )  0
a
a
2
2
b 
b 2
c

 x  ( 2a )   ( 2a )  ( a )  0
2
2
b
c
b


x

(
)
 ( )  ( 2 )


2a 
a
4a
2


c
b
2
2
(2ax  b)  4a  ( )  ( 2 ) 
4a 
 a
(2ax  b)2  b 2  4ac
2ax  b   b 2  4ac
b  b 2  4ac
x
2a
Complete the Square
Arc Length and Radians
r  radius
D  diameter
C  circumfrance
2r  D
C
   3.14159
D
C

2r
C  2 r
C
r
2
C S
 r
2 
r

S
S  r
 is measured in radians
  2
S  r2  C
2 rad  360 o
360 o
1rad 
 57.3 deg rad
2
Small Angle Approximation
Small-angle approximation is a useful simplification of the laws of trigonometry
which is only approximately true for finite angles.
o
FOR   10
sin 
EXAMPLE
sin(10 )  0.173648178
o
10  0.174532925 radians
o
Scalars and Vectors
Vectors and Unit Vectors
• Representation of a vector : has magnitude
and direction. In 2 dimensions only two
numbers are needed to describe the vector
– i and j are unit vectors
– angle and magnitude
– x and y components
• Example of vectors
• Addition and subtraction
• Scalar or dot product
Vectors
A
ĵ
θ
iˆ
A  2iˆ  4 ĵ
Red arrows are the i
and j unit vectors.
Magnitude =
A  2 2  4 2  20  4.47
Angle between A and x axis = θ
tan   y / x  4 / 2  2
  63.4 deg
Adding Two Vectors
A  2iˆ  4 ĵ
r
B  5iˆ  2 ĵ
A
B
Create a
Parallelogram with
The two vectors
You wish you add.
Adding Two Vectors
A
A B
B
A  2iˆ  4 ĵ
r
B  5iˆ  2 ĵ
r r
A  B  7iˆ  6 ĵ
Note you add
. x and
y components
Vector components in terms of sine and
y
cosine
r
r
y
j

i
x
x
cos   x r
sin   y r
x  r cos
y  r sin 
r  xiˆ  yĵ
r  (r cos )iˆ  (r sin  ) ĵ
tan   y / x
Scalar product = A  B  Ax Bx  Ay By
A  2iˆ  4 ĵ
r
B  5iˆ  2 ĵ
r r
A  B  (2)(5)  (4)(2)  18
A
B

AB
Also
A  B  A B cos
18
cos 
 0.748
20 29
  41.63deg
AB is the perpendicular projection of A
on B. Important later.
A
B

90 deg.
AB
A  2iˆ  4 ĵ
r
B  5iˆ  2 ĵ
r r
A  B  (2)(5)  (4)(2)  18
A B
B
18
AB 
 3.34
29
AB 
Also
AB  A cos
AB  20(0.748)
AB  (4.472)(0.748)  3.34
Vectors in 3 Dimensions
For a Right Handed 3D-Coordinate
Systems
y
j
i
k
x
z
iˆ  ĵ  k̂
Right handed rule.
Also called cross
product
r  3iˆ  2 ĵ  5 k̂
Magnitude of
r  32  2 2  5 2
Suppose we have two vectors in 3D and
we want to add them
y
r1  3iˆ  2 ĵ  5 k̂
2 j
5
i
x
k
r1
7
r2
z
1
r2  4 iˆ  1 ĵ  7 k̂
Adding vectors
Now add all 3 components
y
r  r1  r2
r  3iˆ  2 ĵ  5 k̂
1
j
r2  4 iˆ  1 ĵ  7 k̂
r  1iˆ  3 ĵ  12 k̂
i
k
r
r2
r
1
z
x
Scalar product = r1  r2
r1  3iˆ  2 ĵ  5 k̂
r  4 iˆ  1 ĵ  7 k̂
2
r1  r2  (3)(4)  (2)(1)  (5)(7)  25
Cross Product
See your textbook Chapter 3 for more information on vectors
Later on we will need to talk about cross products. Cross
products come up in the force on a moving charge in E/M
and in torque in rotations.
Differential Calculus
Definition of Velocity when it is smoothly
changing
Define the instantaneous velocity
Recall
(x2  x1 ) x
v

(t 2  t1 ) t
x
as t
v  lim
t
Example
x  12 at 2
x  f (t)
(average)
0 = dx/dt (instantaneous)
DISTANCE-TIME GRAPH FOR UNIFORM
ACCELERATION
v x /t
dx/dt = lim x /t as t
0
x + x = f(t + t)
x
x  12 at 2
x  f (t)
x = f(t + t) - f(t)
x, t
.
t

x = f(t)
(t+t)
t

Differential Calculus: an example of a
derivative
x  12 at 2
f (t)  12 at 2
x  f (t)
f (t  t)  12 a(t  t) 2
dx/dt = lim x /t as t
 12 a(t 2  2tt  (t) 2 )
0

2
2
2
1
1
f (t  t)  f (t)
a(t

2tt

(t)
)

at
2

 2

t
t
a(2tt  (t) 2 )

t
1
2
dx
 at
dt
v  at

 12 a(2t  t)  at
t  0
velocity in the x direction



Three Important Rules of Differentiation
Power
Rule
Product Rule
y  cx n
y  30x 5
dy / dx  ncx n1
dy
 5(30)x 4  150x 4
dx
y(x)  f (x)g(x)
dy df
dg

g(x)  f (x)
dx dx
dx
y  3x 2 (ln x)
dy
1
 2(3)x(ln x)  3x 2 ( )  6x ln x  3x
dx
x
dy
 3x(2 ln x  1)
dx
y  (5x 2  1)3  g 3 where g=5x 2  1
y(x)  y(g(x))
Chain Rule
dy dy dg

dx dg dx
dy
2 dg
 3g
 3(5x 2  1)2 (10x)
dx
dx
dy
 30x(5x 2  1)2
dx
Problem 4-7 The position of an electron is given by the following
2
displacement vector r  3tiˆ  4t ĵ  2 k̂ , where t is in s and r is in m.
What is the electron’s velocity v(t)?
dr
v
 3iˆ  8tĵ
dt
What is the electron’s velocity at t= 2 s?
v
dr
 3iˆ  16 ĵ
dt
vx  3m / s
vy  16m / s
+vy
3
+vx
f
-16
What is the magnitude of the velocity or speed?
v  32  16 2  16.28m / s
What is the angle relative to the positive direction of the x axis?
16
f  tan (
)  tan 1 (5.33)  79.3deg
3
1
v
Integral Calculus

How far does it go?
v=dx/dt
v= at
vi
ti
t
N
N
N
i1
i1
i1
x   x i   v it i   at it i
Distance equals area under speed graph regardless of its shape
Area = x = 1/2(base)(height) = 1/2(t)(at) = 1/2at2
Integration:anti-derivative
N
 at t  
i
i
tf
0
atdt where t i  0 and N  
i1

tf
0
1 2
atdt  at
2
x  12 at
2
tf
0
1
1
2
= a (t f  0)  a t f 2
2
2