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Mathematical
Methods
A review and much much more!
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Trigonometry Review
First, recall the
Pythagorean theorem
for a 900 right
triangle

2
2
a +b
=
2
c
c
b
a
Trigonometry Review
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 Next, recall the definitions for
sine and cosine of the angle q.
 sin q = b/c or
 sin q = opposite /
hypotenuse
 cos q = b/c
 cos q = adjacent /
hypotenuse
 tan q = b/a
 tan q = opposite /
adjacent
c
b
q
a
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Trigonometry Review
 Now define in general
terms:
x =horizontal
direction
y = vertical direction
 sin q = y/r or
 sin q = opposite /
hypotenuse
 cos q = x/r
 cos q = adjacent /
hypotenuse
 tan q = y/x
 tan q = opposite /
adjacent
r
y
q
x
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Rotated
sin q = x/r or
 sin q = opposite / hypotenuse
cos q = y/r
 cos q = adjacent / hypotenuse
tan q = x/y
 tan q = opposite / adjacent
y
q
r
 x =horizontal direction
 y = vertical direction
x
If I rotate the
shape, the basic
relations stay the
same but variables
change
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Unit Circle
Now, r can
represent the radius
of a circle and q,
the angle that r
makes with the xaxis
From this, we can
transform from
”Cartesian” (x-y)
coordinates to
plane-polar
coordinates (r-q)
I
II
r
y
q
x
III
IV
The slope of a straight line
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 A non-vertical has the
form of
Positive slope
 y = mx +b
 Where
 m = slope
 b = y-intercept
 Slopes can be positive
or negative
 Defined from whether
y = positive or
negative when x >0
Negative slope
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Definition of slope
m
 y 2  y1 
 x2  x1 
x2 , y2
x1 , y1
The Slope of a Circle
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 The four points picked
on the circle each have
a different slope.
 The slope is determined
by drawing a line
perpendicular to the
surface of the circle
 Then a line which is
perpendicular to the
first line and parallel to
the surface is drawn. It
is called the tangent
The Slope of a Circle
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Thus a circle is a
near-infinite set of
sloped lines.
The Slope of a Curve
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 This is not true for just
circles but any
function!
 In this we have a
function, f(x), and x, a
variable
 We now define the
derivative of f(x) to be
a function which
describes the slope of
f(x) at an point x
 Derivative = f’(x)
f’(x)
f(x)
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Differentiating a straight line
 f(x)= mx +b
 So
 f’(x)=m
 The derivative of a straight line is a constant
 What if f(x)=b (or the function is constant?)
 Slope =0 so f’(x)=0
Power rule
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f(x)=axn
The derivative is :
 f’(x) = a*n*xn-1
A tricky example:
1
f ( x) 
x
or
f ( x)  x

1
2
1
f ' ( x)   x
2
 1 
  1 
 2 
3
1 2
 x
2
Differential Operator
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For x, the operation of differentiation is
defined by a differential operator
d
dx

And the last
example is
formally
given by
1
f ( x) 
x
d
d  1 
f ( x)  

dx
dx  x 
1
f ' ( x) 
2 x3
3 rules
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Constant-Multiple
rule
d
k  f ( x)   k  d  f ( x) , k a constant
dx
dx
Sum rule
d
 f(x)  g(x)  d  f ( x)   d g ( x) 
dx
dx
dx
General power rule


d
 f ( x)n  n   f ( x)n1 d  f ( x) 
dx
dx
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3 Examples
Differentiate the following:




d
d 2
f (t ) 
t 1
dx
dx
d
d 2
f (t ) 
t 1
dt
dt

d 2 2
a t  b 2t  c 2
dt

Note : t  f ( x)
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Functions
 In mathematics, we often define y as
some function of x i.e. y=f(x)
 In this class, we will be more specific
 x will define a horizontal distance
 y will define a direction perpendicular
to x (could be vertical)
 Both x and y will found to be
functions of time, t
 x=f(t) and y=f(t)
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Derivatives of time
Any derivative of a function with
respect to time is equivalent to
finding the rate at which that
function changes with time
Can I take the derivative of a derivative? And then
take its derivative?
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Yep! Look at
f ( x)  x 4
d
f ( x)  4 x 3
dx
d d
 d
f ( x)  
4 x 3  4  3  x 2  12 x 2

dx  dx
 dx
More compactly :
 
Called “2nd derivative”
3rd
derivative
d2
dx 2
d3
dx 3
d4
dx 4
d5
dx 5
f ( x)  12 x 2
f ( x)  24 x
f ( x)  24
f ( x)  0
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Can I reverse the process?
 By reversing, can we take a derivative and find
the function from which it is differentiated?
 In other words go from f’(x) → f(x)?
 This process has two names:
 “anti-differentiation”
 “integration”
Why is it called integration?
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Because I am
summing all the
slopes (integrating
them) into a single
function.
Just like there is a
special differential
operator, there is a
special integral
operator:
th
Called an “indefinite integral”

f ' ( x) dx  f ( x)
18 Century symbol for “s”
Which is now called an integral sign!
What is the “dx”?
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 The “dx” comes from
the differential
operator
 I “multiply” both sides
by “dx”
 The quantity d(f(x))
represents a finite
number of small
pieces of f(x) and I use
the “funky s” symbol
to integrate them
 I also perform the
same operation on
the right side
d
 f ( x)   f ' ( x)
dx
d  f ( x)   f ' ( x)  dx
 d  f ( x)   f ' ( x)  dx
f ( x)   f ' ( x)  dx
Constant of integration
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 Two different functions can have the same
derivative. Consider
 f(x)=x4 + 5
 f(x)=x4 + 6
 f’(x)=4x
 So without any extra information we must write
 4 x dx  x
4
C
 Where C is a constant.
 We need more information to find C
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Power rule for integration
a n1
ax
dx

x

C

n 1
n
Can I integrate multiple times?
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Yes!
 24  dx  24 x  C
1
24 2
2


24
x

C

dx

x

C
x

C

12
x
 C1 x  C2
1
1
2

2
 12 x

12 3 C1 2
x  x  C2 x  C3  4 x 3  C1 x 2  C2 x  C3
3
2
C
C
4 x 3  C1 x 2  C2 x  C3  dx  x 4  1 x 3  2 x 2  C3 x  C4
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2

2
 C1 x  C2  dx 

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Examples
 t
 t
2
2

 1dt
 1 dx Note : t  f ( x)
1
 x 2 dx
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Definite Integral
 The definite integral of f’(x) from x=a to x=b defines the area under
the curve evaluated from x=a to x=b
f(x)
x=a
x=b
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Mathematically
b
 f ' ( x)dx  f (b)  f (a)
a
Note: Technically speaking the integral is equal to f(x)+c and so
(f(b)+c)-(f(a)+c)=f(b)-f(a)
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What to practice on:
 Be able to differentiate using the 4 rules
herein
 Be able to integrate using power rule
herein
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