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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

a2+b2 = c2 a

b

c

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Trigonometry Review

Next, recall the definitions for sine and cosine of the angle .

sin = b/c or sin = opposite / hypotenuse

cos = b/c cos = adjacent / hypotenuse

tan = b/a tan = opposite / adjacent

a

b

c

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Trigonometry Review

Now define in general terms: x =horizontal direction y = vertical direction

sin = y/r or sin = opposite / hypotenuse

cos = x/r cos = adjacent / hypotenuse

tan = y/x tan = opposite / adjacent

x

y

r

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Rotated

If I rotate the shape, the basic relations stay the same but variables change x =horizontal direction y = vertical direction

sin = x/r or sin = opposite / hypotenuse

cos = y/r cos = adjacent / hypotenuse

tan = x/y tan = opposite / adjacent

y

x

r

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Unit Circle

Now, r can represent the radius of a circle and , the angle that r makes with the x-axis

From this, we can transform from ”Cartesian” (x-y) coordinates to plane-polar coordinates (r-)

x

y

r

III

III IV

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The slope of a straight line

A non-vertical has the form of 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

Positive slope

Negative slope

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Definition of slope

x1 , y1

x2 , y2

12

12

xxyym

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The Slope of a Circle

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

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The Slope of a Circle

Thus a circle is a near-infinite set of sloped lines.

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The Slope of a Curve

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)=mThe derivative of a straight line is a constant

What if f(x)=b (or the function is constant?)Slope =0 so f’(x)=0

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Power rule

f(x)=axn

The derivative is : f’(x) = a*n*xn-1

A tricky example:

2

312

1

2

1

2

1

2

1)('

)(

1)(

xxxf

xxf

orx

xf

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Differential Operator

For x, the operation of differentiation is defined by a differential operator

dx

d

And the last example is formally given by

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1)('

1)(

1)(

xxf

xdx

dxf

dx

d

xxf

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3 rules

Constant-Multiple rule

Sum rule

General power rule

)()()(

)()(

constant a,)()(

1 xfdx

dxfnxf

dx

d

xgdx

dxf

dx

dg(x)f(x)

dx

d

kxfdx

dkxfk

dx

d

nn

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3 Examples

Differentiate the following:

2222

2

2

1)(

)(:1)(

ctbtadt

d

tdt

dtf

dt

d

xftNotetdx

dtf

dx

d

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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

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Can I take the derivative of a derivative? And then take its derivative?

Yep! Look at

0)(

24)(

24)(

12)(

:compactly More

12344)(

4)(

)(

5

5

4

4

3

3

22

2

223

3

4

xfdx

d

xfdx

d

xxfdx

d

xxfdx

d

xxxdx

dxf

dx

d

dx

d

xxfdx

d

xxf

Called “2nd derivative”

3rd derivative

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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”

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Why is it called integration?

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:

)()(' xfdxxf

18th Century symbol for “s”Which is now called an integral sign!

Called an “indefinite integral”

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What is the “dx”?

dxxfxf

dxxfxfd

dxxfxfd

xfxfdx

d

)(')(

)(')(

)(')(

)(')( 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

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Constant of integration

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

Where C is a constant. We need more information to find C

Cxdxx 44

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Power rule for integration

Cxn

adxax nn

1

1

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Can I integrate multiple times?

Yes!

4322314

322

13

322

13

32213

212

212

212

1

1

264

423

1212

122

2424

2424

CxCxC

xC

xdxCxCxCx

CxCxCxCxCxC

xdxCxCx

CxCxCxCxdxCx

Cxdx

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Examples

dx

x

dtt

xftdxt

2

2

2

1

1

)( :Note1

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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

x=a x=b

f(x)

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Mathematically

b

a

afbfdxxf )()()('

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