Mathematical MethodsA review and much much more!

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Trigonometry Review

First, recall the Pythagorean theorem for a 900 right triangle

a2+b2 = c2

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a

b

c

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

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a

b

c

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

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x

y

r

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

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y

x

r

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-)

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x

y

r

III

III IV

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

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Positive slope

Negative slope

Definition of slope

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12

xxyym

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x1 , y1

x2 , y2

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)

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f’(x)

f(x)

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

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Power rule

f(x)=axn

The derivative is : f’(x) = a*n*xn-1

A tricky example:

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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

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1)('

1)(

1)(

xxf

xdx

dxf

dx

d

xxf

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And the last example is formally given by

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

2222

2

2

1)(

)(:1)(

ctbtadt

d

tdt

dtf

dt

d

xftNotetdx

dtf

dx

d

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Differentiate the following:

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

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Called “2nd derivative”

3rd derivative

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

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

Called an “indefinite integral”

What is the “dx”?

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

dxxfxf

dxxfxfd

dxxfxfd

xfxfdx

d

)(')(

)(')(

)(')(

)(')(

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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

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x=a x=b

f(x)

Mathematically

b

a

afbfdxxf )()()('

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Note: Technically speaking the integral is equal to f(x)+c and so(f(b)+c)-(f(a)+c)=f(b)-f(a)

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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