Mathematical Methods

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# Mathematical Methods - PowerPoint PPT Presentation

Mathematical Methods. A review and much much more!. Trigonometry Review. First, recall the Pythagorean theorem for a 90 0 right triangle a 2 +b 2 = c 2. c. b. a. Trigonometry Review. Next, recall the definitions for sine and cosine of the angle q . sin q = b/c or

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### Mathematical Methods

A review and much much more!

Trigonometry Review
• First, recall the Pythagorean theorem for a 900 right triangle
• a2+b2 = c2

c

b

a

Trigonometry Review
• 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

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

r

x

q

y

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

r

y

q

x

Unit Circle

I

• Now, r can represent the radius of a circle and q, the angle that r makes with the x-axis
• From this, we can transform from ”Cartesian” (x-y) coordinates to plane-polar coordinates (r-q)

II

IV

III

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

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
The Slope of a Circle
• Thus a circle is a near-infinite set of sloped lines.
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)

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
• f(x)=axn
• The derivative is :
• f’(x) = a*n*xn-1
• A tricky example:
Differential Operator
• For x, the operation of differentiation is defined by a differential operator
• And the last example is formally given by
3 rules
• Constant-Multiple rule
• Sum rule
• General power rule
3 Examples

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)
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
• Yep! Look at

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”
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:

Called an “indefinite integral”

18th Century symbol for “s”

Which is now called an integral sign!

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
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.
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

Mathematically

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