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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

Mathematical Methods

A review and much much more!

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

c

b

a

trigonometry review1
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 review2
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

rotated

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
unit circle

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
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 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 circle1
The Slope of a Circle
  • Thus a circle is a near-infinite set of sloped lines.
the slope of a curve
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
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
Power rule
  • f(x)=axn
  • The derivative is :
    • f’(x) = a*n*xn-1
  • A tricky example:
differential operator
Differential Operator
  • For x, the operation of differentiation is defined by a differential operator
  • And the last example is formally given by
3 rules
3 rules
  • Constant-Multiple rule
  • Sum rule
  • General power rule
3 examples
3 Examples

Differentiate the following:

functions
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
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
Can I take the derivative of a derivative? And then take its derivative?
  • Yep! Look at

Called “2nd derivative”

3rd derivative

can i reverse the process
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
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
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
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
definite integral
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
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
What to practice on:
  • Be able to differentiate using the 4 rules herein
  • Be able to integrate using power rule herein