Mathematical Methods A review and much much more! 1

Trigonometry Review  First, recall the Pythagorean theorem for a 90 0 right triangle  a 2 +b 2 = c 2 2 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 3 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 4 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 5 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-  ) 6 x y r  I II 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 7 Positive slope Negative slope

Definition of slope 8 x 1, y 1 x 2, y 2

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 9

The Slope of a Circle  Thus a circle is a near-infinite set of sloped lines. 10

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) 11 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 12

Power rule  f(x)=ax n  The derivative is :  f’(x) = a*n*x n-1  A tricky example: 13

Differential Operator  For x, the operation of differentiation is defined by a differential operator 14 And the last example is formally given by

3 rules  Constant-Multiple rule  Sum rule  General power rule 15

3 Examples 16 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) 17

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 18

Can I take the derivative of a derivative? And then take its derivative?  Yep! Look at 19 Called “2 nd derivative” 3 rd 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” 20

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: th 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 22

Constant of integration  Two different functions can have the same derivative. Consider  f(x)=x  f(x)=x  f’(x)=4x  So without any extra information we must write  Where C is a constant.  We need more information to find C 23

Power rule for integration 24

Can I integrate multiple times?  Yes! 25

Examples 26

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 27 x=a x=b f(x)

Mathematically 28 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 29

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