1 Mathematical Methods A review and much much more!

2 Trigonometry Review First, recall the Pythagorean theorem for a 90 0 right triangle a 2 +b 2 = c 2 a b c

3 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 

4 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 

5 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 

6 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  I II III IV

7 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

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

9 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

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

11 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)

12 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

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

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

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

16 3 Examples Differentiate the following:

17 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)

18 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

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

20 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”

21 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: 18 th Century symbol for “s” Which is now called an integral sign! Called an “indefinite integral”

22 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

23 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

24 Power rule for integration

25 Can I integrate multiple times? Yes!

26 Examples

27 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)

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

29 What to practice on: Be able to differentiate using the 4 rules herein Be able to integrate using power rule herein