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Prerequisites of Calculus - Coggle Diagram

- - - - Odd functions we have learned how to draw. Odd functions is when -f(x)=f(-x), thus the graph of the function is symmetrical to the origin. All of the following functions in determining odd or even are using its base function
        
        y=x: linear function*(one to one) (D:xER, R:yER)
        
        Standard form: ax+by=c
        Slope Intercept form: y=mx+b
        Point slope form:y-y1=m(x-x1)
        Where m represent slope,
        b represent y intercept in slope intercept form
        
        Two lines are parallel to each other if m1= m2
        Two lines are perpendicular to each other if m1*m2=-1
        
        Linear function is only odd when the y intercept is at (0,0), if the y intercept is at anywhere else, it would become neither odd or even
        
        y=1/x: rational function(can be one to one) (D:xER, x can not be 0, R:yER, y cannot be 0)
        
        y=bx^m/ax^n
        
        if m<n, the horizontal asymptote is y =0
        if m=n, the horizontal asymptote is y=b/a
        if m>n, the horizontal asymptote does not exist, by using numerator to divided by denomenator we can find the oblique asymptote instead of having a horizontal asymptote
        
        Vertical asymptote is such that the x value would make the denomenator 0, given that there are no common factors between the numerator and denomenator
        
        Hole/P.o.d do exist in rational function
        
        y=sinx: sin function(not one to one) (D:xER, R: -1<=y<=1)
        
        y=a sin b(x-c) +d where a is amplitude, period= 2π/b, c is phase shift, d is mid point
        
        Period: 2π / b is the period because as explained previously, a circle is a shape that consists of angles of a sum of 360 degrees, and since the circle repeats itself after a "loop", the period of the base function of sine function would be 2π.
        The range of the function is due to the fact that a unit circle has a radius of 1, and in a triangle the two other sides should never be longer in length than the hypotenuse thus the highest value is +-1.
        
        y=tanx: tan function(nolt one to one) (D: x can not be π/2+n(π) where n is all integer, R:yER)
        
        y=a tan b(x-c) +d where a is amplitude, period= π/b, c is phase shift, d is mid point
        
        Period: Due to the special property of tan being y/x, instead of using the period as 2π, we can simplify it in to π. Not only does every 2π the tangent funciton repeats its y values, every π also does if the b value is 1.
        Range: Different than sine and cosine functions, the tangent function has a range of yER, this is because different than the ratio of sin and cos, tan has the denomenator as one of the side that is not the hypotenuse, thus it can be 0, resulting it to have an asymptote along with a yER range.
        
        The tan function also has vertical asymptote, the equation of the vertical asymptote is x=π/2+n(π) where n is all integer. This is due to the fact that a fraction should have 0 as the denomenator, thus creating asymptote similar to rational function.
      - Even functions we have learned how to draw.
        Even functions is when f(x)=f(-x), thus the graph of the function is symmetrical to the y axis. All of the following functions in determining odd or even are using its base function
        
        y=cosx: cos function(not one to one) (D:xER, R: -1<=y<=1)
        
        y=a cos b(x-c) +d where a is amplitude, period= 2π/b, c is phase shift, d is mid point
        
        Period: 2π / b is the period because as explained previously, a circle is a shape that consists of angles of a sum of 360 degrees, and since the circle repeats itself after a "loop", the period of the base function of cosine function would be 2π.
        The range of the function is due to the fact that a unit circle has a radius of 1, and in a triangle the two other sides should never be longer in length than the hypotenuse thus the highest value is +-1.
        
        y=x^2: quadratic function(not one to one) (D:xER, R:Y=>0)
        
        General Form: y=ax^2 +bx +c Vertex form: y=a(x-h)^2+k
        The vertex of the graph of the vertext form would be (h,k)
        
        If a>0 then the bell opens up, opens down if a<0
        
        Quadratic formula in solving for x intercept: (-b+- sqrtb^2-4ac)/2a
      - Neither Even or Odd
        
        Piece wise function (can be one to one)
        
        A piece wise function is a function that is consist of multiple sub functions, where each sub function is responsible for parts of a domain. Thus, the functions learned previously can all be applied to the piece function
        
        y=b^x: exponential function(one to one) (D:xER, R:y>0)
        
        y=logx. Inverse of the exponential function: logarithmic function(one to one) (D:x>0, R:yER)
        
        y=log a (x)
        log base a to the argument of x
        
        a>1, then as x increases y increases, a<1 as x increases y decreases
        Different than exponential function, it has a vertical asymptote instead of a horizontal one,because of the base exponential function can not obtain a result of 0.
        
        logarithm rules:
        log(xy)=log(x)+log(y) log(x/y)=log(x)−log(y) logx^m=mlog(x)
        logx (y)=logy/logx
        
        1 more item...
        
        Log with the base e would be called ln, which is natural logarithm, base e exponential functions have applications listed above
        
        y=b^x+c
        
        It has horizontal asymptote where y>c, due to the fact that a result of base exponential function can not be 0
        
        When b>1 then it is a exponential growth, if b<1 it is a exponential decay
        
        Exponent laws: a^m*a^n=a^m+n a^m/a^n=a^m-n (a^m)^n=a^mn (ab)^n=a^nb^n (a/b)^n=a^n/b^n
        
        Application learned with exponential functions
        
        Used for compound interest, A= future amount, A0=initial amount, i= interest rate,
        n=how it is compounded(annnually(1 time per year), semi-annually(2 times per year), quaterly(4 times per year),monthly(12 times per year), Daily(365 times) per year,
        t representing number of years elapsed from the intial amount was put in the bank
        
        Used for continuous growth/decay: A= future amount, A0=initial amount, i= type of growth(can be negative if it is a decay), t representing time elapsed
        
        Used for exponential growth/decay: A= future amount, A0=initial amount, p=period for the growth/decay to happen, t representing time elapsed
        
        int function (one to one) (D:xER, R:y can be all of the integers)
        
        The function that is drawn such that for the y value of the given x value will be the integar that is smaller or equal to the x value
        
        y=√x: Radical Function (one to one) (D:x=>0, R:y=>0
        
        Due to the fact radical functions can not have radicand that is negative, thus impacting the y value has to be above 0
      - Polynomial(can be one to one)
        
        Polynomials are sums of terms of the form k⋅xⁿ, thus it can contain any of the function we have learned. As a result, it can be either odd, even or neither depending on the specific polynomial.
    - - Relations we have learned
        
        Parametric equations: A type of equation that uses an independent variable called a parameter (often named as t) and in which dependent variables are defined as continuous functions of the parameter and are not dependent on another existing variable.
        
        By using the equations, we can create all of the functions through subsituting t and manipulate the domain of t. ( the relations can be one to relations)
        
        New function equation learned in this parametric equation section: If the graph includes any of the 6 trignometric ratios, we can us e x^2/a^2+y^2/b^2=1 to check if it is an ellipses. Ellipses are defined as a circle that is stretched in one direction that has a shape of a oval
        
        Circle "function" (not one to one) (Domain and range of a unit circle: D:-1<=x<=1, R:-1<=y<=1)
        
        (x – h)^2 + (y – k)^2 = r^2
        
        The h is x coordinate of the center of the circle, the k is the y coordinate of the center of the circle, the r represent the radius of the circle.
        
        Unit circle is a circle that has the center at (0,0), the radius is 1
- - - - A triangle is a polygon with three edges and three vertices.
    - - Double Angle Identities
        cos(2x) = cos2(x) – sin2(x) = 1 – 2 sin^2(x) = 2 cos^2(x) – 1
        sin(2x) = 2 sin(x) cos(x)
      - Addition and Differenmce Identities
        sin(α + β) = sin(α) cos(β) + cos(α) sin(β) sin(α – β) = sin(α) cos(β) – cos(α) sin(β) cos(α + β) = cos(α) cos(β) – sin(α) sin(β) cos(α – β) = cos(α) cos(β) + sin(α) sin(β)
      - sine law: a/sinA=b/sinB=c/sinC
      - Cosine law: c^2=a^2+b^2-2abcosC
        It not only apply for finding angle C, it can also apply for angle A and B.
      - Area of a triangle: base x height x 0.5
      - Rules of right angle triangle
        
        Sin theta = opposite side/hypotenuse side
        Cos theta = adjacent side/hypotenuse side
        Tan theta = opposite side/adjacent side
        The recipricals of the three rules above
        csc thata =1/Sin theta = hypotenuse side/opposite side
        sec thata = 1/Cos theta = hypotenuse side/adjacent side side
        cot theta = 1/ Tan theta =adjacent side/opposite side
        Note: Since this portion is closely related to circle, depending on the angle position on the cartesian plane, it will can have a positive or negative value.
        
        Pythagorean theorem: a^2=b^2+c^2, The squared length value of the hypotenuse is equal to the sum of the squared length value of the base and the height of the triangle
      - Pythagorean Identities
        sin^2(t) + cos^2(t) = 1 tan^2(x) + 1 = sec^2(x) 1 + cot^2(x) = csc^2(x)
  - - - Circumfrence formula for circle: 2 x radius of the circle x π
        
        arc formula of a circle: 2 x radius of the circle x π x angle in radians / 2π. As mentioned in the "What are they" section, circles are shapes that is consisted of 360 degree worth of angle, the proportion of angle to 360 degree is the same as the arc length to the circumference. We can simplify the equation of arc length into arc length=r x angle in radians
      - Area formula for circle: π x r^2
        
        area formula for any given portion of a circle:π x r^2 x angle in radians / 2π. As mentioned in the "What are they" section, circles are shapes that is consisted of 360 degree worth of angle, the proportion of angle to 360 degree is the same as the arc length to the circumference. We can simplify the equation of arc length into area formula for any given portion of a circle: 1/2 x r^2 x angle in radians
    - - A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
        
        circles are shapes that consists 360 degree worth of angle, thus the rules mentioned in triangle regarding angles can also be used when a angle is taken from the circle
    - - Radians is the SI unit of measuring angles in math, number of radians is counted through how many radius is needed for the given arc length of the anlge in the circle. It can be converted into angle through radians x 180/ π. Angles can be converted to radians through angle measure x π/ 180. Radians can be also applied to the 6 trignometric ratios