Please enable JavaScript.

Coggle requires JavaScript to display documents.

Calculus (Integrals (Indefinite Integral (Notation f(x) dx Anti…

- - - - Relationship To Derivative
        Integrals are also known as
        anti-derivatives, because say f(x) dx = g(x)
        then f(x) is the derivative of g(x)
        
        Integral Rules
        Constant Term: a dx = ax + C
        Power Term: axn dx = axn+1 / (n+1) + C
        eu du = eu + C
        1 / u du = lnlul + C
        au du = (au / ln(a)) +C
        du / lulSQRT(u2-a2) = (1/a) sec-1(lul/a) +C
        du / SQRT(a2-u2) = sin-1(u/a) +C
        1 / (u2+a2) du = (1/a)tan-1(u/a) +C
        cosx dx = sinx +C secxtanx dx = secx +C
        sinx dx = -cosx +C cscxcotx dx = -cscx +C
        sec2x dx = tanx +C csc2x dx = -cotx +C
        
        Why +C?
        Consider the derivative of 2x + 2,
        it is just 2. The constant ( + 2)
        became a 0 when we took the
        derivative, we’d get the same
        result if it were +3 or +4596.
        So when we take the
        ANTI-DERIVATIVE, we have to
        account that the constant can be
        anything, and we do that with a +C.
        
        Applications
        Taking the integral of an acceleration
        equation, gets you the velocity equation.
        Taking the integral of a velocity equation,
        gets you its position equation.
        Taking the integral of any rate equation,
        get you an equation where you can find
        the exact amount at x.
  - - - Definite Integral vs. Indefinite Integral
        The biggest difference between the two is
        that an indefinite integral gets you an
        equation, while a definite integral gets
        you an actual numerical answer
        
        How to Evaluate Definite Integrals
        -Graphing the equation and creating
        geometric shapes and using area
        equations (like A of square = side^2)
        to find the area under the curve.
        However you must make sure that
        these are actual, perfect geometric shapes,
        and once you do this gets you an
        EXACT answer.
        
        Riemann Summs: However, can be a
        pain to evaluate w/o a calculator, and
        also results in only an approximate answer.
        
        Fundamental Theorem of Calculus:
        Must be given the equations to work
        with, and can be near impossible/take
        forever if the equations are extremely
        complicated. However, produces an exact
        value (unless there are decimals).
        
        Calculator FnInt: Obviously calculator
        needs to be allowed, and equation must
        also be given. Gives you an exact value.
        
        Applications
        Taking a definite integral of an
        acceleration equation gets you
        the net change in velocity from
        a to b. Taking a definite integral
        of a velocity equation gets you
        the net change in position from
        a to b. Taking the integral of a
        rate equation (say for example
        rate of water being pumped into
        a tank), you get the exact
        amount of water that was
        pumped from time a to b.
  - - - Area?
        First off, a definite integral
        of an equation, finds the
        ‘area under the curve’ that
        we’ve heard so much of.
        On top of that integrals
        can be used to find the area
        of a region between two
        curves = (f(x) - g(x)) dx
        (Top equation minus
        bottom equation)
        
        Volume
        The Volume of a solid
        resulting from a region
        between curves being
        rotated around an axis
        can be found with
        Disk Method: V = pi[r(x)]2 dx
        where r(x) = f(x) - axis of rotation
        Washer Method: V = pi[R2(x) - r2(x)]
        where R(X) = (outer eqn) - (axis) &
        r(x) = (inner eqn) - (axis).
        OR Solids with geometric cross-sections
        where V = (area of cross-section)dx
        
        Derivative of Anti-Derivative
        & FTC
        Say we want to take the derivative
        of the definite integral x dx. FTC
        says the evaluation of that integral
        would be F(X) - F(a), and we want
        the derivative of that. Well since
        F(a) is a constant that becomes zero,
        and we’re left with just F’(x) which is
        basically how we take the derivative
        of an anti-derivative.
        
        Applications
        In addition to everything mentioned
        in the other applications segments,
        there are a couple more real life
        applications. My personal favorite is
        the amusement park-type problem
        where you have two equations which
        give the rate of people entering, and
        people leaving (or like the raffle
        question). Water being pumped into
        a tank at a rate, and also pumped
        out is another common situation.
  - - - Steps to Solve:
        Differential Equations
        
        Separate the Variables
        
        Integrate each side
        (Only put +C on one side)
        
        Rework equation around
        until you get y = f(x)
  - - - How can we
        use them?
        If enough miniature
        tangents are plotted,
        an image is formed!
        This image is the
        the shape of the
        ORIGINAL
        function! We know
        what the original
        function looks like
        without actually
        doing any actually
        integration.
- - - - Not Differentiable
        
        & Discontinuous
      - - Tangent cannot
        
        be drawn
      - - f'(x) is undf.
    - - Not Differentiable
        
        & Continuous
      - For Tangent @ x=c
      - given f(x) isn't changing
      - slope is undf.
      - (f '(c) is undefined)
      - Vertical Tangent
- - - - Slope is Undefined
        f(x) is not changing
        f'(x) is undefined
    - - Positive Slope
        f(x) is increasing
        f'(x) is positive
      - Negative Slope
        f(x) is decreasing
        f'(x) is negative
      - Slope = 0
        f(x) is not changing
        f'(x) = 0
- - - - Graphical Approach
        Looking at the graph
        to determine L
        
        Hole or Not
        When there is a hole
        or if the entire function
        is continous, the #1 rule
        is to 'Stay on the graph.'
        If you follow this rule,
        the y-value at c is L,
        the limit.
        
        Non-Removable
        Discontinuities
        If there is a non-
        removable discon-
        tinuity at c, then NO
        LIMIT EXISTS (NLE)
        Examples include:
        Steps, Jumps &
        Vertical Asymptotes
        
        Horizontal
        Asymptotes
        If there is a HA
        @ y = a, then we
        know that the limit
        as x -> infinity of
        f(x) is equal to a.
        
        No Limit Exists
        The universal way to
        tell if No limit exists, is
        to check to see if the
        right side limit is the same
        as the left side limit. If they
        are, then the limit exists,
        if they aren't then no limit
        exists.
        
        Algebraic Approach
        
        Simple Stuff
        On the simplist of
        graphs, finding f(c)
        will find the limit of
        f(x) as x->c
        
        No Limit Exists
        If when finding f(c)
        you get an undefined
        answer (ex. 4/0) then
        no limit exists!
        
        No Limit Exists
        The universal way to
        tell if No limit exists, is
        to check to see if the
        right side limit is the same
        as the left side limit. If they
        are, then the limit exists,
        if they aren't then no limit
        exists. This is important when
        dealing with piece-wise
        functions, where you have
        equations for a left side &
        a right side.
        
        Indeterminant
        However if you get
        f(c) = 0/0, this is
        an indeterminant case.
        You must find the
        extended function by
        tweaking f(x) & then
        finding the limit as x->c
        of the adjusted function.
        OR use L'Hopital's Rule:
    - - Defining e
        text
      - Creating a Derivative
        Definition of Derivative:
        text
      - Determining Continuity
        Rules of Continuity:
        f(a) is defined
        f(a) = L as x->a
        limit as x-> a exists
      - Instantaneous
        Rate
        =
        text
      - Calculating a
        Riemann Sum w/
        infinite rectangles
        
        text
- - - - First Derivative f'(x)
        -Equation denoted f'(x)
        -Finds slope of tangent
        at a point on f(x)
        -Finds instant rate
        
        When f'(x) = ?
        
        Derivative Rules
        Constant Term: y = c, y' = 0
        Power Term: y = ax^n, y' = nax^(n-1)
        Sum of Terms: y = u + v, y' = u' + v'
        Product Rule: y = uv, y' = vu' + uv'
        Quotient Rule: y = u/v, y' = (vu' - uv')/v^2
        Chain Rule: y = f(g(x)), y' = g'(x)f'(g(x))
        Trig Derivatives
        y = sinx, y' = cosx
        y = cosx, y' = -sinx
        y = tanx, y' = (secx)^2
        y = cscx, y' = -cscxcotx
        y = secx, y' = secxtanx
        y = cotx, y' = -(cscx)^2
        Exponential Derivatives
        y = ln(u), y' = u'/u
        y = ln(x), y' = 1/x
        y = b^u, y' = ln(b)u'b^u
        y = e^u, y' = u'e^u
        
        f'(x) is positive
        -f(x) is continuous
        -f(x) is increasing
        -can draw tangent
        f'(x) is negative
        -f(x) is continuous
        -f(x) is decreasing
        -can draw tangent
        f'(x) = 0
        -f(x) is continuous
        -f(x) is not changing
        -can draw tangent
        
        f'(x) is undf.
        
        f(x) is Continuous
        -f(x) is not changing
        -vertical tangent
        
        f(x) is Discontinuous
        -cannot draw tangent
        
        When f'(a) = 0 or is undefined
        there is an EVP @ x=a on f(x)
        EVP: Extreme Value Point
        EVPs are maxs or mins
        also known as turning points
        Use Derivative sign chart to
        determine max or min
        
        Gradual Change
        -Horizontal Tan
        -When f'(x) = 0
        
        Abrupt Change
        -Vertical Tan
        -When f'(x) is undf.
      - Apply Definition of
        Derivative a 2nd time
        or Derivative Rules again
        Second Derivative f''(x)
        -Equation denoted f''(x)
        -used to find shape of f(x)
        either concave up or
        concave down
        
        When f''(x) = ?
        
        f''(x) is positive
        -f(x) is c.c. up
        f''(x) is negative
        -f(x) is c.c. down
        
        When f''(a) = 0 or is undefined
        there is a PI @ x=a on f(x)
        PI: Point of Inflection
        A point where f(x) changes shape
        Use derivative sign chart to
        determine the segments of f(x)
        that are c.c. up & c.c. down