1. Function
   1. Definitions
      1. domain and range
         1. domian
            1. horizontal asyomptope
         2. range
            1. vertical aspomptope
         3. input
         4. output
      2. composition
      3. Symmetry
         1. odd f(-x) = -f(x)
         2. even f(-x) = f(x)
      4. One-to-one function
         1. definition
         2. horizonal line test
         3. inverse function
            1. f^-1(y) = x, f(x) = y
            2. if g(x) = f^-1(x), f(g(x)) = x
            3. find inverse
            4. different than reflection
      5. zeros
         1. x-intercept
   2. Special Functions
      1. absolute value f(x) = |x|
      2. greatest-integer function g(x) = [x]
   3. Polynomial and Other Rational Function
      1. polynomila function
         1. linear function
            1. degree 1
            2. y-intercept = b
         2. quadratic function
            1. degree 2
            2. open up if a>0, down if a<0
         3. cubic function
            1. degree 3
         4. the domain for every polinomial is the set of all real
      2. rational function
         1. domain of g(x)/f(x) is the set of all real except f(x) = 0
   4. Trigonometric Function
      1. definition
         1. periodic function
         2. period
         3. amplitude
         4. midpoint
      2. sin, cos
      3. tan, cot
      4. recipical function
         1. sec, csc
         2. cot
      5. inverse trig function
         1. arccos, arcsin
         2. arctan
   5. Exponential and Logarithmic Function
      1. exponential funcition
         1. laws
         2. domian
      2. loarthmic function
         1. laws
         2. domain
         3. laws
   6. Parametrically Defined Function
      1. definition
      2. parameter
2. Limits and Continuity
   1. Definitions and Examples
      1. one-sided limits
         1. for greatest-integer function
         2. endpoint
         3. infinite limit
            1. ❌ limit doesn't exist
            2. ❌ limit still doesn't exist
            3. ❌ limit doesn't exist
            4. end behavior of polynomials
            5. does not exist, but can show the behavior of the function
      2. definded
         1. function is defineded if f(x) exist
         2. funciton is not defineded if f(x) is a empty hole
   2. Asymptotes
      1. horizontal asymptote
      2. vertical asymptote
         1. limit doesn't exist!
   3. Theorems on Limits
      1. if lim f(x) and lim g(x) are finite number
      2. Sandwish (Squeeze) Theorem
         1. definition
         2. examples
   4. Limit of a Quotient Polynomials
      1. the rational function theorem
         1. the degree of P(x) is less than that of Q(x)
         2. the degree of P(x) is higher than that of Q(x)
         3. the degree of P(x) is equal to Q(x)'s degree
   5. Other Basic Limits
      1. this formula can only work if and only if the x and the coefficient of x IS THE SAME
      2. limit definition of e
      3. Sandwish (Squeeze) Theorem
   6. Continuity
      1. continuouos functions
         1. polynomials function
            1. continuous everywhere, because it is a polunomial
            2. continuous at each point in their domian except Q(x) = 0
         2. absolute value function
            1. continuous everywhere
         3. trigonometric function
         4. inverse trigonometric function
         5. exponential function
         6. logarithmic function
         7. n%2==0
         8. continuous if and only if x>0
         9. n%2!=0
         10. continuous everywhere
         11. continuouos at each point which function is defined (or legal)
         12. greatest-integer function
             1. discontinuous at each integer, since it doesn't have a limit at any integer
             2. continuous on the interval (x, x+ Δx), when x is a integer and Δx >1
         13. continuous at each point in their domain
      2. continuous doesn't effect the availability of limit
         1. definition of general limit
            1. definition of general continounity
      3. kinds of discontinuous
         1. not defined at x = -2
         2. jump discontinuouity at x = 0
         3. removeable discontinuouity at x = 2
            1. even if the numerator can cancel with denominator, the removeabe discontinuouoity still exist!
         4. infinite discontinuity, not in this graph but you know what I'm talking about ;)
         5. examples
      4. Theorems on Continuous Function
         1. the extreme value theorem
         2. the intermedia value theorem
         3. the continuous function theorem
            1. if function f and g are both continuous at x=c
            2. kf
            3. continuous at x = c
3. Differentiation
   1. Definition of Derivative
      1. derivative
      2. differentiable
         1. limit exists
      3. difference quotient
         1. definition: average rate of change from a to a+h
         2. difference between the instantaneous rate of change, also known as the derivative at f(x)
      4. the second derivative
   2. Formulas
      1. basic rules
      2. trig rules
         1. be careful with the negative sign!
      3. exponention and logarithmic
      4. inverse function
         1. inverse trig function
            1. be careful with the negative sign
   3. The Chain Rule
   4. differentiability and continuity
      1. continouity
      2. differentiability
         1. exception
            1. hole
            2. jump discontinuouity
            3. infinite discontinoutity / asmptote
            4. tangent = 0
            5. sharp turn
            6. cusp
            7. continuous but not differentiability
            8. not continuous at the first place
   5. Estimating a Derivative
      1. Numerically
         1. esitmating the avergae rate of change
            1. Symmetric difference quotient
      2. Graphically
         1. slope = 0
         2. f(x): decrease->increase
            1. f'(x): negative->positive
   6. Implicit Differentiation
   7. Derivative of the Inverse of a Function
      1. definition approach
      2. numerical approach
   8. The Mean Value Theorem / secant line
      1. Rolle's Theorem
         1. at lease one turning point between any two roots
   9. Indeterminate Forms and L’Hôpital’s Rule
      1. indeterminate
         1. taking the natural logarithm first
            1. using the logarithmic laws to move the exponencial to the front
      2. L’Hôpital’s Rule
   10. Recognizing a Given Limit as a Derivative
4. Application of Differential Calculus
   1. Slope and Critical Point
      1. slope of a curve
         1. definition
      2. critical point
         1. definition
      3. average and instataneous rate of change
         1. average rate of change
         2. insrtataneous rate of change
   2. Tangents and Normals
      1. tangent to the curve
         1. horizontal tangent line
         2. vertical tangent line
      2. tangents to parametrically defined curves
   3. Increasing and Decreasing Functions
      1. Function with Continuous Derivatives
         1. function is increasing, f'(x)>0
         2. function is neither increasing nor decreasing, critical points
         3. function is decreasing, f'(x)<0
      2. Function Whose Derivative Have Discountinuities
         1. only examine the sign when f'(x) exist
   4. Maximum, Minimum and Inflection Point
      1. Definition
         1. Local (relative)
            1. max
            2. rising -> falling
            3. f'(a)>0 for a<x, f'(x)=0 and then f'(b)<0 for b>x
            4. Subtopic 3
            5. min
            6. falling -> rising
            7. f'(a)<0 for a<x, f'(x)=0 and then f'(b)>0 for b>x
            8. concave up when f'(x) =0
            9. extremum/extrema
         2. Global (absolute)
         3. concavity
            1. y''>0, upward
            2. y''=0, inflection point
            3. y''<0, downward
      2. Curve Sketching
         1. Functions That Are Everywhere Differentiable
            1. 1. find y' and y''
            2. 2. find all critical points where y'=0
            3. 3. second derivative test
            4. if y''(x)= 0 and y'(x) =0 at the same time, find sign changes using the number line
            5. if sign changed, local max/min
            6. if sign is not changed, horiontal segement
            7. 4. find all inflection point where y''=0
         2. Functions Whose Derivatives May Not Exist Everywhere
            1. separating the function into several segments that are differetiable everywhere
      3. Global Maximum or Minimum
         1. Differentiable Functions
            1. extreme value therome
            2. intervcal test
         2. Functions That Are Not Everywhere Differentiable
            1. also test those point which f'(x) doesn't exist
         3. checking the endpoints if in a closed interval
   5. Further Aids in Sketching
      1. intercepts
         1. x-intercept
         2. y-intercept
      2. symmetry
         1. symmetry about x-axis if (x, -y) satifies the equation
         2. symmetry about y-axis if (-x, y) satifies the equation
         3. symmetry about line y=x if (-x, -y) satifies the equation
         4. odd f(-x) = -f(x)
         5. even f(-x) = f(x)
      3. asymptote
         1. horizontal asymptote
         2. vertical asymptote
            1. limit doesn't exist!
      4. point of discontinuity
         1. Jump discontinuity
         2. removeable discontinuoty
         3. not defined
         4. infinity discontinuity = asymptote
   6. Optimization: Problems Involving Maxima and Minima
      1. find the general equation that express the answer
         1. find max/min over the given interval
   7. Relating a Function and Its Derivatives Graphically
   8. Motion Along a Line
      1. position p(t)
      2. volecity v(t) = p'(t)
         1. v(t) > 0, partical is moving right
         2. v(t) < 0, partical is moving left
      3. acceleration a(t) = v'(t) = p''(t)
         1. a(t)>0, velocity is increasing
            1. v{t)>0, partical is accelerating
            2. v(t)<0, partical is deccerating
         2. a(t)<0, velocity is decreasing
            1. v{t)<0, partical is accelerating
            2. v(t)>0, partical is deccerating
         3. accelerating if the v(t) and a(t) has a same sign! (can be negative)
   9. Tangent-Line Approximations
      1. local linear approximation
      2. the equation of tangent line at x=a
      3. tangent line approximation
   10. Related Rates
5. Antidifferentiation
   1. Antiderivatives
      1. integrand
      2. antiderivatives
   2. Basic Formulas
      1. basic laws
      2. trig
      3. exponention and logarithmic
      4. inverse function
         1. inverse trig
   3. special techique
      1. rational function
         1. seprating the rational with integer
            1. by long division
            2. by manual seperation
      2. u-subsitution
         1. u-sub with trig
            1. make your life easier by convert trig into expo with u-sub
            2. reduce complexity by replacing trig with u so that we can treat it like expo
         2. u-sub with exponentia
            1. you can also convert exponential by u-sub e^x
         3. trig indentity
            1. the point here is to reduce the degree of sin^2 to cos so that we can use normal way to calculate this single cos(2y) with a easy u-sub that u=2y
      3. grouping
         1. usually result as inverse funciton
      4. Breaking a gaint ass rational function into several small funcions, and integerate them one by one
   4. Integration by Partial Fractions
      1. 1. making sure the degree of numerator is lower than degree of denominator
      2. 2.set up the equation with each term sepreated
      3. 3. from the equation, find the expression for each term using the cross multipalcation
      4. 4. finding the coefficient
         1. method 1
         2. method 2, for A(x-1)(x-2)+B x(x-2)+C x(x-1), we know above. and then we can solve for them
      5. 5. rewrite the equation and solve for integral
   5. Integration by Parts
      1. Parts Formula
         1. Because we know this from the power rule
         2. we have
      2. priority: LIPET
         1. L- logarithmic function
         2. I - Inverse function
         3. P - polynomial function
         4. E: Exponential function
         5. T: Trigonomitric function
      3. example
         1. u =x and dv = cosxdx
         2. we have du=dx and v=sinx
         3. u = x^2 and dv e^xdx
         4. du = 2xdx and v = e^x
         5. u = x and dv = e^xdx
         6. du = dx and v = e^x
         7. u = e^x and dv = cosxdx
         8. du = e^xdx and v = sinx
         9. u = e^x and dv = sinxdx
         10. using trig as dv!
         11. du = e^xdx and v = -cosx
         12. be careful with the negative sign
         13. u = x^4 and dv = lnx
         14. du = 4x^3 dx and v = 1/x
      4. The Tic-Tac-Toe Method
         1. example
   6. Applications of Antiderivatives; Differential Equations
      1. given f'(x) at certain point, find f(x)
      2. motion problem
         1. v'(t) = a(t)
         2. p;(t) = v(t)
6. Definite Integrals
   1. Fundamental Theorem of Calculus (FTC);
      1. Definite Integral
         1. integrand
         2. limits
            1. lower limit
            2. upper limit
   2. Properties of Definite Integrals
      1. Fundamental Theorem of Calculus
         1. FTC with the definition of derivative
         2. U-subsitution with FTC
         3. when x=3, u=1, and when x=6, u=2
         4. k is a constant
         5. If f and g are both integrable functions of x on [a,b]
            1. not integrable function
            2. not continuouos not differentiable
      2. Mean Value Theorem for Integrals
         1. If f(x) ≤ g(x) for all x in the closed interval [a,b]
   3. Definition of definite integral as the limit of a riemann sum
      1. definitions
         1. 1. there exist a function f(x)
         2. 2. integralable over the closed interval [a,b]
            1. upper limit
            2. lower limit
         3. 3. divide the interval into n subintervals
         4. 4. each subinterval has length 👆
            1. not equal length
            2. equal length
         5. 5. sum up the area of all the small subintervals
         6. 6. we have the definition of riemann sum
            1. 6.5 dummy's version
      2. example
         1. Solution 1
         2. 1. recall the definition of the Remann S4um
         3. Solution 2
         4. a = 1, b = 4
   4. Definition of Definite Integral as the Limit of a Sum: The Fundamental Theorem Again
      1. Area
         1. positive
         2. negative
   5. Approximations of the Definite Integral; Riemann Sums
      1. Using Rectangles
         1. left sum
         2. right sum
         3. midpoint sum
      2. Using Trapezoids
      3. Comparing Approximating Sums
         1. if f(x) is an increasing function on [a,b], then left ≤ actual ≤ right
         2. if f(x) is a decreasing function on [a,b], then right ≤ actual ≤left
         3. if f(x) si concave down, then trapezoid ≤ actual ≤ midpoint
         4. if f(x) si concave up, then midpoint ≤ actual ≤ trapezoid
      4. Graphing a Function from Its Derivative; Another Look
   6. Graph a funciton from its derivativ
      1. let u=x-3, so upper limit will be 7-3 = 4 and lower limit will be 2-3 = -1, dx = du
   7. Interpreting ln x as an Area
      1. Subtopic 2
   8. Average Value
      1. average value of a cuntion
      2. examples
         1. Identifying following functions
         2. identifying following functions
      3. average value of the function vs. average rate of change
7. Application of Integration to Geometry
   1. Area
      1. general method for finding the area
         1. 1. draw a sketch of the given region and of a typical element.
            1. perpendicular to x-axis
            2. perpendicular to y-axis
         2. 2. write the expression for the area of a typical rectangle
            1. perpendicular to x-axis
            2. perpendicular to y-axis
         3. 3. set up the definite integral that is the limit of the Riemann Summ of n areas as n -> infinity
            1. perpendicular to x-axis
            2. perpendicular to y-axis
      2. Area Between Curves
         1. sketch
         2. equation
      3. Using Symmetry
         1. symmetry in y-axis
         2. symmetry in x-axis
         3. symmettic to both axes
      4. Evaluaying area using a graphing calculator
   2. Volume
      1. Solids with Known Cross Sections
         1. 1. sketch the function
         2. 2. establish one single rectangle
         3. 3. etsablish one single block
         4. 4. accumulate infinity number of blocks
         5. example: A solid has as its base the circle x^2 + y^2 = 9, and all cross sections parallel to the x-axis are squares. Find the volume of the solid.
            1. 1.
            2. 2.
            3. 3.
            4. 4.
      2. Solids of Revolution
         1. Disks Method
            1. sketch
            2. equation
            3. example
            4. 0.determein the equation
            5. 1. sketch the graph
            6. 2. find a single area
            7. 3. find a singler block
            8. 4. accumulating all the blocks
         2. Washer Method
            1. sketch
            2. equation
            3. example
            4. 1. sketch it
            5. 2. Finding the intersection
            6. a = 0, b = 2
            7. 2. Finding a singe area
            8. 2.1 outter area
            9. 2.2 inner area
            10. 3. Fiding a single block
            11. 4. accumulating all the blockas
         3. Shells Method
            1. sketch
            2. equation
            3. example
            4. 1. sketch it
            5. 2. finding the area of a single shell
            6. 2.1 width
            7. 2.2 height
            8. 3. finding a single block of shell
            9. 4. accumulating all blocks of shell
   3. Improper Integrals
      1. one of the liimts is infinity
         1. evalueate function with one limit is infinity
            1. whatever a funciton is defined when upper limis is inifnity
            2. converge
            3. diverges
            4. examples
            5. the given integral converges to 1
            6. Subtopic 1
            7. Subtopic 2
            8. the given function disverges
            9. the given funciton converge to pi/6
            10. the given funciton converge to 1
            11. the given funciton disverge
            12. the given funciton disverges
      2. function is infinity discontiuntinuity in the interval
         1. evaluating the function with infinity discontinuty in the interval
            1. if f(x) become inifinity at x=a, we define
            2. if f(x) become infinity at x=c, where a < c < b
            3. if f(x) become infinity at x=b, we define
            4. examples
            5. the given function converge in pi/2
            6. the given funciton deverge
            7. the given function converge in 6
            8. the given function disverge
   4. The Comperhasion Test
      1. Convergence
      2. Divergence
      3. example
         1. Subtopic 1
         2. \text{Find the volume, if it exists, of the solid generated by rotating the region in the first quadrant bounded above by } y= \frac{1}{x} \text{ at the left by} x=2 \text{, and below by } y=0 \text{, about the axis}
            1. 1. sketch it
            2. 2. using the disk method, find the area
            3. 3. find the volume
            4. 4. accumulate all volume
         3. \text{ Find the volume, if it exists, of the solid generated by rotating the region in the first quadrant bounded above by } y=\frac{1}{x} \text{ ,at the left by } x=1 \text{, and below by } y=0 \text{, about the y-axis}
            1. 1. sketech it
            2. 2. using the shell method, find the length and height of one single shell
            3. length
            4. height
            5. 3. find the volume of a single shell
            6. 4. accumulate all the shells
8. Further Application of Integration
   1. Motion Along a Straight Line
   2. Motion Along a Plane Curve
   3. Other Applications of Riemann Sums
   4. FTC: Definite Integral of a Rate Is Net Change
9. Differential Equations
   1. Basic Definitions
   2. Slope Fields
   3. Euler’s Method
   4. Solving First-Order Differential Equations Analytically
   5. Exponential Growth and Decay
      1. Exponential Growth
      2. Restricted Growth
      3. Logistic Growth
10. describing the asymptote of f(x) using limits