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  • Algebra I eCourse
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  • Simulations
    • Ohm’s law simulation
    • Wave on a string simulation
  • Completing an ANOVA table
  • AP Calculus BC solved problem
    • Calculus Made Simple PDF
    • solved calculus bc math problems on arc length, differential equations and the integral of the second derivative
    • Limit x sin(1/x) as x goes to infinity
  • solved math problems
    • algebra
      • -2^3, -3^2,-4^3 (Examples of finding exponential expressions with negative bases and positive, whole number exponents)
      • (-2)^3,(-2)^2,(-3)^2 (Examples of finding exponential expression values)
      • 240 is 30% of what number?
      • A rectangular room is 3 times as long as it is wide
      • age problem algebra | algebra age problem
      • age problem involving number of years ago ages were equal
      • solving quadratic equations by using factoring (ac method with grouping, illustrated)
      • solving quadratic equations using the quadratic formula
      • solving quadratic equations by completing the square, no steps skipped
      • solving a radical equation with two radicals on one side
      • examples of solving equations with negative exponents
      • graphing a parabola by plotting points and connecting, graphing an absolute value graph by points
      • Ax+By=C facts and examples
      • graphing a line by xy intercepts and making the equation of a line from two points
      • two numbers add up to 33 and the difference is 77
      • simplifying experssions of the form a to the p over q
    • precalculus
      • domain and range of a square root function with a detailed graph and explanation
      • finding the domain and range of a square root function with a graph
      • domain and range of a quadratic function with a graph
      • domain and range of simple linear function
      • Find a polynomial of least degrees with real coefficients.
      • domains of functions examples
      • domain and range of an inverted quadratic function(negative leading coefficient)
      • finding the domain and range of a cubic function with a graph and a detailed explanation
      • solving rational equations examples with graphs
      • detailed examples of solving radical equations ,including using graphs
      • symmetry, even /odd functions, examples with graphs and detailed explanations
      • solving an inequality with three linear factors already in factored form (x+2)(x-3)(x+1) greater than or equal to 0
      • solving a radical equation by squaring and solving a quadratic quadratic equation with an extranous solution
      • solving a quadratic equation with complex roots detailed example with a graph
      • domain /range of a relation, evaluating functions with numbers and expressions, graphical domain and range of a function, vertical line test
      • examples of checking whether a function is even odd or neither
      • function composition example with value plugged in
      • function composition example with linear functions
      • solving a system of three linear equations step by step with annotations for better understanding
      • write a linear function to model a situation where w is the number of weeks and g w is the value in dollars on a card
      • sample solutions for 1.5 math 201 mercy fall 2021 average rate of change function graph comparing average rates of change
      • notes on multiplfying using foil and simple factoring factoring by grouping
      • detailed example of performing function transformation with reflections translations shifts
      • adding function, multiplying functions, dividing functions, subtrcting functions, rational functions, square root and linear function composition
    • calculus
      • relationship between the natural log and exponential functions and an application to derivatives
      • where are the graphs of x^2 and x^3 concave up/down?
      • concave up versus concave down
      • concavity of 1/x^2 with the second derivative
      • concavity of cubic polynomial
      • finding the concavity of sqrt(x) with the second derivative
      • finding the derivative of the absolute value of x
      • finding the derivative of |x-1| using the chain rule
      • examples of using the chain rule in calculus I
    • multivariable calculus
      • constructing a 3D coordinate system
      • sketching planes in three dimensional space
      • sketching a cylinder in 3D x^2+y^2=1
      • deriving the distance formula in three-dimensional space
      • deriving and using the equation of a sphere
      • region between two spheres
      • examples of inequalities in 3D
      • vector concepts
      • adding vectors
      • scalar multiplication of vectors
      • magnitude of a vector in 2D and 3D
      • forming linear combinations of vectors
      • unit vectors
      • decomposing a vector into components from the length and angle of the vector
      • finding the tensions in two ropes for a hanging object
      • dot product definition and properties
      • proof of Cauchy-Schwarz inequality, proof of Triangle Inequality
      • directions cosines for a vector in 3D
      • scalar , vector projections, work
      • distance from point to line using vectors, proof of formula, with several pictures
      • deriving the formula for finding the cross product of two vectors and vector product properties and applications with pictures
      • vector triple product proof and example
      • torque from vector cross product
      • equations of lines and planes in 3D, with examples and pictures, skew lines
      • equation of a plane in 3d formula and example
      • equation of a plane from three points in 3D
      • where does a line in parametric form cross a plane in 3D?
      • when are two planes in 3D space parallel (direction vectors are scalar multiples)
      • finding the angle between two planes using their direction vectors and the dot product, finding line of intersection of two planes
      • derive the formula for the distance between a point and a plane in 3D space
      • finding the distance between two skew lines in 3D space
      • cylinders and quadric surfaces
      • vector valued functions definition, domain examples
      • finding the limit of a vector-valued function, includes a graph and detailed calculations
      • identify the curve defined by a vector-valued function
      • constructing a picture of a toroidal spiral
      • concepts behind vector-valued function derivatives and an example
      • parametrizing a segment in 3D over 0<=t<=1
      • find parametric equations of tangent line in 3D find velocity vector and derivative of space curve
      • find the derivative and second derivative of a vector valued function at a point including a graph of the curve velocity and acceleration in three space
      • differentiation rules for vector-valued functions with proofs
      • reparametrize a curve example with a detailed explanation
      • integration of vector-valued functions, examples
      • arc length formula development and example for vector-valued functions
      • arc length integral in 3D development with an example for a helix raparametrization by arc length example
      • velocity and acceleration concepts and examples for space curves with projectile motion and tangential and normal components of acceleration using dot and cross products
      • multivariable functions, domains, ranges of graphs, level curves level surfaces definitions
      • examples of level curves for different types of functions.
      • example of limits in multivariable calculus one exists and one doesn’t exist
      • when does the limit of a multivariable function exist case of a functio of two variables
      • example of when a limit does not exist for a function of two variables 1
      • continuity of multivariable functions examples with detailed graphs
      • finding the limit of a function of two variables using polar coordinates
      • partial derivatives concepts graphs tangent lines examples
      • Understanding second partial derivatives with graphs
      • finding implicit partial derivatives examples
      • tangent plane approximations part 1
      • multivariable chain rule case 1 where z depends only on t with graphs
      • gradient vector meaning directional derivative level curves
      • max min concepts and examples in multivariable calculus
      • max min exampls with graph in multivariable calculus
      • max min examples in multivariable calculus 2
      • examples of max/min questions with lagrange multipliers
      • Finding a point of intersection between two planes.
      • solving a system of three linear equations, illustrated
      • calculus III notes
      • solving a system of three linear equations
      • examples of solving systems of equations with three variables
      • application of a system of three linear equations to investments
      • 3 by 3 Gaussian elimination two row operations applied in one step
      • 2 by 2 example and 3 by 3 example of solving systems using Gaussian elimination
      • using gaussian eliminination to show a system is inconstitent 3 by 3 and 2 by 2 examples with graphs of lines and planes
      • APPLICATION OF A GAUSSIAN ELIMINATINO TO SOLVING A SYSTEM OF THREE EQUATION WITH THEATER TICKETS
      • find specific combination of tofu, rice and peanuts to make a certain type of diet
        • solving a linear equation
        • pdf on identifying matrix columns, rows, doing matrix addition, subtraction
        • solving linear equation
      • solving sin squared plus cos x plus 1 equals 0 step by step with unit circle
      • Analysis of Rational Function
      • understanding the slope/derivative of e^x
      • linear dependence/independence for functions with examples
      • moving wave concepts illustrated with lots of graphs
      • rational function analysis
      • linear regression example found using matrices
      • best linear approximation
      • exponential function with a fraction in the base , horizontal asymptote, domain, range, points
      • discrete compounding sample calculations, continuous compounding included.
      • distance traveled by a particle found using calculus
      • solving simple differential equations with initial condition built into integral
      • Understanding Singularities in Complex Variables
      • two detailed examples of variations of parameters
      • Using a unit vector with tension.
      • detailed explanation of direction cosines
      • three examples of evaluating a piece-wise defined function
      • Vector Projections in 3D Space: Intuitive Explanations and Examples
      • How to graph 3D Vectors
  • Set Builder Notation Quiz

understanding the concept of a hole in a rational function graph. detailed steps and a graph.

understanding holes in rational function

Holes In Functions Understanding

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