Precalculus

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Algebraic Modeling

• Graphing Data
• Analyzing Data
• Solving Problems Algebraically and Graphically

Functions Defined and Notation

• Domain
• Range
• Boundedness
• Local and Absolute Extrema
• Symmetry
• Asymptotes
• End Behavior
• Introduction to Twelve Basic Functions
• Function Composition
• Modeling with Functions

Linear and Quadratic Functions

• Linear Functions and Graphs
• Average Rate of Change
• Linear Correlation and Modeling
• Graphing Quadratic Functions
• Applications of Quadratic Functions
• Linear and Quadratic Functions on a Graphing Calculator
• Completing the Square
• The Quadratic Formula

Power Functions and Variation

• Graphing Power Functions
• Modeling with Power Functions
• Power Functions and Variation on a Graphing Calculator

Polynomial Functions of Higher Degree

• Graphing Polynomial Functions
• End Behavior
• Zeros
• Intermediate Value Theorem
• Polynomial Functions of Higher Degree on a Graphing Calculator

Real Zeros of Polynomials

• Zero Factor Property
• Long Division of Polynomials
• Remainder and Factor Theorems
• Synthetic Division
• Rational Zeros
• Upper and Lower Bounds
• Real Zeros of Polynomials on a Graphing Calculator

Complex Zeros

• Fundamental Theorem of Algebra
• Complex Conjugate Zeros
• Factoring Real Number Coefficients
• Complex Zeros on a Graphing Calculator

Graphing Rational Functions

• Transformations of the Reciprocal Function
• Limits - End Behavior and Asymptotes
• Graphing Rational Functions on a Graphing Calculator

Solving Rational Equations

• Extraneous Solutions
• Solving Rational Equations on a Graphing Calculator

Solving Rational Inequalities

• Sign Charts
• Polynomial Inequalities
• Solving Rational Inequalities on a Graphing Calculator

Exponential and Logistic Functions

• Exponential and Logistic Graphs
• The Natural Base e
• Population Models
• Exponential and Logistic Functions on a Graphing Calculator
• Scientific Notation

Exponential and Logistic Modeling

• Constant Percentage and Exponentials
• Exponential Growth and Decay
• Modeling Population with Regression on a Graphing Calculator
• Other Logistic Models
• Exponential and Logistic Modeling on a Graphing Calculator

Properties of Logarithmic Functions

• Logarithm-- Inverse of an Exponential Function
• Common Logs
• Functions with Base b
• Natural Logs

Solving Exponential and Logarithmic Equations

• Orders of Magnitude
• Logarithmic Models
• Newton's Law of Cooling

Vectors in the Plane

• 2-D Vectors
• Vector Operations
• Unit Vectors
• Direction Angles

Dot Product of Vectors

• The Dot Product
• Angle between Vectors
• Vector Projection
• Work

Polar Coordinates

• The Polar Coordinate System
• Converting Coordinates from Rectangular to Polar
• Converting Coordinates from Polar to Rectangular
• Converting Equations from Polar to Rectangular
• Finding Distance Between Polar Coordinates
• Rose Curves
• Cardioid Curves
• Limacon Curves

Complex Numbers in Trigonometric Form

• Complex Number Plane
• Trigonometric Form of Complex Numbers
• Multiplication of Complex Numbers
• Division of Complex Numbers
• Powers of Complex Numbers
• Roots of Complex Numbers

Solving Systems of Two Equations

• Solving by Substitution
• Solving by Elimination
• Solving Graphically

Matrix Algebra

• Addition of Matrices
• Subtraction of Matrices
• Multiplication of Matrices
• Identity Matrix
• Inverse Matrix
• Determinant of a Square Matrix
• Cramer's Rule

Matrix Row Operations

• Gaussian Elimination
• Elementary Row Operations
• Reduced Row Echelon Form
• Solving a System of Equations Using a Matrix
• Partial Fraction Decomposition (Linear Denominators)
• Partial Fraction Decomposition (Irreducible Quadratic Denominators)

Systems of Inequalities in Two Variables

• Graphing Systems of Inequalities
• Linear Programming

Geometry of a Parabola

• Standard Form of the Equation
• Vertex Form of the Equation
• Identify Critical Points
• Graphing Parabolas

Geometry of an Ellipse

• Standard Form of the Equation
• General Form of the Equation
• Identify Critical Points
• Graphing Ellipses

Geometry of a Hyperbola

• Standard Form of the Equation
• General Form of the Equation
• Identify Critical Points
• Graphing Hyperbolas

Translation and Rotation of Axis

• Graphing Conic Sections Algebraically
• Graphing Conic Sections on a Graphing Calculator
• Translation of a Conic Section
• Rotation of a Conic Section
• Finding the Angle of Rotation
• Finding the Coefficients for a Conic in a Rotated System

Polar Equations of Conic Sections

• Writing Polar Equations for Conic Sections
• Analyzing Polar Equations for Conic Sections

3-D Cartesian Coordinate System

• 3-D Coordinates
• Finding Distance and Midpoint
• Equation of a Sphere
• Planes
• Vectors in Space
• Lines in Space

The Binomial Theorem

• Powers of the Binomial
• Pascal's Triangle and Binomial Expansion
• The Binomial Theorem
• Factorial Identities

Sequences

• Infinite Sequences
• Limits of Infinite Sequences
• Arithmetic Sequences
• Geometric Sequences
• Working with Sequences on a Graphing Calculator

Series

• Summation Notation
• Sums of Arithmetic Sequences
• Sums of Geometric Sequences
• Infinite Series
• Convergence of Geometric Series

Limits, Motion, and the Tangent Line

• Average Velocity
• Instantaneous Velocity
• The Derivative by Definition
• Definition of the Tangent Line

Limits, Motion, and Areas

• Distance from a Constant Velocity
• Distance from a Changing Velocity
• Connection to Areas
• The Definite Integral

Limits

• Concepts and Informal Definition of a Limit
• Properties of Limits
• Limits of Continuous Functions
• One-Sided Limits
• Two-Sided Limits
• Limits Involving Infinity

Graphs of Trigonometric Functions

• Graphing Trigonometric Functions with Domain and Range
• Graphing Trigonometric Functions with Critical Points
• Graphing Trigonometric Functions with Translations and Asymptotes
• Graphing Sine and Cosine
• Translations of Sinusodial Graphs

Uncategorized Questions

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• An open box is made from a #30"cm" xx 30"cm"# sheet of metal by removing a square from each corner and folding up the sides. To the nearest #"cm"# what is the size of the removed squares to make a box of volume #1000"cm"^3# ?
• F(x)=-√(1-x) and g(x)=ln x (x-2). TRUE OR FALSE.explain. (i)none algebraic operation between f and g. (ii)g(f(x)) exists but not f(g(x)) ?
• Question #60356
• How to find first derivative of f(x)=2 sin (3x) + x?
• What are #A nn B'#, #A nn B# and #A' nn B# in diagrammatic form?
• Question #7a8d5
• How do you solve #y^2-8y-11 = x# for #y# in terms of #x# ?
• If you are told that #x^7-3x^5+x^4-4x^2+4x+4 = 0# has at least one repeated root, how might you solve it algebraically?
• Question #c9838
• Question #32016
• If 4(x-y^3)-xy=16 ,evaluate dy/dx at the point (2,3)?
• Question #0dd83
• Have I calculated the following correctly?
• Question #5deae
• Question #85911
• Question #1f618
• Question #129e0
• Question #17a90
• Question #e09fa
• Question #d9e88
• Question #3af82
• Question #2d82c
• Which of the following four equations defines a function?
• Question #b3a8b
• Is #y=|x^3+9x|# symmetric around origin, #x#-axis or #y#-axis?
• Question #25994
• How to determine whether the given matrix is invertible ? first row (2,0) second row(0,-5)
• Question #22ea9
• Question #2331e
• Question #aa009
• The equation of line m is 8x-7y+10=0. a. For what value of k is the graph kx-7y+10=0 parallel to line m? b. What is k if the graphs of m and kx-7y+10=0 are perpendicular?
• Write an equation for a line that is parallel to the equation y= -9. The line must pass through the points (4, -11). How do I do this?
• Question #191f2
• How to find all the minors and cofactors of the matrix #A=((1, -2, 3), ( 6, 7, -1 ), (-3, 1, 4))#?
• If A and B are 3×2 matrices, and C is a 7×3 matrix, which of the following are defined? A) C*A B) C^T C) B*C D) A+B E) B^T*C^T F) C−A
• I am doing polynomial functions, for example i have two roots 0 and 2 how would i make it a polynomial function?
• Degree of 3, positive leading coefficient, 2 zeros, 2 turning points. I sketched the condition and the zeros are -1 and 1 and the y-int i made it as 2. How do i make a polynomial function?
• Question #688f2
• When #p# is #-4<p<=3#, #(x+2)# is a factor of the function #f(x) =x^2-(p+1)x +p+4#. Moreover, #g(x) = xf(2x) - f(x - 1/3) +2/3x#. Determine the factors of #g(x)# without finding #f(2x)# and #f(x-1/3)#?
• Is this matrix symmetric ? first row ( 0,-7) second row (-7,7)
• How to determine x and y such that ?
• Question #10acb
• How to find the determinant of the given elementary matrix by inspection? First row (1 0 0 0) , second row (0 1 0 0) , third row (0 0 -5 0), fourth row (0 0 0 1)
• How to evaluate the determinant of the given matrix by reducing the matrix to row echelon form ? first row ( 0 3 1 ) second row ( 1 1 2 ) and third row ( 3 2 4 )
• How to show that Det(A) = 0 without directly evaluating the determinant ? A= first row ( -2 8 1 4 ) second row ( 3 2 5 1 ) third row ( 1 10 6 5 ) and fourth row ( 4 -6 4 -3 )
• Question #016df
• How do you solve #2^x + 2^(1/x) = 18# ?
• Is the domain of a polynomial always #RR# ?
• Question #7a60e
• Question #86d11
• Question #a9277
• What is the inverse of f(x) if f(x) = x ?
• Evaluate the inverse of f(5) where f(x) = 6 - x ?
• If f(x) = (x+1) / (x-2), find the inverse of f(x) ?
• If f(x) = 7-2x and g(x) = x+3 . A. What is g^-1 (x) ? B. What is f(g^-1(5)) ?
• If f(x) = 3x, g(x) = 2x - 5 and h(x) = f(g(x)) . What is the inverse of h(x) ?
• If #f(x) = sqrt(x - 5)#, for #x ≥ 5#, what is the inverse of #f(2)# ?
• Given that the inverse of h(a) = 3, what is the value of a ?
• What is the inverse of f(x) = -1/5x -1 ?
• Question #4a512
• ∥v∥=3 ∥w∥=4 The angle between v and w is 1.2 radians. How to calculate ? a) v⋅w = ,(b) ∥1v+1w∥= , (c) ∥4v−4w∥=
• How to find a vector A that has the same direction as ⟨−8,7,8⟩ but has length 3 ?
• How would I use the dot product of #u# if #u##=##5i##+##2j#?
• How would I find the angle between vectors #u# & #v# if #u##=##i#+#4j# & #v##=##-i#+#2j#?
• How do you find the absolute value of the complex number #-4+i#?
• How do you find the trigonometric form of the complex number #3##sqrt3##+##3i#?
• How would I perform the operation #14(cos 63° + isin 63°)##/##20(cos 15° + isin 15°)#? Make sure to leave the result in trigonometric form.
• How do I use DeMoivre's theorem to find #(-3+3i)^3#?
• How would I find the standard form of the complex number #9(cos((7pi)/6)=isin((7pi)/6))#?
• Lne^x=5 I know that x=5 but why?
• 1) Log(4)64=x 2) Log(3)243=2x+1 The numbers in parenthesis is the base. I know the first one I can just throw in the calculator, but that's now how we're supposed to solve these ones. What's the value of x?
• 1) Log(2)8^x= - 3 2) 5e^0.2x=7 The 2 in parenthesis is the base. What's the value of x?
• 2*10^2-x=5 What is x?
• Question #f9702
• Question #0a2db
• Question #507fd
• How to find an equation of the plane through the point (4, -4, -5) and parallel to the plane −3x−4y+3z=−5 ?
• Question #22c63
• If an arc of radius #6#cm subtends #60^@#, then what is the length of a chord joining its two ends?
• Given an integer #n# is there an efficient way to find integers #p, q# such that #abs(p^2-n q^2) <= 1# ?
• A ball is dropped from a height of 12m. On every successive bounce, the ball bounces to a height that is 2/3 of the previous height. Find the total vertical distance, that the ball has travelled when it hits the ground for the 8th time?
• Question #479cb
• How would you find the radian measure of the central angle of a circle with a radius of 60 in. that intercepts an arc of a length of 245 in.?
• How would you find the length of the arc on a circle with a radius of 15 cm intercepted by a central angle of 60°?
• How do you determine whether a vector is orthogonal, parallel, or neither?
• Solve 2x - 1 = (x + 1)÷(2x) by factorization?
• Question #284a6
• How to find the linear equation of the plane through the point (−2,4,2) and perpendicular to the line represented by the vector equation r(t)=⟨1+4t,−2−t,6+2t⟩ ?
• How do you simplify #log(x(x^3+9)^-1/2)#?
• Question #d2694
• How to find the linear equation of the plane through the point #(1,2,3)# and contains the line represented by the vector equation #r(t)=⟨3t,6−2t,1−2t⟩# ?
• What is the recursive rule for the sequence 10, 18, 26, 34....?
• What is the third term of the sequence defined by f(n) = 2n — 1 ?
• What is the 99th term of the sequence: 2, #5/2#, 3, #7/2#...?
• How do you simplify #(2x^6y^2)(-2x^5y^-1)#?
• How do you express #log 36# in terms of #log 2# and #log 3#?
• What is x in #log_3x = 0#?
• Question #5a2c1
• The order of magnitude of the mass of Earth's atmosphere is #10^18# kilograms. The order of magnitude of the mass of Earth's oceans is #10^3# times greater. What is the order of magnitude of the mass of Earth's oceans?
• You are building a fence on a rectangular lot. Only three sides of the lot will be fenced. You have 15 meters of wood for the fence, what is the area of the largest lot you could build?
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