math solution

Greatest Common Factor

To factor a polynomial means to rewrite the polynomial as a product of simpler polynomials or of polynomials and monomials. Because polynomials may take many different forms, there are many different techniques available for factoring them.

The first method of factoring is called factoring out the GCF ( greatest common factor)

Example 1: Factor 5 x + 5 y

Since each term in this polynomial involves a factor of 5, then 5 is a common factor of the polynomial.

Example 2: Factor 24 x³ − 16 x² + 8 x

x is a common factor for all three terms. Also, the numbers 24, −16, and 8 all have the common factors of 2, 4, and 8. The greatest common factor is 8 x

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Chapter 0: Preliminaries
0.1 Polynomials and Rational Functions
0.2 Graphing Calculators and Computer Algebra Systems
0.3 Inverse Functions
0.4 Trigonometric and Inverse Trigonometric Functions
0.5 Exponential and Logarithmic Functions
Hyperbolic Functions
Fitting a Curve to Data
0.6 Transformations of Functions

Chapter 1: Limits and Continuity
1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
1.2 The Concept of Limit
1.3 Computation of Limits
1.4 Continuity and its Consequences
The Method of Bisections
1.5 Limits Involving Infinity
Asymptotes
1.6 Formal Definition of the Limit
Exploring the Definition of Limit Graphically
1.7 Limits and Loss-of-Significance Errors
Computer Representation of Real Numbers

Chapter 2: Differentiation
2.1 Tangent Lines and Velocity
2.2 The Derivative
Numerical Differentiation
2.3 Computation of Derivatives: The Power Rule
Higher Order Derivatives
Acceleration
2.4 The Product and Quotient Rules
2.5 The Chain Rule
2.6 Derivatives of Trigonometric Functions
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Implicit Differentiation and Inverse Trigonometric Functions
2.9 The Mean Value Theorem

Chapter 3: Applications of Differentiation
3.1 Linear Approximations and Newton’s Method
3.2 Indeterminate Forms and L’Hopital’s Rule
3.3 Maximum and Minimum Values
3.4 Increasing and Decreasing Functions
3.5 Concavity and the Second Derivative Test
3.6 Overview of Curve Sketching
3.7 Optimization
3.8 Related Rates
3.9 Rates of Change in Economics and the Sciences

Chapter 4: Integration
4.1 Antiderivatives
4.2 Sums and Sigma Notation
Principle of Mathematical Induction
4.3 Area
4.4 The Definite Integral
Average Value of a Function
4.5 The Fundamental Theorem of Calculus
4.6 Integration by Substitution
4.7 Numerical Integration
Error Bounds for Numerical Integration
4.8 The Natural Logarithm as an Integral
The Exponential Function as the Inverse of the Natural Logarithm

Chapter 5: Applications of the Definite Integral
5.1 Area Between Curves
5.2 Volume: Slicing, Disks, and Washers
5.3 Volumes by Cylindrical Shells
5.4 Arc Length and Surface Area
5.5 Projectile Motion
5.6 Applications of Integration to Physics and Engineering
5.7 Probability

Chapter 6: Integration Techniques
6.1 Review of Formulas and Techniques
6.2 Integration by Parts
6.3 Trigonometric Techniques of Integration
Integrals Involving Powers of Trigonometric Functions
Trigonometric Substitution
6.4 Integration of Rational Functions Using Partial Fractions
Brief Summary of Integration Techniques
6.5 Integration Tables and Computer Algebra Systems
6.6 Improper Integrals
A Comparison Test

Chapter 7: First Order Differential Equations
7.1 Modeling with Differential Equations
Growth and Decay Problems
Compound Interest
7.2 Separable Differential Equations
Logistic Growth
7.3 Direction Fields and Euler's Method
7.4 Systems of First Order Differential Equations
Predator-Prey Systems

Chapter 8: Infinite Series
8.1 Sequences of Real Numbers
8.2 Infinite Series
8.3 The Integral Test and Comparison Tests
8.4 Alternating Series
Estimating the Sum of an Alternating Series
8.5 Absolute Convergence and the Ratio Test
The Root Test
Summary of Convergence Tests
8.6 Power Series
8.7 Taylor Series
Representations of Functions as Series
Proof of Taylor’s Theorem
8.8 Applications of Taylor Series
The Binomial Series
8.9 Fourier Series

Chapter 9: Parametric Equations and Polar Coordinates
9.1 Plane Curves and Parametric Equations
9.2 Calculus and Parametric Equations
9.3 Arc Length and Surface Area in Parametric Equations
9.4 Polar Coordinates
9.5 Calculus and Polar Coordinates
9.6 Conic Sections
9.7 Conic Sections in Polar Coordinates

Chapter 10: Vectors and the Geometry of Space
10.1 Vectors in the Plane
10.2 Vectors in Space
10.3 The Dot Product
Components and Projections
10.4 The Cross Product
10.5 Lines and Planes in Space
10.6 Surfaces in Space

Chapter 11: Vector-Valued Functions
11.1 Vector-Valued Functions
11.2 The Calculus of Vector-Valued Functions
11.3 Motion in Space
11.4 Curvature
11.5 Tangent and Normal Vectors
Tangential and Normal Components of Acceleration
Kepler’s Laws
11.6 Parametric Surfaces

Chapter 12: Functions of Several Variables and Differentiation
12.1 Functions of Several Variables
12.2 Limits and Continuity
12.3 Partial Derivatives
12.4 Tangent Planes and Linear Approximations
Increments and Differentials
12.5 The Chain Rule
12.6 The Gradient and Directional Derivatives
12.7 Extrema of Functions of Several Variables
12.8 Constrained Optimization and Lagrange Multipliers

Chapter 13: Multiple Integrals
13.1 Double Integrals
13.2 Area, Volume, and Center of Mass
13.3 Double Integrals in Polar Coordinates
13.4 Surface Area
13.5 Triple Integrals
Mass and Center of Mass
13.6 Cylindrical Coordinates
13.7 Spherical Coordinates
13.8 Change of Variables in Multiple Integrals

Chapter 14: Vector Calculus
14.1 Vector Fields
14.2 Line Integrals
14.3 Independence of Path and Conservative Vector Fields
14.4 Green's Theorem
14.5 Curl and Divergence
14.6 Surface Integrals
14.7 The Divergence Theorem
14.8 Stokes' Theorem
14.9 Applications of Vector Calculus

Chapter 15: Second Order Differential Equations
15.1 Second-Order Equations with Constant Coefficients
15.2 Nonhomogeneous Equations: Undetermined Coefficients
15.3 Applications of Second Order Equations
15.4 Power Series Solutions of Differential Equations

Appendix A: Proofs of Selected Theorems

Appendix B: Answers to Odd-Numbered Exercises

Mathematics Is Fun
Algebra - Fun with Calendars
May 08
S M T W TH F S




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A fun mathematical puzzle to play with your friends.
(Or teachers, with your class.)
18 19
25 26
Take any calendar. Tell your friend to choose 4 days that form a square like the four to the right. Your friend should tell you only the sum of the four days, and you can tell her what the four days are.
How does the puzzle work? You know how people always want to see a use for algebra? Well this puzzle uses algebra. Here's what I mean.
Let's pretend that the 4 numbers that the person chose were the highlighted ones here - 18, 19, 25, and 26. She adds up the four numbers and tells you only that the sum is 88.
You make a couple of calculations and tell her the numbers. What calculations? Lets figure that out with algebra. Let's call the first number n. Then you know that the next number would be n + 1 and the next number would be n + 7 and the next number would be n + 8. We had our friend add up the four numbers, so let's add our four numbers:
n + n + 1 + n + 7 + n + 8
And since our friend got 88 when she added them, let's make our sum equal 88:
n + n + 1 + n + 7 + n + 8 = 88
Simplify our equation by adding like terms:
4n + 16 = 88
How would you solve this equation? Subtract 16 from both sides?
4n = 72
Divide both sides by 4?
n = 18
Subtract 16 and divide by 4. That's exactly how you solve the puzzle. When your friend tells you the sum, you subtract 16 then divide by 4. This gives you the first number n. (Then add 1 and 7 and 8 for the other numbers).
Alternate and easier method: Subtracting 16 mentally isn't so easy. Go back to that equation:
4n + 16 = 88
I think I see a better way. Factor 4 from the left side of the equation:
4(n + 4) = 88
Now, I could divide both sides by 4:
(n + 4) = 22
Subtract 4 from both sides.
n = 18
That's a lot easier to do mentally. Divide by four and then subtract 4.
Summary: So how does the puzzle work again? Your friend adds any 4 numbers that form a square on the calendar and tells you the sum. You divide by four and then subtract 4. That gives you the first number. You add 1, 7, and 8 to get the other numbers.
And algebra makes it all possible.

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Friedrich und Lochner Statik v2008.2 SL1 *MULTiLANGUAGE* → SoftWare - Graphics & Design
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