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Linear algebra

Calculus 1 Exam 3 2of4

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E
Calculus
Carl Friedrich Gauss
GAUSS

8) Find the inverse function of f. 9) Find the exact solution and rounded to four decimal places solution of the exponential equation. 10) Find the solution using Gaussian elimination or Gauss ? Jordan elimination. 11) Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. Find the solution using Gaussian elimination or Gauss ? Jordan elimination 12) Graph the solution set of the system of inequality. 13) Use Cramer?s Rule to solve the system.
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the quadratic formula

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Marvel Fact Files

?n C2v0Z1q2v wKzu2t8az aSPopfptvwDaAruet FLKLfC2.S s KANltlH trIiAgPhKtJsI prgeFsXeQrJv9e8dM.E F fMOavdqe7 fwxintLhg DI0nIfgiRnui2tgeQ OAKlMgdecb0rBa9 01i.I Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name___________________________________ Period____Date________________Using the Quadratic Formula Solve each equation with the quadratic formula. 1) m 2 ? 5 m ? 14 = 0 2) b2 ? 4 b + 4 = 0 3) 2 m2 + 2 m ? 12 = 0 4) 2 x2 ? 3 x ? 5 = 0 5) x2 + 4 x + 3 = 0 6) 2 x2 + 3 x ? 20 = 0 7) 4 b2 + 8 b + 7 = 4 8) 2 m2 ? 7 m ? 13 = ?10 -1-

Algebra Fill In Notes 2.6

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Algebra
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Fréchet inequalities

02.06 Compound Inequalities Essential Questions How can you create inequalities in one variable and use them to solve problems? How can you represent constraints by inequalities? How can you interpret solutions as viable or nonviable options in a modeling context? Absolute Value Inequalities: Absolute Value Inequalities are problems that involve ranges. For example: On public stairs, handrails must be installed. The height of the handrails must be within a 3 inch range of 35 inches. Compound inequality Key words ?and? / ?or? Scenario 1 A fish has to measure between 18 and 24 inches in length. b>= 18 and b<=24 Writing the compound inequality like this makes it easier to understand that the solutions for b must fall between 18 and 24 inches Scenario 2

Algebra Fill In Notes 2.5

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Algebra
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Triangle inequality

02.05 Inequalities Essential Questions How can you create inequalities in one variable and use them to solve problems? How can you solve linear inequalities in one variable? An?inequality?means the value of the variable is not equal to one number (like in equations), but instead may be greater than or less than a number. There are four primary symbols you need to know when working with inequalities. Indicate what type of circle goes with each inequality symbol > Greater than ______________ < Less than _______________ ? Greater than or equal to ________________ ? Less than or equal to ________________ When you graph an inequality on a number line, there are two questions you must answer. Open or closed circle

Algebra Fill In Notes 2.2

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02.02 Two-Variable Equations Essential Questions How can we create equations in two or more variables to represent relationships between quantities? How can we represent constraints by equations or inequalities and interpret solutions as viable or nonviable options in the model? Main Idea (page #) DEFINITION OR SUMMARY EXAMPLE Two-Variable Equations p.2 Steps to solving problems Read and understand the situation within the word problem. Identify and pull out ______________ ______________ from the problem. Assign ____________to unknown values. Set up and solve the equation. Check that your answer makes sense within the context of the problem. Consecutive integer problems p.2 Label the first integer with x, the next with x + 1, the next with x + 2, and so on.

Elementary Linear Algebra

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ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 10th February 2010 Comments to the author at [email protected] Contents 1 LINEAR EQUATIONS 1 1.1 Introduction to linear equations . . . . . . . . . . . . . . . . . 1 1.2 Solving linear equations . . . . . . . . . . . . . . . . . . . . . 5 1.3 The Gauss?Jordan algorithm . . . . . . . . . . . . . . . . . . 8 1.4 Systematic solution of linear systems. . . . . . . . . . . . . . 9 1.5 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 16 1.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 MATRICES 23 2.1 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 23

Cross Product

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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?

Cross Product

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Calculus
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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?
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