The document discusses writing equations of lines. It provides two forms for writing equations of lines: slope-intercept form (y=mx+b) and point-slope form (y-y1=m(x-x1)). An example problem finds the slope and y-intercept of two lines given by their equations in slope-intercept form and graphs the lines.
1. The document introduces the basic linear model relating two variables Y and X. The model is Y=α+βX+u, where α and β are parameters to be estimated and u is an error term representing other factors not included in the model.
2. The least squares method is introduced to estimate α and β. This method finds the values of α and β that minimize the sum of squared errors between the observed Y values and the estimated Y values from the model.
3. The least squares estimators for α and β are derived by taking the partial derivatives of the sum of squared errors function and setting them equal to zero. This results in estimators for α and β in terms of the sample means
1. The document contains a 4 part engineering mathematics exam with multiple choice and numerical problems.
2. Problems involve differential equations, Taylor series approximations, numerical methods like Euler's method and Picard's method, complex analysis, probability, and statistics.
3. Questions range from deriving equations like the Cauchy-Riemann equations, to evaluating integrals using Cauchy's integral formula, to finding confidence intervals and performing hypothesis tests on statistical data.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
This document discusses quantifying measurement uncertainty. There are two main sources of uncertainty: a repeatable component and a random component. The random component incorporates all factors affecting measurement precision and leads to uncertainty in measured and calculated values. There are two approaches to quantifying standard uncertainty: Type A uses statistical analysis of replicates, while Type B uses best estimates from other factors like instrument specifications. Standard uncertainty is reported with measured values to indicate the precision of the measurement.
D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi KoivistoCosmoAIMS Bassett
The document discusses disformal relations between the physical and gravitational geometry. It begins by introducing the most general form such a relation could take, with two arbitrary functions C and D of a scalar field and its derivative.
It then discusses how this type of relation naturally arises in many modified gravity and scalar-tensor theories. Specific examples mentioned include f(R) gravity and the Dirac-Born-Infeld (DBI) string scenario.
The document outlines how a disformal coupling could have interesting phenomenological implications and be detectable through effects on cosmology and structure formation. It concludes by stating the disformal relation is an important generalization worth further study.
This document discusses the analysis of statically indeterminate plane frames using the force method. It provides examples of how to: (1) analyze rigid frames by selecting redundant reactions, writing compatibility equations, and solving for member forces; (2) account for support settlements by including the predicted support displacement in the compatibility equation; and (3) calculate joint rotations and displacements to sketch the deformed shape of the frame under loading. The method involves releasing constraints to determine a primary structure, using the unit load method to calculate displacements, and applying the principle of superposition to determine final member forces and deformations.
1. The document introduces the basic linear model relating two variables Y and X. The model is Y=α+βX+u, where α and β are parameters to be estimated and u is an error term representing other factors not included in the model.
2. The least squares method is introduced to estimate α and β. This method finds the values of α and β that minimize the sum of squared errors between the observed Y values and the estimated Y values from the model.
3. The least squares estimators for α and β are derived by taking the partial derivatives of the sum of squared errors function and setting them equal to zero. This results in estimators for α and β in terms of the sample means
1. The document contains a 4 part engineering mathematics exam with multiple choice and numerical problems.
2. Problems involve differential equations, Taylor series approximations, numerical methods like Euler's method and Picard's method, complex analysis, probability, and statistics.
3. Questions range from deriving equations like the Cauchy-Riemann equations, to evaluating integrals using Cauchy's integral formula, to finding confidence intervals and performing hypothesis tests on statistical data.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
This document discusses quantifying measurement uncertainty. There are two main sources of uncertainty: a repeatable component and a random component. The random component incorporates all factors affecting measurement precision and leads to uncertainty in measured and calculated values. There are two approaches to quantifying standard uncertainty: Type A uses statistical analysis of replicates, while Type B uses best estimates from other factors like instrument specifications. Standard uncertainty is reported with measured values to indicate the precision of the measurement.
D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi KoivistoCosmoAIMS Bassett
The document discusses disformal relations between the physical and gravitational geometry. It begins by introducing the most general form such a relation could take, with two arbitrary functions C and D of a scalar field and its derivative.
It then discusses how this type of relation naturally arises in many modified gravity and scalar-tensor theories. Specific examples mentioned include f(R) gravity and the Dirac-Born-Infeld (DBI) string scenario.
The document outlines how a disformal coupling could have interesting phenomenological implications and be detectable through effects on cosmology and structure formation. It concludes by stating the disformal relation is an important generalization worth further study.
This document discusses the analysis of statically indeterminate plane frames using the force method. It provides examples of how to: (1) analyze rigid frames by selecting redundant reactions, writing compatibility equations, and solving for member forces; (2) account for support settlements by including the predicted support displacement in the compatibility equation; and (3) calculate joint rotations and displacements to sketch the deformed shape of the frame under loading. The method involves releasing constraints to determine a primary structure, using the unit load method to calculate displacements, and applying the principle of superposition to determine final member forces and deformations.
The document provides three questions from a past exam on Engineering Mathematics IV. Question 1a asks to find the third order Taylor approximation of the differential equation dy/dx = y + 1 with the initial condition y(0) = 0. Question 1b asks to solve a differential equation using the modified Euler's method at two points. Question 1c asks to find the value of y(0.4) using Milne's predictor-corrector method for a given differential equation.
1. Indices, also known as exponents, are used to succinctly represent extremely large or small numbers. They allow writing numbers like 1.44 x 108 km instead of 144,000,000 km.
2. Standard form is a notation where a number is written as A x 10n, where 1 ≤ A < 10 and n is an integer. This puts all numbers in a normalized form and makes them easier to comprehend.
3. Laws of indices allow mathematical operations like multiplication and division to be performed on expressions with common bases or exponents.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
11.application of matrix algebra to multivariate data using standardize scoresAlexander Decker
This document discusses applying matrix algebra to estimate parameters in a regression equation using standardized scores. It presents a methodology for standardizing multivariate data measured in different units. The methodology is demonstrated by applying it to sample data to estimate the regression plane. The results using standardized scores match those obtained in previous studies using original and mean-corrected scores. Standardizing converts data to approximately normal, unit-less scores, addressing issues that arise when data is measured in different units.
Application of matrix algebra to multivariate data using standardize scoresAlexander Decker
This document discusses applying matrix algebra to estimate parameters in a regression equation using standardized scores. It presents a methodology for standardizing multivariate data measured in different units. The methodology is demonstrated by applying it to sample data to estimate the regression plane. The results using standardized scores match those obtained in previous studies using original and mean-corrected scores. Standardizing converts data to unit-less, approximately normal scores, allowing comparison across different measurement units.
This document contains the questions from an Engineering Mathematics examination from December 2012. It covers topics like:
- Using Taylor's series method and Runge-Kutta method to solve initial value problems
- Using Milne's method, Adams-Bashforth method, and Picard's method to solve differential equations
- Properties of analytic functions and bilinear transformations
- Evaluating integrals using Cauchy's integral formula and finding Laurent series
- Expressing polynomials in terms of Legendre polynomials
- Concepts related to probability distributions like binomial, exponential and normal distributions
- Hypothesis testing and confidence intervals
The questions test the students' understanding of numerical methods for solving differential equations, complex analysis topics, orthogonal
Math 1300: Section 5-1 Inequalities in Two VariablesJason Aubrey
This document discusses how to graph linear inequalities in two variables. It begins by introducing linear equations and noting that linear inequalities need to be graphed differently. Key terminology for half-planes divided by boundary lines is provided. The procedure is then outlined as graphing the boundary line and testing a point to determine which half-plane satisfies the inequality. An example of graphing the inequality y ≤ x - 1 is worked through step-by-step.
The document discusses matrix factorization methods for solving systems of linear equations. It covers direct methods like LU, Cholesky, and QR factorizations that decompose a matrix into products of lower and upper triangular matrices. It also explains how to iteratively factorize a matrix A into A = LU by repeatedly subtracting outer products of rows/columns from submatrices. Examples are provided to demonstrate the factorization process.
Math 1300: Section 5-2 Systems of Inequalities in two variablesJason Aubrey
This document discusses how to graphically solve systems of linear inequalities in two variables. It provides steps to (1) graph each boundary line of an inequality and indicate which side of the plane satisfies it, (2) find the intersection points of boundary lines, which are called "corner points", and (3) clearly mark the feasible region that satisfies all the inequalities. An example problem graphically solves the system 2x + y ≥ 4 and 3x - y ≤ 7 by following these steps.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
Geodesic Method in Computer Vision and GraphicsGabriel Peyré
This document discusses geodesic methods in computer vision and graphics. It begins with an overview of topics including Riemannian data modelling, numerical computations of geodesics, geodesic image segmentation, geodesic shape representation, geodesic meshing, and inverse problems with geodesic fidelity. It then provides details on parametric surfaces, Riemannian manifolds, anisotropy and geodesics, the eikonal equation and viscosity solution, discretization methods, and numerical schemes for solving the fixed point equation.
Math 1300: Section 4-1 Review: Systems of Linear Equations in Two VariablesJason Aubrey
This document discusses solving systems of linear equations in two variables through three methods: graphing, substitution, and elimination. It defines key terms like consistent, independent, and dependent when describing systems of linear equations. The document also presents a theorem stating the possible solutions a linear system can have and provides an example of graphing a system to find the solution set.
Scatter diagrams and correlation and simple linear regresssionAnkit Katiyar
The document discusses scatter diagrams, correlation, and linear regression. It defines key terms like predictor and response variables, positively and negatively associated variables, and the correlation coefficient. It also describes how to calculate the linear correlation coefficient and interpret it. The document shows an example of using least squares regression to fit a line to productivity and experience data. It provides formulas to calculate the slope and intercept of the regression line and how to make predictions with the line. However, predictions should stay within the scope of the observed data used to fit the model.
The document provides information on trigonometric ratios including:
1) It defines the six main trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) using coordinates on a unit circle.
2) It explains the signs of the ratios in the four quadrants of the coordinate plane.
3) It discusses the range of values that the six ratios can take.
4) It provides a table of the exact ratio values for 0, π/6, π/4, π/3, π/2, π, 3π/2, and 2π.
5) It covers fundamental trigonometric
Lecture 4 3 d stress tensor and equilibrium equationsDeepak Agarwal
* Shear stress, τ = 50 N/mm2
* Shear modulus, C = 8x104 N/mm2
* Strain energy per unit volume = τ2/2C
= (50)2 / 2(8x104)
= 0.3125 J/mm3
Therefore, the local strain energy per unit volume stored in the material due to shear stress is 0.3125 J/mm3.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
This document provides definitions and examples related to parallel and perpendicular lines. It defines parallel lines as lines in the same plane that do not intersect, and skew lines as lines in different planes that do not intersect. It also defines angles formed by parallel lines cut by a transversal, such as corresponding angles and alternate interior angles. Examples classify line segments and planes as parallel or skew, and identify angle relationships.
This document discusses ways to prove that two lines are parallel using angle relationships. It introduces the converse of corresponding angles postulate, parallel postulate, and converses of alternate exterior angles, consecutive interior angles, and alternate interior angles postulates. Examples are provided to demonstrate using these theorems to prove lines are parallel or not parallel based on given angle information. Readers are asked to complete practice problems applying these concepts.
The document provides three questions from a past exam on Engineering Mathematics IV. Question 1a asks to find the third order Taylor approximation of the differential equation dy/dx = y + 1 with the initial condition y(0) = 0. Question 1b asks to solve a differential equation using the modified Euler's method at two points. Question 1c asks to find the value of y(0.4) using Milne's predictor-corrector method for a given differential equation.
1. Indices, also known as exponents, are used to succinctly represent extremely large or small numbers. They allow writing numbers like 1.44 x 108 km instead of 144,000,000 km.
2. Standard form is a notation where a number is written as A x 10n, where 1 ≤ A < 10 and n is an integer. This puts all numbers in a normalized form and makes them easier to comprehend.
3. Laws of indices allow mathematical operations like multiplication and division to be performed on expressions with common bases or exponents.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
11.application of matrix algebra to multivariate data using standardize scoresAlexander Decker
This document discusses applying matrix algebra to estimate parameters in a regression equation using standardized scores. It presents a methodology for standardizing multivariate data measured in different units. The methodology is demonstrated by applying it to sample data to estimate the regression plane. The results using standardized scores match those obtained in previous studies using original and mean-corrected scores. Standardizing converts data to approximately normal, unit-less scores, addressing issues that arise when data is measured in different units.
Application of matrix algebra to multivariate data using standardize scoresAlexander Decker
This document discusses applying matrix algebra to estimate parameters in a regression equation using standardized scores. It presents a methodology for standardizing multivariate data measured in different units. The methodology is demonstrated by applying it to sample data to estimate the regression plane. The results using standardized scores match those obtained in previous studies using original and mean-corrected scores. Standardizing converts data to unit-less, approximately normal scores, allowing comparison across different measurement units.
This document contains the questions from an Engineering Mathematics examination from December 2012. It covers topics like:
- Using Taylor's series method and Runge-Kutta method to solve initial value problems
- Using Milne's method, Adams-Bashforth method, and Picard's method to solve differential equations
- Properties of analytic functions and bilinear transformations
- Evaluating integrals using Cauchy's integral formula and finding Laurent series
- Expressing polynomials in terms of Legendre polynomials
- Concepts related to probability distributions like binomial, exponential and normal distributions
- Hypothesis testing and confidence intervals
The questions test the students' understanding of numerical methods for solving differential equations, complex analysis topics, orthogonal
Math 1300: Section 5-1 Inequalities in Two VariablesJason Aubrey
This document discusses how to graph linear inequalities in two variables. It begins by introducing linear equations and noting that linear inequalities need to be graphed differently. Key terminology for half-planes divided by boundary lines is provided. The procedure is then outlined as graphing the boundary line and testing a point to determine which half-plane satisfies the inequality. An example of graphing the inequality y ≤ x - 1 is worked through step-by-step.
The document discusses matrix factorization methods for solving systems of linear equations. It covers direct methods like LU, Cholesky, and QR factorizations that decompose a matrix into products of lower and upper triangular matrices. It also explains how to iteratively factorize a matrix A into A = LU by repeatedly subtracting outer products of rows/columns from submatrices. Examples are provided to demonstrate the factorization process.
Math 1300: Section 5-2 Systems of Inequalities in two variablesJason Aubrey
This document discusses how to graphically solve systems of linear inequalities in two variables. It provides steps to (1) graph each boundary line of an inequality and indicate which side of the plane satisfies it, (2) find the intersection points of boundary lines, which are called "corner points", and (3) clearly mark the feasible region that satisfies all the inequalities. An example problem graphically solves the system 2x + y ≥ 4 and 3x - y ≤ 7 by following these steps.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
Geodesic Method in Computer Vision and GraphicsGabriel Peyré
This document discusses geodesic methods in computer vision and graphics. It begins with an overview of topics including Riemannian data modelling, numerical computations of geodesics, geodesic image segmentation, geodesic shape representation, geodesic meshing, and inverse problems with geodesic fidelity. It then provides details on parametric surfaces, Riemannian manifolds, anisotropy and geodesics, the eikonal equation and viscosity solution, discretization methods, and numerical schemes for solving the fixed point equation.
Math 1300: Section 4-1 Review: Systems of Linear Equations in Two VariablesJason Aubrey
This document discusses solving systems of linear equations in two variables through three methods: graphing, substitution, and elimination. It defines key terms like consistent, independent, and dependent when describing systems of linear equations. The document also presents a theorem stating the possible solutions a linear system can have and provides an example of graphing a system to find the solution set.
Scatter diagrams and correlation and simple linear regresssionAnkit Katiyar
The document discusses scatter diagrams, correlation, and linear regression. It defines key terms like predictor and response variables, positively and negatively associated variables, and the correlation coefficient. It also describes how to calculate the linear correlation coefficient and interpret it. The document shows an example of using least squares regression to fit a line to productivity and experience data. It provides formulas to calculate the slope and intercept of the regression line and how to make predictions with the line. However, predictions should stay within the scope of the observed data used to fit the model.
The document provides information on trigonometric ratios including:
1) It defines the six main trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) using coordinates on a unit circle.
2) It explains the signs of the ratios in the four quadrants of the coordinate plane.
3) It discusses the range of values that the six ratios can take.
4) It provides a table of the exact ratio values for 0, π/6, π/4, π/3, π/2, π, 3π/2, and 2π.
5) It covers fundamental trigonometric
Lecture 4 3 d stress tensor and equilibrium equationsDeepak Agarwal
* Shear stress, τ = 50 N/mm2
* Shear modulus, C = 8x104 N/mm2
* Strain energy per unit volume = τ2/2C
= (50)2 / 2(8x104)
= 0.3125 J/mm3
Therefore, the local strain energy per unit volume stored in the material due to shear stress is 0.3125 J/mm3.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
This document discusses angles and parallel lines. It presents three postulates and theorems about corresponding angles, alternate interior angles, and consecutive interior angles when two parallel lines are cut by a transversal. It then provides three multi-part examples that apply these postulates and theorems to find angle measurements or variables in diagrams. The examples demonstrate using vertical angles, supplementary angles, congruent corresponding and alternate interior angles, and algebra to solve for variables.
This document provides definitions and examples related to parallel and perpendicular lines. It defines parallel lines as lines in the same plane that do not intersect, and skew lines as lines in different planes that do not intersect. It also defines angles formed by parallel lines cut by a transversal, such as corresponding angles and alternate interior angles. Examples classify line segments and planes as parallel or skew, and identify angle relationships.
This document discusses ways to prove that two lines are parallel using angle relationships. It introduces the converse of corresponding angles postulate, parallel postulate, and converses of alternate exterior angles, consecutive interior angles, and alternate interior angles postulates. Examples are provided to demonstrate using these theorems to prove lines are parallel or not parallel based on given angle information. Readers are asked to complete practice problems applying these concepts.
This document discusses finding distances between lines and points. It defines equidistant lines as lines where the distance between them is the same when measured along a perpendicular. It explains that the distance between a point and line is the length of the perpendicular segment from the point to the line, and the distance between parallel lines is the length of the perpendicular segment between the lines. The document provides an example problem that finds the distance between a line and point by first finding the equations of the given line and perpendicular line through the point, then solving the system of equations.
The document discusses slopes of lines. It defines slope as the ratio of the vertical change to the horizontal change between two points. The slope formula is given as m=(y2-y1)/(x2-x1). An example problem finds the slopes of lines passing through two pairs of points by applying the slope formula.
This document summarizes key concepts about proving segment relationships using postulates and theorems of geometry. It includes:
1) An introduction describing essential questions about segment addition and congruence proofs, as well as relevant postulates and theorems.
2) An example proof that if two segments are congruent, their endpoints are also congruent.
3) A proof of the transitive property of segment congruence.
4) An example proof using the properties of congruence and equality to show that if corresponding sides of a badge are congruent or equal, then the bottom and left sides are also congruent.
This document contains a chapter about inductive reasoning and making conjectures based on observations. It includes examples of writing conjectures to describe patterns in sequences and geometric relationships. It also discusses using counterexamples to show when a conjecture is not true. The document contains vocabulary definitions and practice problems for students to work through.
The document defines key terms related to conditional statements, including conditional statement, if-then statement, hypothesis, conclusion, converse, inverse, and contrapositive. It provides examples of identifying the hypothesis and conclusion of conditional statements and determining the truth value of conditionals. Examples are worked through step-by-step to identify related conditionals and write statements in if-then form. Key terms are defined over multiple slides as examples are introduced.
The document discusses postulates and paragraph proofs in geometry. It defines key terms like postulate, axiom, proof, theorem, and paragraph proof. It provides examples of using postulates to determine if statements are always, sometimes, or never true. It also gives examples of writing paragraph proofs, including using the midpoint postulate to prove two line segments are congruent. The document emphasizes building proofs using accepted postulates and definitions.
This document defines key logic terms like statements, truth values, negation, conjunction, disjunction, and truth tables. It then provides examples of writing compound statements using these operators and determining their truth values, including using Venn diagrams. Truth tables are constructed to represent the truth values of compound statements.
The document discusses proving angle relationships through postulates and theorems. It introduces the protractor postulate, angle addition postulate, and theorems regarding supplementary, complementary, congruent, and right angles. Examples are provided to demonstrate using these concepts to prove and determine angle measures.
This document provides an overview of deductive reasoning, including the Law of Detachment and Law of Syllogism. It defines deductive reasoning as using facts, rules, and definitions to reach a logical conclusion. The Law of Detachment states that if p implies q is true and p is true, then q must be true. The Law of Syllogism states that if p implies q and q implies r are both true, then p implies r must be true. The document provides examples and problems applying these laws of deductive reasoning.
The document provides information on algebraic proofs and two-column proofs. It defines an algebraic proof as using a series of algebraic steps to solve problems and justify steps. A two-column proof is defined as having one column for statements and a second column for justifying each statement. Properties of equality like addition, subtraction, multiplication, and division properties are also defined for writing algebraic proofs. An example problem is worked through step-by-step to demonstrate an algebraic proof.
This document summarizes key concepts about isosceles and equilateral triangles. It defines important vocabulary like the legs and vertex angle of an isosceles triangle. It states theorems like if two sides of a triangle are congruent, the angles opposite them are congruent. It also defines properties of equilateral triangles, like they are equiangular and each angle is 60 degrees. Examples demonstrate using these concepts to find missing angle and side measures. The document concludes with assigning practice problems.
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
The document discusses two postulates for proving triangle congruence: the Angle-Side-Angle (ASA) postulate and the Angle-Angle-Side (AAS) postulate. It includes examples of using these postulates to prove triangles are congruent by showing that corresponding angles and sides are congruent. The examples demonstrate how to set up and solve proofs of triangle congruence using angle and side relationships.
This document provides information about angles of triangles, including essential questions, vocabulary, theorems, examples, and a problem set. It defines key terms like auxiliary line, exterior angle, and remote interior angles. It presents the Triangle Angle-Sum Theorem stating the sum of interior angles is 180 degrees and the Exterior Angle Theorem relating exterior and remote interior angles. Examples demonstrate using these theorems to find angle measures. The final problem set directs working additional practice problems.
This document provides examples and explanations for proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with definitions of key vocabulary like included angle. Example 1 uses SSS to prove two triangles congruent by showing that corresponding sides are congruent. Example 2 has students graph triangles on a coordinate plane and determine if they are congruent. Example 3 uses the midpoint theorem and vertical angles theorem to prove triangles congruent via SAS. The document concludes with practice problems for students.
This document provides information about surface areas and volumes of spheres. It defines key terms like great circle, pole, and hemisphere. It presents the formulas for calculating the surface area and volume of spheres and hemispheres. It then works through 6 examples applying these formulas to find surface areas and volumes given radius or diameter values. The examples demonstrate how to set up and solve the related equations.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step to model how to set up and solve for the line equation in different forms.
This document provides examples and explanations for writing and graphing linear equations. It defines key vocabulary like linear equation, slope, y-intercept, slope-intercept form, and point-slope form. Examples demonstrate finding the slope and y-intercept of lines given in various forms and writing the equation of a line given the slope and a point. The examples are worked through step-by-step with explanations of the process.
The document discusses solving the 2D wave equation using separation of variables and superposition. It separates the wave equation into ordinary differential equations for the spatial and temporal parts. The solutions to the spatial equations give normal modes, which are combined using superposition to satisfy the initial conditions. As an example, the document finds the solution for a rectangular membrane with a given initial shape.
The document discusses straight line graphs on the number plane and their equations.
1. The red lines are parallel with equations that have -1/2, which determines the angle. The sign of the constant a indicates whether the line increases or decreases as x increases.
2. The blue lines are also parallel, with equations sharing the constant 2, which determines their angle.
3. The lines y=-1/2x, y=2x, and y=x intersect at the origin because their equations have b=0, making the y-value 0 when x=0.
The document then provides two alternative ways to graph a line: i) finding the x- and y-
The document discusses finding the equation of a straight line given different pieces of information. It explains that the equation of a line can be expressed in the forms y=mx+c or ax+by+c=0, where m is the gradient and c is the y-intercept. It also discusses how to determine the gradient and y-intercept from the equation of a line, and how to write an equation in standard form y=mx+c given information like two points or the gradient and a point.
The document provides an overview of acoustics and sound waves presented in a lecture. It discusses the wave equation, acoustic tubes, reflections, resonances, and standing waves. Key concepts covered include traveling waves, wave velocity, terminations, transfer functions, scattering junctions, and modeling the vocal tract as a concatenated tube system.
This document discusses beam deflections and summarizes a method for calculating beam deflection using multiple integration. It provides an example of using this method to calculate the deflection of a beam under three-point bending. The maximum deflection occurs at the beam's midpoint and is given by the equation P L3/48EI. It also discusses analyzing statically indeterminate beams by writing slope and deflection equations with unknown reaction forces and solving for the forces using boundary conditions. An example is provided of calculating the deflection of a beam supported at three points.
This document discusses fundamental concepts in analytic geometry related to lines. It defines key terms like slope, the different forms of a line equation, and how to find the distance from a point to a line. It also covers properties and elements of triangles, including how to calculate the length of a median, altitude, and angle bisector.
The document discusses calculating and interpreting the gradient of a straight line segment.
1. The gradient of a line segment is a measure of how steep the line is, and can be calculated as the change in y over the change in x between two points on the line.
2. Parallel lines have the same gradient, while perpendicular lines have gradients that are negative reciprocals of each other.
3. Examples are provided to demonstrate calculating gradients of lines and using gradients to determine whether lines are parallel or perpendicular.
This document contains instructions for 5 assignment questions involving numerical integration and solving differential equations. Question 1 involves using the quad function to evaluate several integrals. Question 2 involves using quad to evaluate Fresnel integrals and plot the results. Question 3 involves using Monte Carlo methods to estimate volumes and double integrals. Question 4 involves using Euler's method to solve an initial value problem and analyze errors. Question 5 involves using lsode to solve a system of differential equations modeling atmospheric circulation and experimenting with initial conditions.
The document discusses key topics in microwave engineering including:
1. Maxwell's equations which describe the fundamentals of electromagnetics.
2. Explanations of important concepts like electric and magnetic fields, vectors, divergence, curl and boundary conditions.
3. An overview of industries utilizing RF components and analysis of the RF components market including development trends, major companies and factors changing the industry.
This document contains a 9 question final exam for an applied ordinary differential equations engineering course. The questions cover a range of topics including: finding general and particular solutions to 1st and 2nd order differential equations using various methods; solving initial value problems; solving systems of differential equations; power series solutions; and modeling an LRC circuit with a differential equation. Students are given 3 hours to complete the exam worth a total of 100 points.
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1) Finding even and odd components of signals, determining if signals are energy or power signals, and plotting shifted versions of a signal.
2) Proving properties of LTI systems based on impulse response and input, determining output of LTI systems given various inputs and impulse responses.
3) Finding Fourier series coefficients and representations of signals, determining Fourier transforms and properties.
4) Determining difference/differential equation descriptions and impulse/frequency responses of systems based on given input-output relations or equations.
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- Telegrapher's equation and circuit model for transmission lines
- Wave propagation and characteristic impedance calculations
- Reflection coefficient and standing wave ratio definitions
- Comparisons of transmission line, circuit, and field theories
2. Specific transmission line types are analyzed, including planar lines, coaxial cables. Equations are given for calculating the capacitance, conductance, inductance, resistance, and characteristic impedance of these common line configurations.
3. Simulation and modeling techniques for transmission lines are briefly mentioned, such as the transmission line matrix method for modeling microstrip lines in antennas and circuits.
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This document contains text and examples about linear functions and equations. It defines linear equations as having terms that are either constants or the product of a constant and a single variable. It discusses identifying linear and nonlinear functions. It also covers identifying and graphing the slope and y-intercept of a line, writing equations in slope-intercept, standard, and intercept form, graphing lines from their equations, and finding x- and y-intercepts of a line.
This document contains the questions from an engineering mathematics exam with 8 questions divided into 2 parts (A and B). Part A contains 3 multi-part questions on topics related to differential equations, including using Taylor's series, Runge-Kutta method, and Milne's predictor-corrector method to solve initial value problems. Part B contains 5 multi-part questions covering additional topics such as Legendre polynomials, Bessel's differential equation, probability, hypothesis testing, and confidence intervals. The exam tests knowledge of numerical analysis techniques for solving differential equations as well as topics in advanced calculus, probability, and statistics.
The document is a math exam containing 5 questions regarding differential equations. Question 1 has 3 parts asking about integrating factors, values of m, and solutions to a differential equation. Question 2 asks for the general solution to a given differential equation. Question 3 has 7 parts testing knowledge of rate equations, linear dependence, differential equation types, undetermined coefficients methods, uniqueness of solutions, damped motion, and Euler's method. Question 4 asks for the position function of a damped spring system. Question 5 asks to use the variation of parameters method to find the particular solution to a given nonhomogeneous differential equation.
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Geometry Section 3-4 1112
1. SECTION 3-4
Equations of Lines
Tuesday, January 3, 2012
2. ESSENTIAL QUESTIONS
How do you write an equation of a line given
information about the graph?
How do you solve problems by writing equations?
Tuesday, January 3, 2012
4. VOCABULARY
1. Slope-intercept Form: y = mx + b, where m = slope
and b = y coordinate of the y-intercept -- (0, b)
2. Point-slope form:
Tuesday, January 3, 2012
5. VOCABULARY
1. Slope-intercept Form: y = mx + b, where m = slope
and b = y coordinate of the y-intercept -- (0, b)
2. Point-slope form: y - y1 = m(x - x1), where m = slope
and (x1, y1) is a point on the line
Tuesday, January 3, 2012
6. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
Tuesday, January 3, 2012
7. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
Tuesday, January 3, 2012
8. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
Tuesday, January 3, 2012
9. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
10. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
11. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
12. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
13. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
14. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
15. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
16. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
17. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
1
a. y = x
2
1
m=
2
b=0
y-intercept: (0, 0)
Tuesday, January 3, 2012
18. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
Tuesday, January 3, 2012
19. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x
Tuesday, January 3, 2012
20. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
Tuesday, January 3, 2012
21. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
Tuesday, January 3, 2012
22. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
Tuesday, January 3, 2012
23. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
Tuesday, January 3, 2012
24. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
Tuesday, January 3, 2012
25. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m=
2
Tuesday, January 3, 2012
26. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
27. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
28. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
29. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
30. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
31. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
32. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
33. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
34. EXAMPLE 1
Identify the slope and y-intercept for each line and
graph the lines.
b. 2y − x = −6
+x +x
2y = x − 6
2 2
1
y = x −3
2
1
m= y-intercept = (0, -3)
2
Tuesday, January 3, 2012
35. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Tuesday, January 3, 2012
36. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
Tuesday, January 3, 2012
37. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
Tuesday, January 3, 2012
38. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
39. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
40. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
41. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
42. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
43. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
44. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
45. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
46. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
a. m = -2, (0, 4)
Use slope-intercept
y = mx + b
y = −2x + 4
Tuesday, January 3, 2012
47. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Tuesday, January 3, 2012
48. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
Tuesday, January 3, 2012
49. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
Tuesday, January 3, 2012
50. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8)
3
4
Tuesday, January 3, 2012
51. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
Tuesday, January 3, 2012
52. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
y = x−8
3
4
Tuesday, January 3, 2012
53. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
y = x−8
3
4
Tuesday, January 3, 2012
54. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
y = x−8
3
4
Tuesday, January 3, 2012
55. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
y = x−8
3
4
Tuesday, January 3, 2012
56. EXAMPLE 2
Write the equation of the line, choosing which
form you would need before you begin. Then
graph the line.
b. m = 3/4, (8, -2)
Use point-slope
y − y1 = m(x − x1 )
y + 2 = (x − 8) 3
4
y +2= x−6 3
4
y = x−8
3
4
Tuesday, January 3, 2012
57. EXAMPLE 3
Write the equation for the line.
a.
Tuesday, January 3, 2012
58. EXAMPLE 3
Write the equation for the line.
a.
5
m=−
7
Tuesday, January 3, 2012
59. EXAMPLE 3
Write the equation for the line.
a.
5
m=− (4, -1)
7
Tuesday, January 3, 2012
60. EXAMPLE 3
Write the equation for the line.
a. y − y1 = m(x − x1 )
5
m=− (4, -1)
7
Tuesday, January 3, 2012
61. EXAMPLE 3
Write the equation for the line.
a. y − y1 = m(x − x1 )
y +1= − (x − 4)
5
7
5
m=− (4, -1)
7
Tuesday, January 3, 2012
62. EXAMPLE 3
Write the equation for the line.
a. y − y1 = m(x − x1 )
y +1= − (x − 4)
5
7
y +1= − x +
5
7
20
7
5
m=− (4, -1)
7
Tuesday, January 3, 2012
63. EXAMPLE 3
Write the equation for the line.
a. y − y1 = m(x − x1 )
y +1= − (x − 4)
5
7
y +1= − x + 5
7
20
7
5 y=− x+ 5
7
13
7
m=− (4, -1)
7
Tuesday, January 3, 2012
64. EXAMPLE 3
Write the equation for the line.
b.
Tuesday, January 3, 2012
65. EXAMPLE 3
Write the equation for the line.
b.
6
m=
8
Tuesday, January 3, 2012
66. EXAMPLE 3
Write the equation for the line.
b.
6 3
m= =
8 4
Tuesday, January 3, 2012
67. EXAMPLE 3
Write the equation for the line.
b.
6 3
m= = (3, 2)
8 4
Tuesday, January 3, 2012
68. EXAMPLE 3
Write the equation for the line.
b.
y − y1 = m(x − x1 )
6 3
m= = (3, 2)
8 4
Tuesday, January 3, 2012
69. EXAMPLE 3
Write the equation for the line.
b.
y − y1 = m(x − x1 )
y − 2 = (x − 3)
3
4
6 3
m= = (3, 2)
8 4
Tuesday, January 3, 2012
70. EXAMPLE 3
Write the equation for the line.
b.
y − y1 = m(x − x1 )
y − 2 = (x − 3)
3
4
y−2= x− 3
4
9
4
6 3
m= = (3, 2)
8 4
Tuesday, January 3, 2012
71. EXAMPLE 3
Write the equation for the line.
b.
y − y1 = m(x − x1 )
y − 2 = (x − 3)
3
4
y−2= x− 3
4
9
4
6 3 y= x−
3
4
1
4
m= = (3, 2)
8 4
Tuesday, January 3, 2012
72. CHECK YOUR
UNDERSTANDING
Review problems #1-12 on p. 200
Tuesday, January 3, 2012
74. PROBLEM SET
p. 200 #13-41 odd, 55
“We can have facts without thinking, but we cannot
have thinking without facts.” - John Dewey
Tuesday, January 3, 2012