Unit 6.3
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This document discusses parametric equations and motion. It defines parametric curves as graphs of ordered pairs (x,y) where x and y are functions of a parameter t. Parametric equations can be used to model the path of objects in motion. Examples are provided of graphing parametric equations, eliminating the parameter to get a standard equation, and finding parametric equations to represent a line. The document also contains sample problems testing understanding of parametric equations and related concepts.
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2. It asks students to define sets, prove properties of sets, solve problems involving sets and functions, write symbolic logic statements, and determine if logic arguments are valid or not.
3. Several questions also involve topics like tautologies, propositional logic, and predicate logic.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance. It defines a line segment as a portion of a line distinguished by endpoints, and defines betweenness of points and the term "between." It also defines congruent segments, constructions, distance, and irrational numbers. Several examples are provided to demonstrate calculating distances between points on a number line, using a ruler to measure segments, and applying the Pythagorean theorem.
Geometry Section 1-2
This document defines key vocabulary terms related to line segments and distance, including line segment, betweenness of points, congruent segments, and distance formula. It provides examples of calculating distances between points on number lines and using the Pythagorean theorem to find distances between points graphed on a coordinate plane. Examples include measuring line segments with a ruler, finding distances by adding or subtracting measures, and applying the distance formula and Pythagorean theorem to solve for unknown distances.
Straight line
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
Coordinate geometry
The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.
3rd semester Civil Engineering ( 2013-June) Question Papers
3rd semester Civil Engineering ( 2013-June) Question Papers BGS Institute of Technology, Adichunchanagiri University (ACU)
This document contains questions from engineering mathematics, strength of materials, and surveying exams. Some key questions include:
1) Finding Fourier transforms and series expansions of various functions.
2) Calculating stresses, strains, deflections, and loads in beams, columns, and other structural elements.
3) Explaining surveying concepts like bearings, triangulation, traversing, leveling, contours, and performing related calculations.2013-June 3rd Semester Civil Engineering Question Papers
This document contains information about an engineering mathematics examination, including questions on various topics like Fourier transforms, differential equations, interpolation, and numerical methods. It provides instructions to answer 5 full questions, with at least 2 questions from each part. The first part covers questions on Fourier series, transforms, differential equations, and interpolation. The second part includes questions on numerical methods, matrices, and integration.
Geometry Section 1-3 1112
This document discusses distance and midpoints between points in a coordinate plane. It defines distance as the length of a segment between two points and the Pythagorean theorem. The midpoint of a segment is the point halfway between the two endpoints. Examples are provided to demonstrate calculating distance and midpoints using formulas like the distance formula and midpoint formula.
G coordinate, s tat, c measure
The document discusses various topics in coordinate geometry including: distance between two points, division of line segments, midpoints, the ratio theorem, areas of polygons, equations of straight lines, parallel and perpendicular lines, loci involving distance between two points. It also provides notes to candidates on how to approach questions involving diagrams, using formulas correctly, and not accepting solutions by scale drawing alone.
Coordinate geometry
The document discusses various concepts in coordinate geometry including:
(1) Calculating the gradient of a straight line from its equation in slope-intercept form or given two points on the line. Parallel and perpendicular lines have related gradients.
(2) Finding the midpoint and length of a line segment given two points.
(3) Determining the equation of a straight line given its gradient and a point, or given two points, or from its graph.
(4) Interpreting the x- and y-intercepts from the equation of a straight line.
5.8 Modeling with Quadratic Functions
This document discusses modeling with quadratic functions. It provides examples of writing quadratic functions in vertex form, intercept form, and standard form given certain information about the parabola. It also discusses finding a quadratic model from a data set and using it to find the maximum point.
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Geometry 201 Unit 4.1
This document discusses identifying congruent figures and using congruence to find missing measures. It defines congruent figures as having the same shape and size. The Side-Side-Side Rule for triangles is explained - if the sides of two triangles are congruent, then the triangles are congruent. Examples are provided of identifying congruent triangles and polygons based on corresponding sides and angles. The document also provides examples of using known congruent figures to find missing side lengths or angle measures.
Algebra 302 unit 11.4
This document discusses conditional probability and how to calculate and interpret joint, marginal, and conditional relative frequencies using two-way frequency tables. It provides examples of finding conditional probabilities based on frequency tables of data involving variables like whether students are taking ballet or tap dancing classes. One example determines the probability a student is not taking tap dancing given they are taking ballet class is about 69%.
Algebra 2 unit 10.6
This document discusses double-angle and half-angle identities in trigonometry. It provides examples of using these identities to evaluate expressions and prove trigonometric identities. Key double-angle identities covered are sin(2θ), cos(2θ), and tan(2θ). The document also derives the half-angle identities and provides examples of using them to find exact trigonometric values.
Algebra 2 unit 10.7
This document discusses double-angle and half-angle identities in trigonometry. It provides examples of using these identities to evaluate expressions and prove trigonometric identities. Key double-angle identities covered are sin(2θ), cos(2θ), and tan(2θ). The document also derives the half-angle identities and provides examples of using them to find exact trigonometric values.
Algebra 2 unit 10.5
The document discusses the Law of Cosines and Heron's Formula. It begins by explaining that the Law of Cosines can be used to find side lengths and angle measures of a triangle when given side-angle-side or side-side-side information. It then provides examples of using the Law of Cosines to solve for missing values of triangles. It also introduces Heron's Formula, which can be used to find the area of a triangle based on its side lengths. Examples are given of using both the Law of Cosines and Heron's Formula to solve multi-step word problems involving triangles.
Algebra 2 unit 10.4
This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
Algebra 2 unit 10.3
This document provides instruction on right triangles and trigonometric ratios. It begins with examples of finding missing angles and side lengths in right triangles using trigonometric functions like sine, cosine and tangent. Special right triangles involving 30-60-90 and 45-45-90 triangles are discussed. Real world applications include finding the height a skier leaves a ramp and the height of a tree using an angle of elevation. The document also covers cosecant, secant and cotangent functions and has practice problems for students.
Algebra 2 unit 10.2
This document discusses inverse trigonometric functions and how to use them to solve problems. It defines the inverse sine, cosine, and tangent functions and explains that they find the measure of angles given the value of a trigonometric function. Examples are provided to demonstrate evaluating inverse trigonometric functions and using them to solve equations and application problems like finding the angle a ladder should make with the ground given its length. Restricting the domains of the trigonometric functions is necessary to define the inverse functions since more than one angle can produce the same output value.
11.1 combination and permutations
This document discusses permutations and combinations. It provides examples of using formulas and the fundamental counting principle to calculate the number of possible outcomes for different scenarios involving selecting items with or without regard for order. Examples include determining the number of possible sandwiches, voicemail passwords, bead necklace designs, and ways to select people or items from a group. The key concepts of permutations, combinations, and factorials are explained.
Unit 11.3 probability of multiple events
This document summarizes key concepts about probability of multiple events from Holt Algebra 1. It discusses independent and dependent events and provides examples of calculating probabilities. For independent events, the probability is calculated by multiplying the individual probabilities. For dependent events, the probability of the second event is based on the sample space after the first event occurs. Examples include flipping coins, drawing marbles from a bag with and without replacement, and selecting items randomly. Diagrams are used to illustrate how the sample space changes for dependent events.
Unit 11.2 experimental probability
This document provides an overview of experimental probability:
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- It explains how to calculate experimental probability by taking the number of times an event occurs over the total number of trials.
- It discusses how experimental probability can be used to make predictions about future experiments based on past results.
Unit 11.2 theoretical probability
This document provides an overview of theoretical probability concepts including:
- Theoretical probability is calculated as the number of favorable outcomes divided by the total number of equally likely outcomes.
- Experiments are considered "fair" if all outcomes are equally likely.
- The probability of an event occurring plus the probability of it not occurring equals 1.
- Complement probability refers to outcomes not included in the main event.
- Odds can be used to express likelihood and converted to and from probabilities using ratios of favorable to total outcomes.
11.1 11.1 combination and permutations
This document discusses permutations and combinations. It provides examples of using the fundamental counting principle and formulas for permutations and combinations to calculate the number of possible outcomes for various events and selections. Examples include determining the number of possible meal combinations, voicemail passwords, bead stringing patterns, sandwich ingredients, swimming race outcomes, people lining up, and selecting groups from a class. The key concepts of permutations versus combinations and using factorials are also explained.
Geometry 201 unit 5.7
1) The document discusses using inequalities to compare angles and side lengths in two triangles. It provides examples of applying the hinge theorem and its converse to determine relationships between angles and sides.
2) One example involves comparing the distances traveled by two people leaving school at different routes to determine who is farther from school.
3) The document also demonstrates how to write two-column proofs to prove triangle relationships based on given information about the triangles. It includes practice problems for the reader to check their understanding.
Geometry 201 unit 5.5
This document provides examples and explanations of indirect proofs. It begins with examples of writing indirect proofs to show that a triangle cannot have two right angles or that a number is greater than 0. It then discusses using inequalities in indirect proofs involving triangles. Examples demonstrate ordering triangle sides and angles, applying the triangle inequality theorem to determine if a triangle can exist with given side lengths, and using indirect proofs to find possible side lengths. The document concludes with practice problems applying these concepts.
Geometry 201 unit 5.4
1) A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The medians of any triangle are concurrent at the centroid.
2) An altitude of a triangle is a perpendicular segment from a vertex to the opposite side. The three altitudes of a triangle are concurrent at the orthocenter.
3) This document discusses properties of medians and altitudes of triangles, including using the centroid and orthocenter to solve problems involving lengths of segments and finding coordinates of special points like the centroid and orthocenter. Examples are provided to illustrate these concepts.
Geometry 201 unit 5.3
This document discusses bisectors in triangles. It defines perpendicular bisectors as lines that bisect and are perpendicular to a side of a triangle. Angle bisectors bisect an angle of a triangle. The three perpendicular bisectors and angle bisectors of a triangle are both concurrent, meaning they intersect at a single point.
The point of concurrency of the perpendicular bisectors is called the circumcenter and is equidistant from the triangle's vertices. The point of concurrency of the angle bisectors is called the incenter and is always inside the triangle, equidistant from its sides. Examples show using bisectors to find distances and angles in triangles, as well as applications like placing a building or monument equidistant from three
Geometry 201 unit 4.7
This document provides information about using the Hypotenuse-Leg Theorem to prove triangles congruent. It gives examples of when the theorem can and cannot be used. It also asks what additional information is needed to prove triangles congruent using the theorem. The document then discusses how to identify and separate overlapping triangles when proving congruence. It provides examples of writing paragraph proofs to show triangles are congruent using angle-angle-side, side-side-side, and angle-side-angle criteria.
Geometry 201 unit 4.4
1) The document discusses using corresponding parts of congruent triangles (CPCTC) to prove that parts of triangles are congruent. CPCTC states that if two triangles are congruent, then their corresponding parts are congruent.
2) It provides examples of using CPCTC to prove that corresponding angles and sides of congruent triangles are congruent. Proofs involve showing triangles are congruent using criteria like SSS, SAS, ASA etc. and then invoking CPCTC.
3) The document emphasizes working backwards in proofs - looking for congruent angles/sides and then identifying congruent triangles to apply CPCTC. Several practice examples are provided to illustrate this technique.
Geometry 201 unit 4.3
This document discusses triangle congruence using the ASA, AAS, and Hypotenuse-Leg (HL) theorems. It provides examples of applying these theorems to determine if triangles are congruent and to prove triangles congruent. The examples include real world applications involving directions and distances between locations. Key terms like included side are also defined. Practice problems are included for students to determine if a triangle congruence theorem can be used or additional information is needed.
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Unit 6.3
- 1. 6.3 Parametric Equations and Motion Copyright © 2011 Pearson, Inc.
- 2. What you’ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher … and why These topics can be used to model the path of an object such as a baseball or golf ball. Copyright © 2011 Pearson, Inc. Slide 6.3 - 2
- 3. Parametric Curve, Parametric Equations The graph of the ordered pairs (x,y) where x = f(t) and y = g(t) are functions defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval. Copyright © 2011 Pearson, Inc. Slide 6.3 - 3
- 4. Example Graphing Parametric Equations For the given parametric interval, graph the parametric equations x t 2 2, y 3t. (a) 3 t 1 (b) 2 t 3 (c) 3 t 3 Copyright © 2011 Pearson, Inc. Slide 6.3 - 4
- 5. Example Graphing Parametric Equations For the given parametric interval, graph the parametric equations x t 2 2, y 3t. (a) 3 t 1 (b) 2 t 3 (c) 3 t 3 Copyright © 2011 Pearson, Inc. Slide 6.3 - 5
- 6. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x t 1, y 2t, t . Copyright © 2011 Pearson, Inc. Slide 6.3 - 6
- 7. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x t 1, y 2t, t . Solve one equation for t: x t 1 t x 1 Substitute t into the second equation: y 2t 2(x 1) y 2x 2 The graph of y 2x 2 is a line. Copyright © 2011 Pearson, Inc. Slide 6.3 - 7
- 8. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x 3cost, y 3sint, 0 t 2 . Copyright © 2011 Pearson, Inc. Slide 6.3 - 8
- 9. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x 3cost, y 3sint, 0 t 2 . x2 y2 9cos2 t 9sin2 t 9cos2 t sin2 t 9(1) The graph of x2 y2 9 is a circle with the center at (0,0) and a radius of 3. Copyright © 2011 Pearson, Inc. Slide 6.3 - 9
- 10. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x 5 t 2 , y 6t. Copyright © 2011 Pearson, Inc. Slide 6.3 - 10
- 11. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x 5 t 2 , y 6t. Solve the second equation for t: t y 6 Substitute that result into the first equation: x 5 t 2 x 5 y 6 2 x 5 y2 36 The graph of this equation is a parabola that opens to the left with vertex 5,0. 36x 180 y2 y2 180 36x y2 36x 5 Copyright © 2011 Pearson, Inc. Slide 6.3 - 11
- 12. Example Eliminating the Parameter Eliminate the parameter and identify the graph of the parametric curve x 5 t 2 , y 6t. Confirm Graphically This is consistent with the graph. Copyright © 2011 Pearson, Inc. Slide 6.3 - 12
- 13. Example Finding Parametric Equations for a Line Find a parametrization of the line through the points A (2,3) and B (3,6). Copyright © 2011 Pearson, Inc. Slide 6.3 - 13
- 14. Example Finding Parametric Equations for a Line Find a parametrization of the line through the points A (2,3) and B (3,6). Let P(x, y) be an arbitrary point on the line through A and B. Vector OP is the tail-to-head vector sum of OA and AP. AP is a scalar multiple of AB. Let the scalar be t and OP OA AP OP OA t AB x, y 2,3 t 3 2,6 3 x, y 2,3 t 5,3 x, y 2 5t,3 3t Copyright © 2011 Pearson, Inc. Slide 6.3 - 14
- 15. Quick Review 1. Find the component form of the vectors (a) OA, (b) OB, and (c) AB where O is the origin, A (3,2) and B (-4,-6). 2. Write an equation in point-slope form for the line through the points (3,2) and (-4,-6). 3. Find the two functions defined implicitly by y2 2x. 4. Find the equation for the circle with the center at (2,3) and a radius of 3. 5. A wheel with radius 12 in spins at the rate 400 rpm. Find the angular velocity in radians per second. Copyright © 2011 Pearson, Inc. Slide 6.3 - 15
- 16. Quick Review Solutions 1. Find the component form of the vectors (a) OA, (b) OB, and (c) AB where O is the origin, A (3,2) and B (4, 6). (a) 3,2 (b) 4, 6 (c) 7, 8 2. Write an equation in point-slope form for the line through the points (3,2) and ( 4, 6). y 2 8 7 (x 3) 3. Find the two functions defined implicitly by y2 2x. y 2x; y 2x Copyright © 2011 Pearson, Inc. Slide 6.3 - 16
- 17. Quick Review Solutions 4. Find the equation for the circle with the center at (2,3) and a radius of 3. x 22 y 32 9 5. A wheel with radius 12 in spins at the rate 400 rpm. Find the angular velocity in radians per second. 40 / 3 rad/sec Copyright © 2011 Pearson, Inc. Slide 6.3 - 17