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Math Calculus Calculus

PUTNAM EXAM CHALLENGE For i = 1,2, let T i be a triangle with side lengths a i , b i , c i , and area A i , Suppose that a 1 ≤ a 2 , b 1 ≤ b 2 , c 1 ≤ c 2 , and that T 2 is an acute triangle. Does it follow that A 1 ≤ A 2 ? This problem was composed by the Committee on the Putnam Prize Competition. © The Mathematical Association of America. All rights reserved.

PUTNAM EXAM CHALLENGE For i = 1,2, let T i be a triangle with side lengths a i , b i , c i , and area A i , Suppose that a 1 ≤ a 2 , b 1 ≤ b 2 , c 1 ≤ c 2 , and that T 2 is an acute triangle. Does it follow that A 1 ≤ A 2 ? This problem was composed by the Committee on the Putnam Prize Competition. © The Mathematical Association of America. All rights reserved.

Solution Summary: The author explains that if T_i be a triangle with side lengths, then the area is less than or equal to area.

Calculus

10th Edition

ISBN: 9781285948133

Author: Ron Larson; Bruce H. Edwards

Publisher: Cengage Learning US

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Textbook Question

Chapter P.4, Problem 17E

PUTNAM EXAM CHALLENGE

For i = 1,2, let T_i be a triangle with side lengths a_i, b_i, c_i, and area A_i, Suppose that a₁ ≤ a₂, b₁ ≤ b₂, c₁ ≤ c₂, and that T₂ is an acute triangle. Does it follow that A₁ ≤ A₂?

This problem was composed by the Committee on the Putnam Prize Competition.

© The Mathematical Association of America. All rights reserved.

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Chapter P.4, Problem 16E

Chapter P, Problem 1RE

Chapter P Solutions

Calculus

Ch. P.1 - Matching In Exercises 3-6, match the equation with...Ch. P.1 - Matching In Exercises 3-6, match the equation with...Ch. P.1 - Matching In Exercises 3-6, match the equation with...Ch. P.1 - Matching In Exercises 3-6, match the equation with...Ch. P.1 - Prob. 5E Ch. P.1 - Sketching a Graph by Point Plotting In Exercises...Ch. P.1 - Sketching a Graph by Point Plotting In Exercises...Ch. P.1 - Prob. 8E Ch. P.1 - Sketching a Graph by Point Plotting In Exercises...Ch. P.1 - Sketching a Graph by Point Plotting In Exercises...

Ch. P.1 - Sketching a Graph by Point Plotting In Exercises...Ch. P.1 - Prob. 12E Ch. P.1 - Prob. 13E Ch. P.1 - Prob. 14E Ch. P.1 - Approximating Solution Points Using Technology In...Ch. P.1 - Approximating Solution Points Using Technology In...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Prob. 20E Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Finding Intercepts In Exercises 19-28, find any...Ch. P.1 - Prob. 27E Ch. P.1 - Prob. 28E Ch. P.1 - Prob. 29E Ch. P.1 - Prob. 30E Ch. P.1 - Prob. 31E Ch. P.1 - Prob. 32E Ch. P.1 - Prob. 33E Ch. P.1 - Prob. 34E Ch. P.1 - Prob. 35E Ch. P.1 - Prob. 36E Ch. P.1 - Prob. 37E Ch. P.1 - Prob. 38E Ch. P.1 - Prob. 39E Ch. P.1 - Prob. 40E Ch. P.1 - Prob. 41E Ch. P.1 - Prob. 42E Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Prob. 44E Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Prob. 46E Ch. P.1 - Prob. 47E Ch. P.1 - Prob. 48E Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Prob. 51E Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Prob. 53E Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Using Intercepts and Symmetry to Sketch a Graph In...Ch. P.1 - Prob. 56E Ch. P.1 - Finding Points of Intersection In Exercises 57-62....Ch. P.1 - Finding Points of Intersection In Exercises 57-62....Ch. P.1 - Prob. 59E Ch. P.1 - Finding Points of Intersection In Exercises 57-62,...Ch. P.1 - Prob. 61E Ch. P.1 - Finding Points of Intersection In Exercises 57-62....Ch. P.1 - Finding Points of Intersection Using Technology In...Ch. P.1 - Prob. 64E Ch. P.1 - Prob. 65E Ch. P.1 - Finding Points of Intersection Using Technology In...Ch. P.1 - Modeling Data The table shows the Gross Domestic...Ch. P.1 - Modeling Data The table shows the numbers of...Ch. P.1 - Break-Even Point Find the sales necessary to break...Ch. P.1 - Copper Wire The resistance y in ohms of 1000 feet...Ch. P.1 - Using Solution Points For what values of k does...Ch. P.1 - Using Solution Points For what values of k does...Ch. P.1 - WRITING ABOUT CONCEPTS Writing Equations In...Ch. P.1 - EXPLORING CONCEPTS Using Intercepts Write an...Ch. P.1 - Prob. 75E Ch. P.1 - HOW DO YOU SEE IT? Use the graphs of the two...Ch. P.1 - True or False ? In Exercises 75-78, determine...Ch. P.1 - Prob. 78E Ch. P.1 - True or False? In Exercises 75-78, determine...Ch. P.1 - True or False? In Exercises 75-78, determine...Ch. P.2 - Estimating Slope In Exercises 36, estimate the...Ch. P.2 - Prob. 2E Ch. P.2 - Prob. 3E Ch. P.2 - Prob. 4E Ch. P.2 - Prob. 5E Ch. P.2 - Prob. 6E Ch. P.2 - Finding the Slope of a Line In Exercises 7-12,...Ch. P.2 - Finding the Slope of a Line In Exercises 7-12,...Ch. P.2 - Finding the Slope of a Line In Exercises 7-12,...Ch. P.2 - Finding the Slope of a Line In Exercises 7-12,...Ch. P.2 - Sketching Lines In Exercises 13 and 14. sketch the...Ch. P.2 - Sketching Lines In Exercises 13 and 14, sketch the...Ch. P.2 - Prob. 13E Ch. P.2 - Finding Points on a Line In Exercises 1518, use...Ch. P.2 - Prob. 15E Ch. P.2 - Finding Points on a Line In Exercises 1518, use...Ch. P.2 - Finding an Equation of a Line In Exercises 19-24,...Ch. P.2 - Prob. 18E Ch. P.2 - Prob. 19E Ch. P.2 - Prob. 20E Ch. P.2 - Prob. 21E Ch. P.2 - Prob. 22E Ch. P.2 - Prob. 23E Ch. P.2 - Modeling Data The table shows the populations y...Ch. P.2 - Prob. 25E Ch. P.2 - Finding the Slope and y-Intercept In Exercises...Ch. P.2 - Prob. 27E Ch. P.2 - Prob. 28E Ch. P.2 - Prob. 29E Ch. P.2 - Prob. 30E Ch. P.2 - Sketching a Line in the Plane In Exercises 35-42,...Ch. P.2 - Prob. 32E Ch. P.2 - Prob. 33E Ch. P.2 - Prob. 34E Ch. P.2 - Prob. 35E Ch. P.2 - Prob. 36E Ch. P.2 - Prob. 37E Ch. P.2 - Prob. 38E Ch. P.2 - Prob. 39E Ch. P.2 - Prob. 40E Ch. P.2 - Prob. 41E Ch. P.2 - Prob. 42E Ch. P.2 - Prob. 43E Ch. P.2 - Prob. 44E Ch. P.2 - Prob. 45E Ch. P.2 - Finding an Equation of a Line In Exercises 3946,...Ch. P.2 - Find an equation of the vertical line with...Ch. P.2 - Prob. 48E Ch. P.2 - Writing an Equation in General Form In Exercises...Ch. P.2 - Prob. 50E Ch. P.2 - Prob. 51E Ch. P.2 - Prob. 52E Ch. P.2 - Writing an Equation in General Form In Exercises...Ch. P.2 - Prob. 54E Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Finding Parallel and Perpendicular Lines In...Ch. P.2 - Prob. 62E Ch. P.2 - Prob. 63E Ch. P.2 - Prob. 64E Ch. P.2 - Prob. 65E Ch. P.2 - Prob. 66E Ch. P.2 - Prob. 67E Ch. P.2 - Prob. 68E Ch. P.2 - Prob. 69E Ch. P.2 - Prob. 70E Ch. P.2 - Prob. 71E Ch. P.2 - Prob. 72E Ch. P.2 - Analyzing a Line A line is represented by the...Ch. P.2 - Tangent Line Find an equation of the line tangent...Ch. P.2 - Prob. 82E Ch. P.2 - Prob. 74E Ch. P.2 - Prob. 75E Ch. P.2 - Reimbursed Expenses A company reimburses its sales...Ch. P.2 - Prob. 77E Ch. P.2 - Straight-Line Depreciation A small business...Ch. P.2 - Apartment Rental A real estate office manages an...Ch. P.2 - Prob. 80E Ch. P.2 - Prob. 83E Ch. P.2 - Prob. 84E Ch. P.2 - Prob. 85E Ch. P.2 - Prob. 86E Ch. P.2 - Prob. 87E Ch. P.2 - Prob. 88E Ch. P.2 - Prob. 89E Ch. P.2 - Prob. 90E Ch. P.2 - Prob. 91E Ch. P.2 - Prob. 92E Ch. P.2 - Prob. 93E Ch. P.2 - Prob. 94E Ch. P.2 - True or False? In Exercises 85 and 86, determine...Ch. P.2 - Prob. 96E Ch. P.3 - Evaluating a Function In Exercises 110, evaluate...Ch. P.3 - Prob. 2E Ch. P.3 - Prob. 3E Ch. P.3 - Prob. 4E Ch. P.3 - Evaluating a Function In Exercises 1-10, evaluate...Ch. P.3 - Prob. 6E Ch. P.3 - Prob. 7E Ch. P.3 - Prob. 8E Ch. P.3 - Prob. 9E Ch. P.3 - Prob. 10E Ch. P.3 - Prob. 11E Ch. P.3 - Prob. 12E Ch. P.3 - Prob. 13E Ch. P.3 - Finding the Domain and Range of a Function In...Ch. P.3 - Prob. 15E Ch. P.3 - Finding the Domain and Range of a Function In...Ch. P.3 - Finding the Domain and Range of a Function In...Ch. P.3 - Prob. 18E Ch. P.3 - Prob. 19E Ch. P.3 - Finding the Domain and Range of a Function In...Ch. P.3 - Prob. 21E Ch. P.3 - Prob. 22E Ch. P.3 - Prob. 23E Ch. P.3 - Prob. 24E Ch. P.3 - Prob. 25E Ch. P.3 - Prob. 26E Ch. P.3 - Prob. 27E Ch. P.3 - Prob. 28E Ch. P.3 - Prob. 29E Ch. P.3 - Prob. 30E Ch. P.3 - Prob. 31E Ch. P.3 - Prob. 32E Ch. P.3 - Prob. 33E Ch. P.3 - Prob. 34E Ch. P.3 - Prob. 35E Ch. P.3 - Prob. 36E Ch. P.3 - Prob. 37E Ch. P.3 - Prob. 38E Ch. P.3 - Prob. 39E Ch. P.3 - Prob. 40E Ch. P.3 - Prob. 41E Ch. P.3 - Sketching a Graph A student who commutes 27 miles...Ch. P.3 - Prob. 43E Ch. P.3 - Using the Vertical Line Test In Exercises 39-42,...Ch. P.3 - Prob. 45E Ch. P.3 - Prob. 46E Ch. P.3 - Prob. 47E Ch. P.3 - Prob. 48E Ch. P.3 - Prob. 49E Ch. P.3 - Prob. 50E Ch. P.3 - Prob. 51E Ch. P.3 - Prob. 52E Ch. P.3 - Prob. 53E Ch. P.3 - Prob. 54E Ch. P.3 - Prob. 55E Ch. P.3 - Matching In Exercises 51-56, use the graph of...Ch. P.3 - Prob. 57E Ch. P.3 - Prob. 58E Ch. P.3 - Prob. 59E Ch. P.3 - Prob. 60E Ch. P.3 - Sketching Transformations Use the graph of f shown...Ch. P.3 - Sketching Transformations Use the graph of f shown...Ch. P.3 - Prob. 63E Ch. P.3 - Prob. 64E Ch. P.3 - Prob. 65E Ch. P.3 - Prob. 66E Ch. P.3 - Finding Composite Functions In Exercises 63-66,...Ch. P.3 - Prob. 68E Ch. P.3 - Prob. 69E Ch. P.3 - Prob. 70E Ch. P.3 - Evaluating Composite Functions Use the graphs of f...Ch. P.3 - Ripples A pebble is dropped into a calm pond,...Ch. P.3 - Prob. 73E Ch. P.3 - Prob. 74E Ch. P.3 - Think About It In Exercises 71 and 72, find the...Ch. P.3 - Prob. 76E Ch. P.3 - Ever, and Odd Functions The graphs of f, g, and h...Ch. P.3 - Prob. 78E Ch. P.3 - Prob. 79E Ch. P.3 - Prob. 80E Ch. P.3 - Prob. 81E Ch. P.3 - Prob. 82E Ch. P.3 - Prob. 83E Ch. P.3 - Prob. 84E Ch. P.3 - Prob. 85E Ch. P.3 - Prob. 86E Ch. P.3 - Prob. 87E Ch. P.3 - Prob. 88E Ch. P.3 - Prob. 89E Ch. P.3 - Prob. 90E Ch. P.3 - Domain Find the value of c such that the domain of...Ch. P.3 - Domain Find all values of c such that the domain...Ch. P.3 - Graphical Reasoning An electronically controlled...Ch. P.3 - HOW DO YOU SEE IT? Water runs into a vase of...Ch. P.3 - Prob. 96E Ch. P.3 - Prob. 95E Ch. P.3 - Prob. 97E Ch. P.3 - Prob. 98E Ch. P.3 - Proof Prove that the function is odd...Ch. P.3 - Proof Prove that the function is even....Ch. P.3 - Prob. 101E Ch. P.3 - Prob. 102E Ch. P.3 - Length A right triangle is formed in the first...Ch. P.3 - Volume An open box of maximum volume is to be made...Ch. P.3 - Prob. 105E Ch. P.3 - Prob. 106E Ch. P.3 - Prob. 107E Ch. P.3 - Prob. 108E Ch. P.3 - Prob. 109E Ch. P.3 - Prob. 110E Ch. P.3 - Prob. 111E Ch. P.3 - Prob. 112E Ch. P.4 - Prob. 1E Ch. P.4 - Prob. 2E Ch. P.4 - Hooke's Law Hookes Law states that the force F...Ch. P.4 - Prob. 4E Ch. P.4 - Prob. 5E Ch. P.4 - Prob. 6E Ch. P.4 - Beam Strength Students in a lab measured the...Ch. P.4 - Car Performance The time t (in seconds) required...Ch. P.4 - Engine Performance A V8 car engine is coupled to a...Ch. P.4 - Boiling Temperature The table shows the...Ch. P.4 - Prob. 11E Ch. P.4 - Prob. 12E Ch. P.4 - Harmonic Motion The motion of an oscillating...Ch. P.4 - Temperature The table shows the normal daily high...Ch. P.4 - Prob. 15E Ch. P.4 - Prob. 16E Ch. P.4 - PUTNAM EXAM CHALLENGE For i = 1,2, let Ti be a...Ch. P - Finding Intercepts In Exercises 1-4, find any...Ch. P - Finding Intercepts In Exercises 1-4, find any...Ch. P - Finding Intercepts In Exercises 1-4, find any...Ch. P - Prob. 4RE Ch. P - Prob. 5RE Ch. P - Prob. 6RE Ch. P - Prob. 7RE Ch. P - Prob. 8RE Ch. P - Prob. 9RE Ch. P - Prob. 10RE Ch. P - Prob. 11RE Ch. P - Prob. 12RE Ch. P - Prob. 13RE Ch. P - Prob. 14RE Ch. P - Prob. 15RE Ch. P - Prob. 16RE Ch. P - Prob. 17RE Ch. P - Prob. 18RE Ch. P - Prob. 19RE Ch. P - Prob. 20RE Ch. P - Prob. 21RE Ch. P - Prob. 22RE Ch. P - Prob. 23RE Ch. P - Prob. 24RE Ch. P - Prob. 25RE Ch. P - Prob. 26RE Ch. P - Prob. 27RE Ch. P - Sketching a Line in the Plane In Exercises 27-30,...Ch. P - Prob. 29RE Ch. P - Prob. 30RE Ch. P - Finding Equations of Lines Find equations of the...Ch. P - Prob. 32RE Ch. P - Rate of Change The purchase price of a new machine...Ch. P - Break-Even Analysis A contractor purchases a piece...Ch. P - Prob. 35RE Ch. P - Prob. 36RE Ch. P - Evaluating a Function In Exercises 37-40, evaluate...Ch. P - Prob. 38RE Ch. P - Prob. 39RE Ch. P - Prob. 40RE Ch. P - Prob. 41RE Ch. P - Prob. 42RE Ch. P - Prob. 43RE Ch. P - Prob. 44RE Ch. P - Prob. 45RE Ch. P - Prob. 46RE Ch. P - Transformations of Functions Use a graphing...Ch. P - Conjecture (a) Use a graphing utility to graph the...Ch. P - Prob. 49RE Ch. P - Think About It What is the minimum degree of the...Ch. P - Prob. 51RE Ch. P - Median Income The data in the table show the...Ch. P - Prob. 53RE Ch. P - Prob. 1PS Ch. P - Finding Tangent Lines There are two tangent lines...Ch. P - Heaviside Function The Heaviside function H(x) is...Ch. P - Sketching Transformations Consider the graph of...Ch. P - Prob. 5PS Ch. P - Prob. 6PS Ch. P - Prob. 7PS Ch. P - Prob. 8PS Ch. P - Slope of a Tangent Line One of the fundamental...Ch. P - Slope of a Tangent Line Sketch the graph of the...Ch. P - Prob. 11PS Ch. P - Graphing an Equation Explain how you would graph...Ch. P - Sound Intensity A large room contains two speakers...Ch. P - Sound Intensity Suppose the speakers in Exercise...Ch. P - Lemniscate Let d1 and d2 be the distances from the...

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