Explanation Formula used: Area of the equilateral triangle, A = 1 2 × base × height . Calculation : Let the side length of the equilateral triangle be x. Obtain the height of the equilateral triangle as follows: Draw a line from any vertex to its opposite side so that it makes two equal right angled triangles. Consider any one of the right angled triangle. Since the side of the equilateral triangle is x , the base of the right angled triangle is x 2 . Let the height of the right angled triangle be h . Note that the height of the right angled triangle is the height of the equilateral triangle. Apply Pythagoras theorem and obtain the value of height. ( x 2 ) 2 + h 2 = x 2 x 2 4 + h 2 =

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 1.1, Problem 59E

To determine

To find: The function of the area of an equilateral triangle in terms of the length of the side and the function domain.

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Chapter 1.1, Problem 58E

Chapter 1.1, Problem 60E

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A factorization A = PDP 1 is not unique. For A= 7 2 -4 1 1 1 5 0 2 1 one factorization is P = D= and P-1 30 = Use this information with D₁ = to find a matrix P₁ such that - -1 -2 0 3 1 - - 1 05 A-P,D,P P1 (Type an integer or simplified fraction for each matrix element.)

Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 30 -1 - 1 0 -1 400 0 0 1 A= 3 4 3 0 1 3 040 3 1 3 0 0 4 1 0 0 003 -1 0 -1 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A basis for the corresponding eigenspace is { A. There is one distinct eigenvalue, λ = B. In ascending order, the two distinct eigenvalues are λ₁ ... = and 2 = Bases for the corresponding eigenspaces are { and ( ), respectively. C. In ascending order, the three distinct eigenvalues are λ₁ = = 12/2 = and 3 = Bases for the corresponding eigenspaces are {}, }, and { respectively.

Chapter 1 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 1.1 - 1. If f(x)=x+2x and g(u)=u+2u, is it true that f =...Ch. 1.1 - If f(x)=x2xx1andg(x)=x is it true that f = g?Ch. 1.1 - The graph of a function f is given. (a) State the...Ch. 1.1 - The graphs of f and g are given. (a) State the...Ch. 1.1 - Figure 1 was recorded by an instrument operated by...Ch. 1.1 - Prob. 7E Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E Ch. 1.1 - Prob. 11E

Ch. 1.1 - Prob. 12E Ch. 1.1 - Prob. 13E Ch. 1.1 - Prob. 14E Ch. 1.1 - The graph shows the power consumption for a day in...Ch. 1.1 - Prob. 16E Ch. 1.1 - Prob. 17E Ch. 1.1 - Prob. 18E Ch. 1.1 - Sketch the graph of the amount of a particular...Ch. 1.1 - You place a frozen pie in an oven and bake it for...Ch. 1.1 - Prob. 21E Ch. 1.1 - Prob. 22E Ch. 1.1 - Prob. 23E Ch. 1.1 - Prob. 24E Ch. 1.1 - Prob. 25E Ch. 1.1 - Prob. 26E Ch. 1.1 - Prob. 27E Ch. 1.1 - Prob. 28E Ch. 1.1 - Prob. 29E Ch. 1.1 - Prob. 30E Ch. 1.1 - Prob. 31E Ch. 1.1 - Find the domain of the function. 32....Ch. 1.1 - Prob. 33E Ch. 1.1 - Prob. 34E Ch. 1.1 - Prob. 35E Ch. 1.1 - Prob. 36E Ch. 1.1 - Prob. 37E Ch. 1.1 - Find the domain and range and sketch the graph of...Ch. 1.1 - Prob. 39E Ch. 1.1 - Prob. 40E Ch. 1.1 - Prob. 41E Ch. 1.1 - Prob. 42E Ch. 1.1 - Prob. 43E Ch. 1.1 - Evaluate f(3), f(0), and f(2) for the piecewise...Ch. 1.1 - Prob. 45E Ch. 1.1 - Prob. 46E Ch. 1.1 - Prob. 47E Ch. 1.1 - Prob. 48E Ch. 1.1 - Prob. 49E Ch. 1.1 - Sketch the graph of the function. 50. g(x) = ||x| ...Ch. 1.1 - Prob. 51E Ch. 1.1 - Prob. 52E Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Prob. 54E Ch. 1.1 - Prob. 55E Ch. 1.1 - Prob. 56E Ch. 1.1 - Prob. 57E Ch. 1.1 - Prob. 58E Ch. 1.1 - Prob. 59E Ch. 1.1 - Prob. 60E Ch. 1.1 - Prob. 61E Ch. 1.1 - A Norman window has the shape of a rectangle...Ch. 1.1 - Prob. 63E Ch. 1.1 - Prob. 64E Ch. 1.1 - Prob. 65E Ch. 1.1 - An electricity company charges its customers a...Ch. 1.1 - In a certain country, income tax is assessed as...Ch. 1.1 - Prob. 68E Ch. 1.1 - Prob. 69E Ch. 1.1 - Prob. 70E Ch. 1.1 - Prob. 71E Ch. 1.1 - Prob. 72E Ch. 1.1 - Prob. 73E Ch. 1.1 - Prob. 74E Ch. 1.1 - Prob. 75E Ch. 1.1 - Prob. 76E Ch. 1.1 - Prob. 77E Ch. 1.1 - Prob. 78E Ch. 1.1 - Prob. 79E Ch. 1.1 - Prob. 80E Ch. 1.2 - Classify each function as a power function, root...Ch. 1.2 - Prob. 2E Ch. 1.2 - Prob. 3E Ch. 1.2 - Prob. 4E Ch. 1.2 - Prob. 5E Ch. 1.2 - Prob. 6E Ch. 1.2 - Prob. 7E Ch. 1.2 - Prob. 8E Ch. 1.2 - Prob. 9E Ch. 1.2 - Prob. 10E Ch. 1.2 - Prob. 11E Ch. 1.2 - Prob. 12E Ch. 1.2 - Prob. 13E Ch. 1.2 - The manager of a weekend flea market knows from...Ch. 1.2 - Prob. 15E Ch. 1.2 - Prob. 16E Ch. 1.2 - Biologists have noticed that the chirping rate of...Ch. 1.2 - Prob. 18E Ch. 1.2 - Prob. 19E Ch. 1.2 - Prob. 20E Ch. 1.2 - Prob. 21E Ch. 1.2 - Prob. 22E Ch. 1.3 - Suppose the graph of f is given. Write equations...Ch. 1.3 - Prob. 2E Ch. 1.3 - The graph of y=f(x) is given. Match each equation...Ch. 1.3 - Prob. 4E Ch. 1.3 - Prob. 5E Ch. 1.3 - Prob. 6E Ch. 1.3 - Prob. 7E Ch. 1.3 - Prob. 8E Ch. 1.3 - Prob. 9E Ch. 1.3 - Prob. 10E Ch. 1.3 - Prob. 11E Ch. 1.3 - Prob. 12E Ch. 1.3 - Prob. 13E Ch. 1.3 - Prob. 14E Ch. 1.3 - Prob. 15E Ch. 1.3 - Prob. 16E Ch. 1.3 - Prob. 17E Ch. 1.3 - Prob. 18E Ch. 1.3 - Prob. 19E Ch. 1.3 - Prob. 20E Ch. 1.3 - Prob. 21E Ch. 1.3 - Prob. 22E Ch. 1.3 - Prob. 23E Ch. 1.3 - Prob. 24E Ch. 1.3 - The city of New Orleans is located at latitude...Ch. 1.3 - Prob. 26E Ch. 1.3 - Prob. 27E Ch. 1.3 - Prob. 28E Ch. 1.3 - Prob. 29E Ch. 1.3 - Prob. 30E Ch. 1.3 - Prob. 31E Ch. 1.3 - Prob. 32E Ch. 1.3 - Prob. 33E Ch. 1.3 - Prob. 34E Ch. 1.3 - Prob. 35E Ch. 1.3 - Prob. 36E Ch. 1.3 - Prob. 37E Ch. 1.3 - Prob. 38E Ch. 1.3 - Prob. 39E Ch. 1.3 - Prob. 40E Ch. 1.3 - Prob. 41E Ch. 1.3 - Prob. 42E Ch. 1.3 - Prob. 43E Ch. 1.3 - Prob. 44E Ch. 1.3 - Prob. 45E Ch. 1.3 - Prob. 46E Ch. 1.3 - Express the function in the form f g. 47. v(t) =...Ch. 1.3 - Prob. 48E Ch. 1.3 - Prob. 49E Ch. 1.3 - Prob. 50E Ch. 1.3 - Prob. 51E Ch. 1.3 - Prob. 52E Ch. 1.3 - Prob. 53E Ch. 1.3 - Use the given graphs of f and g to estimate the...Ch. 1.3 - Prob. 55E Ch. 1.3 - Prob. 56E Ch. 1.3 - A ship is at a speed of 30km/h parallel to a...Ch. 1.3 - Prob. 58E Ch. 1.3 - Prob. 59E Ch. 1.3 - The Heaviside function defined in Exercise 59 can...Ch. 1.3 - Prob. 61E Ch. 1.3 - Prob. 62E Ch. 1.3 - Prob. 63E Ch. 1.3 - Prob. 64E Ch. 1.3 - Prob. 65E Ch. 1.3 - Prob. 66E Ch. 1.4 - Use the Law of Exponents to rewrite and simplify...Ch. 1.4 - Use the Law of Exponents to rewrite and simplify...Ch. 1.4 - Prob. 3E Ch. 1.4 - Use the Law of Exponents to rewrite and simplify...Ch. 1.4 - Prob. 5E Ch. 1.4 - Prob. 6E Ch. 1.4 - Prob. 7E Ch. 1.4 - Prob. 8E Ch. 1.4 - Prob. 9E Ch. 1.4 - Prob. 10E Ch. 1.4 - Prob. 11E Ch. 1.4 - Prob. 12E Ch. 1.4 - Prob. 13E Ch. 1.4 - Prob. 14E Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Prob. 16E Ch. 1.4 - Prob. 17E Ch. 1.4 - Prob. 18E Ch. 1.4 - Prob. 19E Ch. 1.4 - Prob. 20E Ch. 1.4 - Prob. 21E Ch. 1.4 - Find the exponential function f(x) = Cbx whose...Ch. 1.4 - Prob. 23E Ch. 1.4 - Suppose you are offered a job that lasts one...Ch. 1.4 - Prob. 25E Ch. 1.4 - Compare the functions f(x) = x5and g(x) = 5x by...Ch. 1.4 - Compare the functions f(x) = x10 and g(x) = ex by...Ch. 1.4 - Prob. 28E Ch. 1.4 - A bacteria culture starts with 500 bacteria and...Ch. 1.4 - The half-life of bismuth-210, 210Bi, is 5 days....Ch. 1.4 - Prob. 32E Ch. 1.4 - Prob. 33E Ch. 1.4 - Prob. 34E Ch. 1.4 - Use a graphing calculator with exponential...Ch. 1.4 - Prob. 36E Ch. 1.4 - Prob. 37E Ch. 1.4 - Prob. 38E Ch. 1.5 - (a) What is a one-to-one function? (b) How can you...Ch. 1.5 - Prob. 2E Ch. 1.5 - Prob. 3E Ch. 1.5 - Prob. 4E Ch. 1.5 - A function is given by a table of values, a graph,...Ch. 1.5 - A function is given by a table of values, a graph,...Ch. 1.5 - Prob. 7E Ch. 1.5 - A function is given by a table of values, a graph,...Ch. 1.5 - Prob. 9E Ch. 1.5 - Prob. 10E Ch. 1.5 - Prob. 11E Ch. 1.5 - Prob. 12E Ch. 1.5 - Prob. 13E Ch. 1.5 - Prob. 14E Ch. 1.5 - Assume that f is a one-to-one function. (a) If...Ch. 1.5 - Prob. 16E Ch. 1.5 - Prob. 17E Ch. 1.5 - Prob. 18E Ch. 1.5 - Prob. 19E Ch. 1.5 - Prob. 20E Ch. 1.5 - Prob. 21E Ch. 1.5 - Prob. 22E Ch. 1.5 - Prob. 23E Ch. 1.5 - Prob. 24E Ch. 1.5 - Find a formula for the inverse of the function....Ch. 1.5 - Find a formula for the inverse of the function....Ch. 1.5 - Find an explicit formula for f1 and use it to...Ch. 1.5 - Prob. 28E Ch. 1.5 - Prob. 29E Ch. 1.5 - Prob. 30E Ch. 1.5 - Prob. 31E Ch. 1.5 - Prob. 32E Ch. 1.5 - Prob. 33E Ch. 1.5 - Prob. 34E Ch. 1.5 - Find the exact value of each expression. 35. (a)...Ch. 1.5 - Find the exact value of each expression. 35. (a)...Ch. 1.5 - Prob. 37E Ch. 1.5 - Find the exact value of each expression. 38. (a)...Ch. 1.5 - Prob. 39E Ch. 1.5 - Prob. 40E Ch. 1.5 - Express the given quantity as a single logarithm....Ch. 1.5 - Use Formula 10 to evaluate each logarithm correct...Ch. 1.5 - Prob. 43E Ch. 1.5 - Prob. 44E Ch. 1.5 - Prob. 45E Ch. 1.5 - Prob. 46E Ch. 1.5 - Prob. 47E Ch. 1.5 - Make a rough sketch of the graph of each function....Ch. 1.5 - (a) What are the domain and range of f? (b) What...Ch. 1.5 - (a) What are the domain and range of f? (b) What...Ch. 1.5 - Solve each equation for x. 51. (a) e74x=6 (b)...Ch. 1.5 - Prob. 52E Ch. 1.5 - Prob. 53E Ch. 1.5 - Prob. 54E Ch. 1.5 - Prob. 55E Ch. 1.5 - Prob. 56E Ch. 1.5 - Prob. 57E Ch. 1.5 - Prob. 58E Ch. 1.5 - Prob. 61E Ch. 1.5 - When a camera flash goes off, the batteries...Ch. 1.5 - Prob. 63E Ch. 1.5 - Prob. 64E Ch. 1.5 - Prob. 65E Ch. 1.5 - Prob. 66E Ch. 1.5 - Prob. 67E Ch. 1.5 - Prob. 68E Ch. 1.5 - Prob. 69E Ch. 1.5 - Simplify the expression. 70. tan(sin1 x)Ch. 1.5 - Prob. 71E Ch. 1.5 - Simplify the expression. 72. sin(2 arccos x)Ch. 1.5 - Prob. 73E Ch. 1.5 - Prob. 74E Ch. 1.5 - Prob. 75E Ch. 1.5 - Prob. 76E Ch. 1.5 - Prob. 77E Ch. 1 - (a) What is a function? What are its domain and...Ch. 1 - Discuss four ways of representing a function....Ch. 1 - (a) What is an even function? How can you tell if...Ch. 1 - Prob. 4RCC Ch. 1 - Prob. 5RCC Ch. 1 - Prob. 6RCC Ch. 1 - Prob. 7RCC Ch. 1 - Prob. 8RCC Ch. 1 - Suppose that f has domain A and g has domain B....Ch. 1 - Prob. 10RCC Ch. 1 - Prob. 11RCC Ch. 1 - Prob. 12RCC Ch. 1 - Prob. 13RCC Ch. 1 - Prob. 1RQ Ch. 1 - Prob. 2RQ Ch. 1 - Prob. 3RQ Ch. 1 - Prob. 4RQ Ch. 1 - Prob. 5RQ Ch. 1 - Prob. 6RQ Ch. 1 - Prob. 7RQ Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 10RQ Ch. 1 - Prob. 11RQ Ch. 1 - Prob. 12RQ Ch. 1 - Prob. 13RQ Ch. 1 - Prob. 14RQ Ch. 1 - Let f be the function whose graph is given. (a)...Ch. 1 - Prob. 2RE Ch. 1 - Prob. 3RE Ch. 1 - Prob. 4RE Ch. 1 - Prob. 5RE Ch. 1 - Prob. 6RE Ch. 1 - Prob. 7RE Ch. 1 - Prob. 8RE Ch. 1 - Suppose that the graph of .f is given. Describe...Ch. 1 - Prob. 10RE Ch. 1 - Prob. 11RE Ch. 1 - Prob. 12RE Ch. 1 - Prob. 13RE Ch. 1 - Prob. 14RE Ch. 1 - Prob. 15RE Ch. 1 - Prob. 16RE Ch. 1 - Prob. 17RE Ch. 1 - Prob. 18RE Ch. 1 - Prob. 19RE Ch. 1 - Prob. 20RE Ch. 1 - Prob. 22RE Ch. 1 - Prob. 23RE Ch. 1 - Prob. 24RE Ch. 1 - Prob. 25RE Ch. 1 - Prob. 26RE Ch. 1 - The half-life of palladium-100, 100Pd, is four...Ch. 1 - The population of a certain species in a limited...Ch. 1 - One of the legs of a right triangle has length 4...Ch. 1 - Prob. 2P Ch. 1 - Prob. 3P Ch. 1 - Prob. 4P Ch. 1 - Prob. 5P Ch. 1 - Prob. 6P Ch. 1 - Prob. 7P Ch. 1 - Prob. 8P Ch. 1 - The notation max{a, b, } means the largest of the...Ch. 1 - Prob. 10P Ch. 1 - Prob. 11P Ch. 1 - Prob. 12P Ch. 1 - Prob. 13P Ch. 1 - Prob. 14P Ch. 1 - Prob. 15P Ch. 1 - Prob. 16P Ch. 1 - Prob. 17P Ch. 1 - Prove that 1 + 3 + 5 + + (2n l ) = n2.Ch. 1 - Prob. 19P Ch. 1 - Prob. 20P

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