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Math Calculus James Stewart Calculus for MAT 127/128/229 8th edition

(a) If f o ( x ) = 1 2 − x and f n + 1 = f o ∘ f n for n = 0 , 1 , 2 , … , find an expression for f n ( x ) and use mathematical induction to prove it. (b) Graph f 0 , f 1 , f 2 , f 3 on the same screen and describe the effects of repeated composition.

(a) If f o ( x ) = 1 2 − x and f n + 1 = f o ∘ f n for n = 0 , 1 , 2 , … , find an expression for f n ( x ) and use mathematical induction to prove it. (b) Graph f 0 , f 1 , f 2 , f 3 on the same screen and describe the effects of repeated composition.

Solution Summary: The author explains the inductive step of n=k+1 to prove the statement is true.

James Stewart Calculus for MAT 127/128/229 8th edition

James Stewart Calculus for MAT 127/128/229 8th edition

8th Edition

ISBN: 9781305743663

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 1, Problem 20P

(a) If f o ( x ) = 1 2 − x and f n + 1 = f o ∘ f n for n = 0 , 1 , 2 , … , find an expression for f_n(x) and use mathematical induction to prove it.

(b) Graph f₀, f₁, f₂, f₃ on the same screen and describe the effects of repeated composition.

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Chapter 1, Problem 19P

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Chapter 1 Solutions

James Stewart Calculus for MAT 127/128/229 8th edition

Ch. 1.1 - If f(x)=x+2x and g(u)=u+2u, is it true that f = g?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 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Shown is a graph of the global average temperature...

Ch. 1.1 - Trees grow faster and form wider rings in warm...Ch. 1.1 - You put some ice cubes in a glass, fill the glass...Ch. 1.1 - Three runners compete in a 100-meter race. The...Ch. 1.1 - The graph shows the power consumption for a day in...Ch. 1.1 - Sketch a rough graph of the number of hours of...Ch. 1.1 - Sketch a rough graph of the outdoor temperature as...Ch. 1.1 - Sketch a rough graph of the market value of a new...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 - A homeowner mows the lawn every Wednesday...Ch. 1.1 - An airplane takes off from an airport and lands an...Ch. 1.1 - Temperature readings T (in F) were recorded every...Ch. 1.1 - Researchers measured the blood alcohol...Ch. 1.1 - If f(x) = 3x2 x + 2, find f(2), f(2), f(a), f(a),...Ch. 1.1 - A spherical balloon with radius r inches has...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Find the domain of the function. 31. f(x)=x+4x29 Ch. 1.1 - Find the domain of the function. 32....Ch. 1.1 - Find the domain of the function. 33. f(t)=2t13 Ch. 1.1 - Find the domain of the function. 34. g(t)=3t2+t Ch. 1.1 - Find the domain of the function. 35. h(x)=1x25x4 Ch. 1.1 - Find the domain of the function. 36....Ch. 1.1 - Find the domain of the function. 37. F(p)=2p Ch. 1.1 - Find the domain and range and sketch the graph of...Ch. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Evaluate f(3), f(0), and f(2) for the piecewise...Ch. 1.1 - Evaluate f(3), f(0), and f(2) for the piecewise...Ch. 1.1 - Evaluate f(3), f(0), and f(2) for the piecewise...Ch. 1.1 - Evaluate f(3), f(0), and f(2) for the piecewise...Ch. 1.1 - Sketch the graph of the function. 45. f(x) = x +...Ch. 1.1 - Sketch the graph of the function. 46. f(x) = |x +...Ch. 1.1 - Sketch the graph of the function. 47. g(t) = |1 ...Ch. 1.1 - Sketch the graph of the function. 48. h(t) = |t| +...Ch. 1.1 - Sketch the graph of the function. 49....Ch. 1.1 - Sketch the graph of the function. 50. g(x) = ||x| ...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - A Norman window has the shape of a rectangle...Ch. 1.1 - A box with an open top is to be constructed from a...Ch. 1.1 - A cell phone plan has a basic charge of 35 a...Ch. 1.1 - In a certain state the maximum speed permitted on...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 - The functions in Example 10 and Exercise 67 are...Ch. 1.1 - Graphs of f and g are shown. Decide whether each...Ch. 1.1 - Graphs of f and g are shown. Decide whether each...Ch. 1.1 - (a) If the point (5, 3) is on the graph of an even...Ch. 1.1 - A function f has domain [5, 5] and a portion of...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - If f and g are both even functions, is f + g even?...Ch. 1.1 - If f and g are both even functions, is the product...Ch. 1.2 - Classify each function as a power function, root...Ch. 1.2 - Classify each function as a power function, root...Ch. 1.2 - Match each equation with its graph. Explain your...Ch. 1.2 - Match each equation with its graph. Explain your...Ch. 1.2 - Find the domain of the function. 5. f(x)=cosx1sinx Ch. 1.2 - Find the domain of the function. 6. g(x)=11tanx Ch. 1.2 - (a) Find an equation for the family of linear...Ch. 1.2 - What do all members of the family of linear...Ch. 1.2 - What do all members of the family of linear...Ch. 1.2 - Find expressions for the quadratic functions whose...Ch. 1.2 - Find an expression for a cubic function f if...Ch. 1.2 - Recent studies indicate that the average surface...Ch. 1.2 - If the recommended adult dosage for a drug is D...Ch. 1.2 - The manager of a weekend flea market knows from...Ch. 1.2 - The relationship between the Fahrenheit (F) and...Ch. 1.2 - Jason leaves Detroit at 2:00 PM and drives at a...Ch. 1.2 - Biologists have noticed that the chirping rate of...Ch. 1.2 - The manager of a furniture factory finds that it...Ch. 1.2 - At the surface of the ocean, the water pressure is...Ch. 1.2 - The monthly cost of driving a car depends on the...Ch. 1.2 - For each scatter plot, decide what type of...Ch. 1.2 - For each scatter plot, decide what type of...Ch. 1.3 - Suppose the graph of f is given. Write equations...Ch. 1.3 - Explain how each graph is obtained from the graph...Ch. 1.3 - The graph of y=f(x) is given. Match each equation...Ch. 1.3 - The graph of f is given. Draw the graphs of the...Ch. 1.3 - The graph of f is given. Use it to graph the...Ch. 1.3 - The graph of y=3xx2 is given. Use transformations...Ch. 1.3 - The graph of y=3xx2 is given. Use transformations...Ch. 1.3 - (a) How is the graph of y = 2 sin x related to the...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Prob. 10E Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Prob. 18E Ch. 1.3 - Graph the function by hand, not by plotting...Ch. 1.3 - Graph the function by hand, not by plotting...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 - A variable star is one whose brightness...Ch. 1.3 - Some of the highest tides in the world occur in...Ch. 1.3 - In a normal respiratory cycle the volume of air...Ch. 1.3 - (a) How is the graph of y=f(|x|)related to the...Ch. 1.3 - Use the given graph of f to sketch the graph of y...Ch. 1.3 - Prob. 31E Ch. 1.3 - Find (a) f + g, (b) f g, (c) fg, and (d) f/g and...Ch. 1.3 - Find the functions (a) fg, (b) gf, (c) ff, and (d)...Ch. 1.3 - Find the functions (a) fg, (b) gf, (c) ff, and (d)...Ch. 1.3 - Find the functions (a) fg, (b) gf, (c) ff, and (d)...Ch. 1.3 - Prob. 36E Ch. 1.3 - Find the functions (a) f g, (b) g f, (c) f f,...Ch. 1.3 - Find the functions (a) f g, (b) g f, (c) f f,...Ch. 1.3 - Find f g h. 39. f(x) = 3x 2, g (x) = sin x,...Ch. 1.3 - Prob. 40E Ch. 1.3 - Find f g h. 41.f(x)=x3, g(x) = x2, h(x) = x3 + 2 Ch. 1.3 - Find f g h. 42. f(x) = tan x, g(x)=xx1,h(x)=x3 Ch. 1.3 - Express the function in the form f g. 43. F(x) =...Ch. 1.3 - Express the function in the form f g. 44. F(x) =...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 - Use the table to evaluate each expression. (a)...Ch. 1.3 - Use the given graphs of f and g to evaluate each...Ch. 1.3 - Use the given graphs of f and g to estimate the...Ch. 1.3 - A stone is dropped into a lake, creating a...Ch. 1.3 - A spherical balloon is being inflated and the...Ch. 1.3 - Prob. 57E Ch. 1.3 - An airplane is flying at a speed of 350 mi/h at an...Ch. 1.3 - The Heaviside function H is defined by It is used...Ch. 1.3 - The Heaviside function defined in Exercise 59 can...Ch. 1.3 - Let f and g be linear functions with equations...Ch. 1.3 - Prob. 62E Ch. 1.3 - (a) If g(x) = 2x + 1 and h(x) = 4x2 + 4x + 7, find...Ch. 1.3 - If f(x) = x + 4 and h(x) = 4x 1, find a function...Ch. 1.3 - Suppose g is an even function and let h =f g. Is...Ch. 1.3 - Suppose g is an odd function and let h = f g. Is...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 - Use the Law of Exponents to rewrite and simplify...Ch. 1.4 - Use the Law of Exponents to rewrite and simplify...Ch. 1.4 - (a) Write an equation that defines the exponential...Ch. 1.4 - (a) How is the number e defined? (b) What is an...Ch. 1.4 - Graph the given functions on a common screen. How...Ch. 1.4 - Graph the given functions on a common screen. How...Ch. 1.4 - Graph the given functions on a common screen. How...Ch. 1.4 - Graph the given functions on a common screen. How...Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Make a rough sketch of the graph of the function....Ch. 1.4 - Starting with the graph of y = ex, write the...Ch. 1.4 - Starting with the graph of y = ex, find the...Ch. 1.4 - Find the domain of each function. 19. (a)...Ch. 1.4 - Find the domain of each function. 20. (a)...Ch. 1.4 - Find the exponential function f(x) = Cb2 whose...Ch. 1.4 - Find the exponential function f(x) = Cbx whose...Ch. 1.4 - If f(x) = 5x, show that f(x+h)f(x)h=5x(5h1h)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 - Use a graph to estimate the values of x such that...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 - An isotope of sodium, 24Na, has a half-life of 15...Ch. 1.4 - Prob. 33E Ch. 1.4 - After alcohol is fully absorbed into the body, it...Ch. 1.4 - Use a graphing calculator with exponential...Ch. 1.4 - The table gives the population of the United...Ch. 1.4 - If you graph the function f(x)=1e1/x1+e1/x you' ll...Ch. 1.4 - Graph several members of the family of functions...Ch. 1.5 - (a) What is a one-to-one function? (b) How can you...Ch. 1.5 - (a) Suppose f is a one-to-one function with domain...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 - 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 - 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 - 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 - 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 - 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 - Assume that f is a one-to-one function. (a) If...Ch. 1.5 - If f(x) = x5 + x3 + x, find f1(3) and f(f 1(2)).Ch. 1.5 - If g(x) = 3 + x + ex, find g1(4).Ch. 1.5 - The graph of f is given. (a) Why is f one-to-one?...Ch. 1.5 - The formula C=59(F32), where F 459.67, expresses...Ch. 1.5 - In the theory of relativity, the mass of a...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 a formula for the inverse of the function....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 a formula for the inverse of the function....Ch. 1.5 - Find an explicit formula for f1 and use it to...Ch. 1.5 - Find an explicit formula for f1 and use it to...Ch. 1.5 - Prob. 29E Ch. 1.5 - Use the given graph of f to sketch the graph of...Ch. 1.5 - Let f(x)=1x2,0x1. (a) Find f1. How is it related...Ch. 1.5 - Prob. 32E Ch. 1.5 - Prob. 33E Ch. 1.5 - (a) What is the natural logarithm? (b) What is the...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 - Find the exact value of each expression. 37. (a)...Ch. 1.5 - Find the exact value of each expression. 38. (a)...Ch. 1.5 - Express the given quantity as a single logarithm....Ch. 1.5 - Express the given quantity as a single logarithm....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 - Use Formula 10 to graph the given functions on a...Ch. 1.5 - Use Formula 10 to graph the given functions on a...Ch. 1.5 - Suppose that the graph of y = log2 x is drawn on a...Ch. 1.5 - Prob. 46E Ch. 1.5 - Prob. 47E Ch. 1.5 - Prob. 48E 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 - Solve each equation for x. 52. (a) ln(x2 1) = 3...Ch. 1.5 - Solve each equation for x. 53. (a) 2x5 = 3 (b) ln...Ch. 1.5 - Solve each equation for x. 54. (a) ln(ln x) = 1...Ch. 1.5 - Solve each inequality for x. 55. (a) ln x 0 (b)...Ch. 1.5 - Solve each inequality for x. 56. (a) 1 e3x1 2...Ch. 1.5 - (a) Find the domain of f(x) = ln(ex 3). (b) Find...Ch. 1.5 - (a) What are the values of eln 300 and ln(e300)?...Ch. 1.5 - If a bacteria population starts with 100 bacteria...Ch. 1.5 - When a camera flash goes off, the batteries...Ch. 1.5 - Find the exact value of each expression. 63. (a)...Ch. 1.5 - Find the exact value of each expression. 64. (a)...Ch. 1.5 - Prob. 65E Ch. 1.5 - Find the exact value of each expression. 66. (a)...Ch. 1.5 - Find the exact value of each expression. 67. (a)...Ch. 1.5 - Find the exact value of each expression. 68. (a)...Ch. 1.5 - Prove that cos(sin1x)=1x2.Ch. 1.5 - Simplify the expression. 70. tan(sin1 x)Ch. 1.5 - Simplify the expression. 71. sin(tan1 x)Ch. 1.5 - Simplify the expression. 72. sin(2 arccos x)Ch. 1.5 - Prob. 73E Ch. 1.5 - Prob. 74E Ch. 1.5 - Find the domain and range of the function g(x) =...Ch. 1.5 - (a) Graph the function f(x) = sin(sin1 x) and...Ch. 1.5 - (a) If we shift a curve to the left, what happens...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 - What is an increasing function?Ch. 1 - What is a mathematical model?Ch. 1 - Give an example of each type of function. (a)...Ch. 1 - Sketch by hand, on the same axes, the graphs of...Ch. 1 - Draw, by hand, a rough sketch of the graph of each...Ch. 1 - Suppose that f has domain A and g has domain B....Ch. 1 - How is the composite function f g defined? What...Ch. 1 - Suppose the graph of f is given. Write an equation...Ch. 1 - (a) What is a one-to-one function? How can you...Ch. 1 - (a) How is the inverse sine function f(x) = sin1 x...Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Let f be the function whose graph is given. (a)...Ch. 1 - The graph of g is given. (a) State the value of...Ch. 1 - lf f(x) = x2 2x + 3, evaluate the difference...Ch. 1 - Sketch a rough graph or the yield of a crop as a...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Prob. 7RE Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Prob. 9RE Ch. 1 - The graph of .f is given. Draw the graphs of the...Ch. 1 - Prob. 11RE Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Prob. 13RE Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Determine whether f is even, odd, or neither even...Ch. 1 - Find an expression for the function whose graph...Ch. 1 - If f(x) = In x and g(x) = x2 9. find the...Ch. 1 - Express the function F(x)=1/x+x as a composition...Ch. 1 - A small-appliance manufacturer finds that it costs...Ch. 1 - If f(x) = 2x + In x, find f1(2).Ch. 1 - Find the inverse function of f(x)=x+12x+1.Ch. 1 - Find the exact value of each expression. 64. (a)...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 - The altitude perpendicular to the hypotenuse of a...Ch. 1 - Solve the equation |2x 1| |x + 5| = 3.Ch. 1 - Solve the inequality |x 1| |x 3| 5.Ch. 1 - Prob. 5P Ch. 1 - Sketch the graph of the function g(x) = |x2 1 | ...Ch. 1 - Prob. 7P Ch. 1 - Prob. 8P Ch. 1 - The notation max{a, b, } means the largest of the...Ch. 1 - Sketch the region in the plane defined by each of...Ch. 1 - Evaluate (log2 3)(log3 4)(log4 5)(log31 32).Ch. 1 - (a) Show that the function f(x)=ln(x+x2+1) is an...Ch. 1 - Solve the inequality ln(x2 2x 2) 0.Ch. 1 - Use indirect reasoning to prove that log2 5 is an...Ch. 1 - A driver sets out on a journey. For the first half...Ch. 1 - Is it true that f(g+h)=fg+fh?Ch. 1 - Prove that if n is a positive integer, then 7n 1...Ch. 1 - Prove that 1 + 3 + 5 + + (2n l ) = n2.Ch. 1 - If fo(x) = x2 and fn+1(x) = fo(fn(x)) for n = 0,...Ch. 1 - (a) If fo(x)=12x and fn+1=fofnforn=0,1,2,, find an...

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