Math

CalculusSingle Variable CalculusSuppose that a function f is continuous on [0, 1] except at 0.25 and that f (0) = 1 and f (1) = 3. Let N = 2. Sketch two possible graphs of f , one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis).BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Chapter 1.8, Problem 50E

Textbook Problem

Suppose that a function *f* is continuous on [0, 1] except at 0.25 and that *f*(0) = 1 and *f*(1) = 3. Let *N* = 2. Sketch two possible graphs of *f*, one showing that *f* might not satisfy the conclusion of the Intermediate Value Theorem and one showing that *f* might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis).

Single Variable Calculus

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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)=cosx1sinxCh. 1.2 - Find the domain of the function. 6. g(x)=11tanxCh. 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 f(1)...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...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. 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F(x) =...Ch. 1.3 - Express the function in the form f g. 44. F(x) =...Ch. 1.3 - Express the function in the form f g. 45....Ch. 1.3 - Express the function in the form f g. 46....Ch. 1.3 - Express the function in the form f g. 47. v(t) =...Ch. 1.3 - Express the function in the form f g. 48....Ch. 1.3 - Express the function in the form f g h. 49....Ch. 1.3 - Express the function in the form f g h. 50....Ch. 1.3 - Express the function in the form f g h. 51. S(t)...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 - A ship is moving at a speed of 30 km/h parallel to...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...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 - If you invest x dollars at 4% interest compounded...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 - A tank holds 1000 gallons of water, which drains...Ch. 1.4 - A cardiac monitor is used to measure the heart...Ch. 1.4 - The point P(2, 1) lies on the curve y = 1/(1 x)....Ch. 1.4 - The point P(0.5, 0) lies on the curve y = cos x....Ch. 1.4 - If a ball is thrown into the air with a velocity...Ch. 1.4 - If a rock is thrown upward on the planet Mars with...Ch. 1.4 - The table shows the position of a motorcyclist...Ch. 1.4 - The displacement (in centimeters) of a particle...Ch. 1.4 - The point P(1, 0) lies on the curve y = sin(10/x)....Ch. 1.5 - Explain in your own words what is meant by the...Ch. 1.5 - Explain what it means to say that...Ch. 1.5 - Explain the meaning of each of the following. (a)...Ch. 1.5 - Use the given graph of f to state the value of...Ch. 1.5 - For the function f whose graph is given, state the...Ch. 1.5 - For the function h whose graph is given, state the...Ch. 1.5 - For the function g whose graph is given, state the...Ch. 1.5 - For the function A whose graph is shown, state the...Ch. 1.5 - For the function f whose graph is shown, state the...Ch. 1.5 - A patient receives a 150-mg injection of a drug...Ch. 1.5 - Sketch the graph of the function and use it to...Ch. 1.5 - Sketch the graph of the function and use it to...Ch. 1.5 - Use the graph of the function f to state the value...Ch. 1.5 - Use the graph of the function f to state the value...Ch. 1.5 - Sketch the graph of an example of a function f...Ch. 1.5 - Sketch the graph of an example of a function f...Ch. 1.5 - Sketch the graph of an example of a function f...Ch. 1.5 - Sketch the graph of an example of a function f...Ch. 1.5 - Guess the value of the limit (if it exists) by...Ch. 1.5 - Guess the value of the limit (if it exists) by...Ch. 1.5 - Guess the value of the limit (if it exists) by...Ch. 1.5 - Guess the value of the limit (if it exists) by...Ch. 1.5 - Use a table of values to estimate the value of the...Ch. 1.5 - Use a table of values to estimate the value of the...Ch. 1.5 - Use a table of values to estimate the value of the...Ch. 1.5 - Use a table of values to estimate the value of the...Ch. 1.5 - (a) By graphing the function f(x) = (cos 2x cos...Ch. 1.5 - (a) Estimate the value of limx0sinxsinx by...Ch. 1.5 - Determine the infinite limit. 29. limx5+x+1x5Ch. 1.5 - Determine the infinite limit. 30. limx5x+1x5Ch. 1.5 - Determine the infinite limit. 31. limx12x(x1)2Ch. 1.5 - Determine the infinite limit. 32. limx3x(x3)5Ch. 1.5 - Determine the infinite limit. 33. limx2+x1x2(x+2)Ch. 1.5 - Determine the infinite limit. 34. limx0x1x2(x+2)Ch. 1.5 - Determine the infinite limit. 35. limx(/2)+1xsecxCh. 1.5 - Determine the infinite limit. 36. limxcotxCh. 1.5 - Determine the infinite limit. 37. limx2xcscxCh. 1.5 - Determine the infinite limit. 38. limx2x22xx24x+4Ch. 1.5 - Determine the infinite limit. 39....Ch. 1.5 - (a) Find the vertical asymptotes of the function...Ch. 1.5 - Determine limx11x31and limx1+1x31 (a) by...Ch. 1.5 - (a) By graphing the function f(x) = (tan 4x)/x and...Ch. 1.5 - (a) Evaluate the function f(x) = x2 (2x/1000) for...Ch. 1.5 - (a) Evaluate h(x) = (tan x x)/x3 for x = 1, 0.5,...Ch. 1.5 - Graph the function f(x) = sin(/x) of Example 4 in...Ch. 1.5 - Consider the function f(x)=tan1x. 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[Hint:Usexa=xax+a.]Ch. 1.7 - If H is the Heaviside function defined in Example...Ch. 1.7 - If the function f is defined by...Ch. 1.7 - By comparing Definitions 2, 3, and 4. prove...Ch. 1.7 - How dose to 3 do we have to take x so that...Ch. 1.7 - Prove, using Definition 6, that limx31(x+3)4=.Ch. 1.7 - Prove that limx15(x+1)3=.Ch. 1.7 - Suppose that limxa f(x) = and limxa g(x) = c,...Ch. 1.8 - Write an equation that expresses the fact that a...Ch. 1.8 - If f is continuous on (, ), what can you say about...Ch. 1.8 - (a) From the graph of f, state the numbers at...Ch. 1.8 - From the graph of g, state the intervals on which...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - The toll T charged for driving on a certain...Ch. 1.8 - Explain why each function is continuous or...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - How would you remove the discontinuity of f? In...Ch. 1.8 - How would you remove the discontinuity of f? In...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Locate the discontinuities of the function and...Ch. 1.8 - Locate the discontinuities of the function and...Ch. 1.8 - Use continuity to evaluate the limit. 35....Ch. 1.8 - Use continuity to evaluate the limit. 36....Ch. 1.8 - Use continuity to evaluate the limit. 37....Ch. 1.8 - Use continuity to evaluate the limit. 38....Ch. 1.8 - Show that f is continuous on (, ). 39....Ch. 1.8 - Show that f is continuous on (, ). 40....Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - The gravitational force exerted by the planet...Ch. 1.8 - For what value of the constant c is the function f...Ch. 1.8 - Find the values of a and b that make f continuous...Ch. 1.8 - Suppose f and g are continuous functions such that...Ch. 1.8 - Let f(x) = 1/x and g(x) = 1/x2. (a) Find (f ...Ch. 1.8 - Which of the following functions f has a removable...Ch. 1.8 - Suppose that a function f is continuous on [0, 1]...Ch. 1.8 - If f(x) = x2 + 10 sin x, show that there is a...Ch. 1.8 - Suppose f is continuous on [1, 5] and the only...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - (a) Prove that the equation has at least one real...Ch. 1.8 - (a) Prove that the equation has at least one real...Ch. 1.8 - (a) Prove that the equation has at least one real...Ch. 1.8 - (a) Prove that the equation has at least one real...Ch. 1.8 - Prove, without graphing, that the graph of the...Ch. 1.8 - Prove, without graphing, that the graph of the...Ch. 1.8 - Prove that f is continuous at a if and only if...Ch. 1.8 - To prove that sine is continuous, we need to show...Ch. 1.8 - Prove that cosine is a continuous function.Ch. 1.8 - (a) Prove Theorem 4, part 3. (b) Prove Theorem 4,...Ch. 1.8 - For what values of x is f continuous?...Ch. 1.8 - For what values of x is g continuous?...Ch. 1.8 - Is there a number that is exactly 1 more than its...Ch. 1.8 - If a and b are positive numbers, prove that the...Ch. 1.8 - Show that the function f(x)={x4sin(1/x)ifx00ifx=0...Ch. 1.8 - (a) Show that the absolute value function F(x) = |...Ch. 1.8 - A Tibetan monk leaves the monastery at 7:00 am and...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 - Explain what each of the following means and...Ch. 1 - Describe several ways in which a limit can fail to...Ch. 1 - What does it mean to say that the line x = a is a...Ch. 1 - State the following Limit Laws. (a) Sum Law (b)...Ch. 1 - What does the Squeeze Theorem say?Ch. 1 - (a) What does it mean for f to be continuous at a?...Ch. 1 - (a) Give examples of functions that are continuous...Ch. 1 - What does the Intermediate Value Theorem say?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 - 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 - Determine whether each curve is the graph of a...Ch. 1 - If f(x) = x2 2x + 3, evaluate the difference...Ch. 1 - Sketch a rough graph of 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 - Find the domain and range of the function. Write...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Suppose that the graph of f is given. Describe how...Ch. 1 - The graph of f is given. Draw the graphs of the...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 - 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)=x and g(x) = sin x, find the functions (a)...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 - The graph of f is given. (a) Find each limit, or...Ch. 1 - Sketch the graph of an example of a function f...Ch. 1 - Find the limit. 25. limx0cos(x+sinx)Ch. 1 - Find the limit. 26. limx3x29x2+2x3Ch. 1 - Find the limit. 27. limx3x29x2+2x3Ch. 1 - Find the limit. 28. limx1+x29x2+2x3Ch. 1 - Find the limit. 29. limh0(h1)3+1hCh. 1 - Find the limit. 30. limt2t24t38Ch. 1 - Find the limit. 31. limr9r(r9)4Ch. 1 - Find the limit. 32. limv4+4v4vCh. 1 - Find the limit. 33. limu1u41u3+5u26uCh. 1 - Find the limit. 34. limx3x+6xx33x2Ch. 1 - Find the limit. 35. lims164ss16Ch. 1 - Find the limit. 36. limv2v2+2v8v416Ch. 1 - Find the limit. 37. limx011x2xCh. 1 - Find the limit. 38. limx1(1x1+1x23x+2)Ch. 1 - If 2x 1 f(x) x2 for 0 x 3, find limx1 f(x).Ch. 1 - Prove that limx0 x2 cos(1/x2) = 0.Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Let f(x)={xifx03xif0x3(x3)2ifx3 (a) Evaluate each...Ch. 1 - Let g(x)={2xx2if0x22xif2x3x4if3x4ifx4 (a) For each...Ch. 1 - Show that the function is continuous on its...Ch. 1 - Show that the function is continuous on its...Ch. 1 - Use the Intermediate Value Theorem to show that...Ch. 1 - Use the Intermediate Value Theorem to show that...Ch. 1 - Suppose that |f(x)| g(x) for all x, where limxa...Ch. 1 - Let f(x) = x + x. (a) For what values of a does...Ch. 1 - Solve the equation |2x 1| |x + 5| = 3.Ch. 1 - Solve the inequality |x 1| |x 3 | 5.Ch. 1 - Sketch the graph of the function f(x) = |x2 4|x|...Ch. 1 - Sketch the graph of the function g(x) = |x2 1| ...Ch. 1 - Draw the graph of the equation x + |x| = y + |y|.Ch. 1 - Sketch the region in the plane consisting of all...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 - A driver sets out on a journey. For the first half...Ch. 1 - Is it true that f (g + h) = f g + f h?Ch. 1 - Prove that if n is a positive integer, then 7n 1...Ch. 1 - Prove that 1 + 3 + 5+ ...+ (2n 1)=n2.Ch. 1 - If f0(x) = x2 and fn+1(x) = f0(fn(x)) for n = 0,...Ch. 1 - (a) If f0(x)=12x and fn+1 = f0 fn for n = 0, 1,...Ch. 1 - Evaluate limx1x31x1.Ch. 1 - Find numbers a and b such that limx0ax+b2x=1.Ch. 1 - Evaluate limx02x12x+1x.Ch. 1 - The figure shows a point P on the parabola y = x2...Ch. 1 - Evaluate the following limits, if they exist,...Ch. 1 - Sketch the region in the plane defined by each of...Ch. 1 - Find all values of a such that f is continuous on...Ch. 1 - A fixed point of a function f is a number c in its...Ch. 1 - If limxa [f(x) + g(x)] = 2 and limxa [f(x) g(x)]...Ch. 1 - (a) The figure shows an isosceles triangle ABC...Ch. 1 - (a) If we start from 0 latitude and proceed in a...

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