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 | 1. Think about the following “real-life” functions, each of which is a
function of the independent variable time: the height of a falling object, the velocity of an object, the amount of money in a bank account, the cholesterol level of a person, the heart rate of a person, the amount of a certain chemical present in a test tube and a machine's most recent measurement of the cholesterol level of a person. Which of these are continuous functions? For each
function you identify as discontinuous, what is the real-life meaning of the discontinuities? |
| | 2. Whether a process is continuous or not is not always clear-cut. When you watch television or a movie, the action seems to be continuous. This is an optical illusion, since both movies and television consist of individual “snapshots” that are played back at many frames per second. Where does the illusion of continuous motion come from? Given that the average person blinks several times per minute, is our perception of the world actually continuous? [A famous experiment in cognitive psychology showed that the human brain first decides whether a stimulus is important enough to merit conscious consideration. If so, the brain “predates” the stimulus so that the person correctly identifies when the stimulus actually occurred (hence, the different reaction times we have to stimuli)]. |
| | 3. When you sketch the graph of the parabola y = x2 with pencil or pen, is your sketch (at the molecular level) actually the graph of a continuous function? Is your calculator or computer's graph actually the graph of a continuous function? On many calculators, you have the option of a connected or disconnected graph. At the pixel level, does a connected graph show the graph of a function? Does a disconnected graph show the graph of a continuous function? Do we ever have problems correctly interpreting a graph due to these limitations? In the exercises in section 1.6 we examine one case where our perception of a computer graph depends on which choice is made. |
| | 4. For each of the graphs in
Figures 1.16,1.17,1.18, and
1.19, describe (with an example) what the formula for f (x) might look like to produce the given discontinuity. |
| In exercises 5-10, use the given graph to identify all discontinuities of the functions.
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| 5.
| 6.
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| 7.
| 8.
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| 9.
| 10.
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| In exercises 11-16, explain why each
function is discontinuous at the given point by indicating which of the three conditions in
Definition 3.1 are not met.
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| 11. 
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| 12.  at x = 1
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| 13.  at x = 0
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| 14. f (x) = e1/x at x = 0
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| 15. 
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| 16. 
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| In exercises 17-28, find all discontinuities of
f (x). For each discontinuity that is removable, define a new
function that removes the discontinuity.
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| 17.  | 18.  |
| 19.  | 20.  |
| 21. f (x) = x2tan x | 22. f (x) = csc x |
| 23. f (x) = xcot x | 24. 
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| 25. 
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| 26. 
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| 27. 
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| 28. 
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| In exercises 29-36, determine the intervals on which
f (x) is continuous.
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| 29.  | 30.  |
| 31.  | 32. f (x) = (x -1)3/2 |
| 33. f (x) = sin (x2 + 2) | 34.  |
| 35. f (x) = ln (x + 1) | 36. f (x) = ln (4 - x2)
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| 37. Suppose that a state's income tax code states that the tax liability on x dollars of taxable income is given by
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| Determine the constant c that makes this
function continuous at all x. Give a rationale why such a function should be continuous. |
| 38. Suppose a state's income tax code states that tax liability is 12% on the first $20,000 of taxable earnings and 16% on the remainder. Find constants a and b for the tax
function
such that T(x) is continuous at all x. |
| 39. In
Example 3.8, find formulas for T(x) for (a) 128,100 < x  278,450 and (b) x > 278,450. |
| 40. In
Example 3.8, show that T(x) is continuous at x = 61,400. |
| In exercises 41-46, use the Intermediate Value Theorem to verify that
f (x) has a zero in the given interval. Then use the method of bisections to find an interval of length 1/32 which contains the zero.
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| 41. f (x) = x2 -7, [2, 3]
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| 42. f (x) = x3 - 4x -2, [2, 3]
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| 43. f (x) = x3 - 4x - 2 , [ -1, 0]
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| 44. f (x) = x3 - 4x -2, [ -2, - 1]
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| 45. f (x) = cos x - x, [0, 1]
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| 46. f (x) = ex + x, [ -1, 0]
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| A function is continuous from the right at
x = a if  In exercises 47-50, determine if f (x) is continuous from the right at
x = 2.
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| 47. 
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| 48. 
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| 49. 
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| 50. 
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| 51. Define what it means for a
function to be continuous from the left at x = a and determine which of the functions in exercises 47-50 are continuous from the left at x = 2. |
| 52. Suppose that  and h(a) = 0. Determine whether each of the following statements is definitely true, definitely false, or maybe true/maybe false. Explain. (a)  does not exist. (b) f (x) is discontinuous at x = a.
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| 53. The sex of newborn Mississippi alligators is determined by the temperature of the eggs in the nest. The eggs fail to develop unless the temperature is between 26° C and 36° C. All eggs between 26° C and 30° C develop into females and eggs between 34° C and 36° C develop into males. The percentage of females and males decreases from 100% at 30°C to 0% at 34° C. If f (T) is the percentage of females developing from an egg at T° C, then 
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| for some function g(T). Explain why it is reasonable that f (T) be continuous. Determine a
function g(T) such that 0 g(T) 100 for 30 T 34 and the resulting
function f (T) is continuous. [Hint: It may help to draw a graph first and make g(T)
linear.] |
| 54. If 
and g(x) = 2x , show that
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| 55. If you push on a large box resting on the ground, at first nothing will happen because of the static friction force that opposes motion. If you push hard enough, the box will start sliding, although there is again a friction force that opposes the motion. Suppose you are given the following description of the friction force. Up to 100 pounds, friction matches the force you apply to the box. Over 100 pounds, the box will move and the friction force will equal 80 pounds. Sketch a graph of friction as a
function of your applied force based on this description. Where is this graph discontinuous? What is significant physically about this point? Do you think the friction force actually ought to be continuous? Modify the graph to make it continuous while still retaining most of the characteristics described. |
| 56. For  , we have f ( -1) > 0 and f (2) < 0. Does the Intermediate Value Theorem guarantee a zero of f (x) between x = - 1 and x = 2 ? What happens if you try the method of bisections? |
| 57. On Monday morning, a saleswoman leaves on a business trip at 7:13 a.m. and arrives at her destination at 2:03 p.m. The fol-lowing morning, she leaves for home at 7:17 a.m. and arrives at 1:59 p.m. The woman notices that at a particular stoplight along the way, a nearby bank clock changes from 10:32 a.m. to 10:33a.m. on both days. Therefore, she must have been at the same location at the same time on both days. Her boss doesn't believe that such an unlikely coincidence could occur. Use the Intermediate Value Theorem to argue that it must be true that at some point on the trip, the saleswoman was at exactly the same place at the same time on both Monday and Tuesday. |
| 58. Suppose you ease your car up to a stop sign at the top of a hill. Your car rolls back a couple of feet and then you drive through the intersection. A police officer pulls you over for not coming to a complete stop. Use the Intermediate Value Theorem to argue that there was an instant in time when your car was stopped (in fact, there were at least two). What is the difference between this stopping and the stopping that the police officer wanted to see? |
| 59. Suppose a worker's salary starts at $40,000 with $2000 raises every 3 months. Graph the salary
function s(t ) ; why is it discontinuous? How does the function  (t in months) compare? Why might it be easier to do calculations with f (t ) than s(t ) ? |
| 60. Prove the final two parts of
Theorem 3.2. |
| 61. Suppose that f (x) is a
continuous function with consecutive zeros at x = a and x = b ; that is, f (a) = f (b) = 0 and f (x) 0 for a < x < b. Further, suppose that f (c) > 0 for some number c between a and b. Use the Intermediate Value Theorem to argue that f (x) > 0 for all a < x < b. |
| 62. Use the method of bisections to estimate the other two zeros in
Example 3.9. |
 | 63. In the text, we discussed the use of the method of bisections to find an
approximate solution of equations such as f (x) = x3 + 5x -1 = 0. We can start by noticing that f (0) = - 1 and f (1) = 5. Since f (x) is continuous, the Intermediate Value Theorem tells us that there is a solution between x = 0 and x = 1. For the method of bisections, we guess the midpoint, x = 0.5. Is there any reason to suspect that the solution is actually closer to x = 0 than to x = 1 ? Using the
function values f (0) = - 1 and f (1) = 5 , devise your own method of guessing the location of the solution. Generalize your method to using f (a) and f (b) , where one
function value is positive and one is negative. Compare your method to the method of bisections on the problem x3 + 5x -1 = 0 ; for both methods, stop when you are within 0.001 of the solution, x 0.198437. Which method performed better? Before you get overconfident in your method, compare the two methods again on x3 + 5x2 -1 = 0. Does your method get close on the first try? See if you can determine graphically why your method works better on the first problem. |
| | 64. You have probably seen the turntables on which luggage rotates at the airport. Suppose that such a turntable has two long straight parts with semicircles on the end (see the figure). We will model the left/right movement of the luggage. Suppose the straight part is 40 ft long, extending from x = - 20 to x = 20. Assume that our luggage starts at time t = 0 at location x = - 20 , and that it takes 60 s for the luggage to reach x = 20. Suppose the radius of the circular motion is 5 ft, and it takes the luggage 30 s to complete the half-circle. We model the straight-line motion with a linear
function x(t ) = at + b. Find constants a and b so that x(0) = - 20 and x(60) = 20. For the circular motion, we use a cosine (Why is this a good choice?) x(t ) = 20 + d cos (et + f ) for constants d, e and f. The requirements are x(60) = 20 (since the motion is continuous), x(75) = 25 and x(90) = 20. Find values of d, e and f to make this work. Find equations for the position of the luggage along the backstretch and the other semicircle. What would the motion be from then on?
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Luggage
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