Homework Assignments

• HW Section 6.3/6.4 (Joint)
• Not Due: These problems are listed for your practice and to help your understanding. I will not correct them.
• Section 6.3:
  • 0, 2, 5,
  • 7 & 11 (same function, but distinguish derivative at a point with derivative as a function),
  • 17-25 (try several),
  • 26-31 and 32-40 (Try several, don't forget to check where f '(x) DNE)
• Section 6.4:
  • 0, 13, 14,
  • 17 (Tip: Cancel common factors=Hole or asymptote, Use left/right limits at asymptotes to see where the function goes),
  • 22 (Cancel first, Check domain),
  • 24
  • We won't worry about 31-39.
• HW Section 6.1/6.2 (Joint)
• Due Wednesday, December 1
• Required:
  • Section 6.1: 0, 14, 17, 19, 37, 42, 48
  • Section 6.2: 0, 10, 24, 25, 26, 29, 32, 36, 37, 38, 51, 55
• Study tip: Read through all of the problems—it should help you recognize if you understand the material.
• HW Section 5.3/5.4 (Joint)
• Due Tuesday, November 30 (after Thanksgiving)
• Warning: Do not confuse this with the HW 5.2/5.3 joint assignment.
• Note: This homework will be graded and the quiz will use one of the application problems.
• Required:
  • Section 5.3: 43, 47, 52, 56, 60
  • Section 5.4: 0, 3, 10, 12, 15, 19, 21, 23, 28, 30
• HW Section 5.2/5.3 (Joint)
• Due Tuesday, November 23 (before Thanksgiving)
• Required:
  • Section 5.2: 0, 2-4, 29, 30, 37, 40, 44, 52, 59, 63, 66
  • Section 5.3: 0, 3, 7, 21, 25, 34, 38
• Note: Problems after 40 in Section 5.3 will appear on the next homework.
• Suggested:
  • 5.2.5-7 — Looking at turns to estimate degree
  • 5.2.8-14 — Using verbal properties to sketch a graph, or identify when impossible
  • 5.2.19-26 — True/False statements. Understand them.
  • 5.2.27-38 — Different types of limits. Pay attention to what limit is taken!
  • 5.2.39-47 — Derivatives. Sometimes you'll need to expand first (no product rule!)
  • 5.2.48-53 — Antiderivatives and initial conditions
  • 5.2.54-62 — Local behavior of functions by hand. Very important!!
  • 5.2.63-68 — Applications.
  • 5.2.69-78 — Proofs. Check Blackboard's Discussion Board for some comments.
    
  • 5.3.3-5 — Estimate multiplicity of roots.
  • 5.3.6-13 — Create a polynomial based on factors. (Important)
  • 5.3.14-20 — True/False. Important to understand reasons.
  • 5.3.21-30 — Plotting by factoring
  • 5.3.31-39 — Plotting by graph analysis (roots of f, f' and f''). Usually involves factoring.
• HW Section 5.1
• Due: Friday, November 19
• Required: 0, 10, 11, 40, 46, 55, 57, 62, 69, 70, 72, 74, 78
• Note: I have some comments in the Discussion Board on Blackboard that should help.
• Practice: Choose 2 or 3 problems from each section 79-86 and 87-92 to practice synthetic division before class on Friday.
• Suggested:
  • 1-9 — identifying and characterizing polynomials
  • 12-20 — Do you know what these mean?
  • 23, 24
  • 26-38 — Good practice on vocabulary and properties
  • 48-54 — Understand these!
  • More skills problems
• HW Section 4.5
• Due Monday, November 15
• Required: 0, 15, 22, 39, 49, 58, 65, 75
• Suggested: Pay special attention to the following
  • 2-4 provide a good practice of listing properties (including, but not limited to, domain, differentiability, positive/negative, increase/decrease, concavity)
  • 27-32 involve determining if even/odd (by examining f(-x))
  • 33-35 compare functions with their reciprocals
  • 36-41 practice finding inverse functions involving powers
  • 51-56 involve multiple transformations of functions
• Notes:
  • Be sure to show clear work on the required problems.
  • Be responsible and practice additional problems.
• HW Section 4.4
• Due Friday, November 12
• Required: 0, 2, 24, 40, 47, 50, 62, 64
• Suggested: Check each of the problem clusters. Know how you would attempt to solve each problem.
• Notes
  • Be sure to give a complete and thorough solution to each assigned problem.
  • Just because a problem isn't assigned, it doesn't mean you don't need to know how to do it.
  • Recognize the difference between clusters 33-38 and 39-41. Why are they both included?
  • Recall graph transformations (Sec 1.6) when doing 48-56. (Recall how we found compositions indicating the order)
• HW Section 4.3
• Due Wednesday, November 10
• See comment below about assignment!
• Required: 0, 1, 26, 45, 51, 65, 73
• Notes:
  • I am assigning a handful of problems that I expect you to complete carefully.
  • Each problem needs to be neatly presented, showing your work. Your objective is to demonstrate that you understand how to do the problem, not just to get the answer.
  • Use English phrases (not necessarily sentences) to connect your steps. Explain what you are doing. A reader needs to have an idea where you are going with your calculations.
• Suggested: (You need to understand how to do each of the following types of problems. You are the master of your own fate depending on how much time and effort you spend on problems. The required problems are not enough)
  • 5-10 — Rewrite, if possible, as power functions or sums of power functions. (Pay special attention to #12)
  • 16-21 and 22-27 — Catch the distinction here: the definition of the derivative at a point vs. the definition of the derivative as a function.
  • 28-33 — Practice on more complicated functions!!!
  • 34-48 — Learn to identify when our rules so far don't apply.
  • 49-57 — Piecewise functions take special care.
  • 58-60 — Create equations for constants a and b to make a piecewise function differentiable.
  • 61-66 — Recognize a limit as a derivative in disguise to more quickly calculate a limit.
  • 67-72 — Antidifferentiation
  • 76-79 — Proofs
• HW Section 4.2
• Due: Monday, November 8
• Required: 0, 2, 5, 9, 12, 14, 17, 20, 24, 25, 28, 30, 31, 34, 35, 36, 38, 39, 41, 43, 46, 48, 50, 53, 54, 61, 68
• Suggested: 41-55 are important, 58-63 will help for Section 4.3, others are also good.
• Notes:
  • Hint for 68: See the discussion board for an in-depth hint.
  • The following are indeterminate forms for limits:
    • 0/0 and ∞/∞ (both require you to find a common factor and cancel)
    • ∞ - ∞ (factor before taking limit and use product rule for limits)
    • 0⋅∞ (rewrite it as a fraction to be like first case, and then try to cancel something).
  • Adding or subtracting a number to or from infinity is still infinity.
• HW Section 4.1
• Due: Friday, November 5
• Required: 0, 3, 5, 7, 15, 19, 20, 28, 34, 39, 40, 46, 48, 54, 58, 60, 64, 65, 70, 71, 75, 79, 81, 87, 90, 97
• Suggested: 1-9* and all of the Skills section problems. Proofs 94-97 are also very good.
• Notes:
  • Problems 53-61!! These are very common mistakes. Be sure you understand what went wrong. When it says find a counterexample, that means find a number for x and another for y so that the two sides do not equal each other. (Avoid values 0 and 1 for best chances.)
  • Problems 62-67: We don't know that power functions are continuous (yet) except for integer powers, so we don't really have license to use the Intermediate Value Theorem strategy of finding zeros, and then testing the intervals. We'll have this in the next section, however, so go ahead and use it anyway. (What would you do if this wasn't allowed?)
• HW Section 3.8
• Due: Wednesday, November 3
• Required: 0, 2, 6, 10, 14, 19, 22, 25, 27, 30, 36, 42, 44, 51, 52, 55, 59, 63, 66, 72, 73
• Suggested: Use the clusters already established in the book. Problems that really test your understanding: 5, 6-13*, 25, 47, 56, 76, 77
• Notes:
  • Recall that a derivative does not exist at a point if the function does not exist at a point or is not continuous or does not have a well-defined tangent line.
  • Some functions are written as products of various expressions. We do not have a product rule yet, so you must first multiply these functions completely out and then you can differentiate.
• HW Section 3.7 (wrap-up)
• Due: Tuesday, November 2
• Required: 0, 3, 8, 10, 14, 19, 20, 23, 26, 28, 38, 40, 43, 44, 53
• Suggested: 1-8*, 10-15*, 17-18*, 22-27*, 38-43*, 48, 50-52*
• Notes:
  • Warning!! Pay attention to whether a graph gives the graph of f or f'. (Do you know why it matters?)
  • Again, I encourage you to check your answer with the back of the book (after you do the work). Then change your answer and understanding as you need.
• HW Section 3.6 and 3.7 (partial)
• Due: Monday, November 1
• Required:
  • Section 3.6: 0, 2, 4, 7, 14, 16, 20, 23, 24, 31, 33, 36, 37, 41, 43, 48, 54, 56, 63
  • Section 3.7: 2, 4, 5, 8, 29, 33, 36, 46
• Suggested:
  • Section 3.6: 12-17*, 18-26*, any of Skills problems, 61*, 62*
  • Section 3.7: 1*, 9*, 16, 29-37*
• Notes:
  • For the True/False problems, I encourage you to check your answer in the back of the book. But it is up to you to find the counterexamples.
  • Warning!! Know the difference between f(x) and f'(x). Don't use f instead of f' and vice versa, especially in the problems for Section 3.7. If you are given f, you may need to find f'.
• HW Section 3.5
• Due: Wednesday, October 27
• Required: 0, 9, 11, 12, 13, 17, 21, 22, 24, 27, 32, 36, 37, 38, 40, 42, 44, 45, 49-54, 55, 59, 61, 64, 65, 67, 68, 71, 77, 80, 82, 87
• Suggested: 1-8*, 10, 14-15*, 16-20*, 23*, 24-29, 31-36*, 37-42*, 43-48*, 55-63*, 66*, 69*, 70-72*, 73*, 74*, 75, 76-81*, 88*, 89*, 90*
• Notes:
  • Check the domain before finding the derivative. If a function is not continuous at x=c, then the derivative is not defined at x=c, even if the original function "simplifies."
  • If a function is not written as a simple polynomial, then you must first simplify the expression so that you can use the sum rule and the constant multiple rule with the power rule.
• HW Section 3.4
• Due: Tuesday, October 26
• Required: 0, 3, 4, 5, 9, 10, 14, 15, 16, 17, 20, 21, 26, 27, 31, 32, 34, 35, 36, 37, 40, 43, 48, 52, 54, 55, 57, 62, 63, 66, 68
• Suggested: 1, 2, 6-7*, 8-11*, 12*, 13, 18-23*, 24-29*, 30-35*, 36-38*, 39-50*, 51-53*, 54-56*, 57-62*, 63-65*, 67*, 69*, 70-73*
• Notes:
  • In spite of the long list of problems, this probably isn't really enough practice. You need to feel confident in finding the derivative using a limit, in spite of the fact that we'll soon have rules that make it much easier. Honest--I limited the number of problems assigned.
  • Problems 18-23 and 24-29 -- Do NOT find a limit. Just simplify.
  • Problems 30-35 -- Learn to know the difference of how a limit treats the variable x when it considers a second variable hÅ®0 or tÅ®x. Most of these limits are related to derivatives.
  • Problems 36-38 and 39-50 -- Find the derivative using limits involving h as well as limits involving t.
  • Problems 30-35 and 36-38 require you to justify your steps. This means using the basic limit rules (and not just plugging it in). The key is that the variable x is a constant during the limit. So it can factor out using the constant multiple rule, and the limit of x is just x using the limit of a constant rule. (See the Caution on bottom of page 258.)
  • Problems 39-50 don't require you to do limits step by step. Plug it in as appropriate.
• Section 3.3
• Due: Friday, October 22
• Required: 0, 4, 5, 13, 15, 17, 19, 20, 26, 27, 30
• Suggested: 2, 6, 7, 8-12*, 14, 16, 18, 21*, 22-24, 25-30*, 31-34*, 35*, 36, 37
• Notes:
  • Problem 3 describes a function that can not exist. Can you see why?
  • When you have a problem with absolute values, you need to first rewrite it as a piecewise function.
  • If you find a function is not continuous, you still need to algebraically show it is not differentiable (even though we have a theorem that guarantees this)--you need to show how it is infinite.
  • You only need to worry about left and right derivatives separately at "break" points. Otherwise, just compute the derivative as a limit.
• Section 3.2
  
• Due: Monday, October 18
• Required: 0, 3, 4, 6, 8, 9, 12, 15, 17, 19, 21, 22, 24, 26, 30, 32, 36, 37, 38
• Suggested: 1, 2, 5*, 7*, 10*, 11-16*, 17-19*, 20-25*, 26-35*
• Notes:
  • The intuitive application problems (26-37) are probably the most important long-term concepts to understand. This isn't to dismiss the importance of the algebraic ideas, but being able to relate descriptions of function behavior to approximate graphs is incredibly valuable.
  • Again, the proof for this assignment is straight-forward. Be sure that you do it.
• Section 3.1
• Due: Wednesday, October 13
• Required: 0, 5, 8, 9, 12, 13, 19, 21, 22, 29, 34, 36, 38, 39, 40, 42, 44, 45, 46, 50, 55, 57, 59, 66, 72, 76, 77, 81, 90, 96, 97
  
• Suggested: 1-4**, 6-7, 10*, 11*, 14, 15, 16-18, 20, 23-26*, 27-33*, 35-40*, 41-49*, 50-58*, 59-67*, 68-73*, 74-79*, 84-89, 91-93, 94, 95, 98
• Notes:
• Problems 1-4 are much like Problem Zero. I won't grade these, but you really need to know how to do them.
• Several problems are paired so that you do the same work on the same functions, but in different ways. You should check your answers, but be sure you are familiar with the different methods.
• Don't skip the proofs at the end!
  
  
• Section 2.7
• Due: Tuesday, October 12
• Required: 0, 1, 2, 3, 4, 8, 9, 10, 13, 14, 17, 18, 21, 24, 26, 30, 32, 35, 37, 38, 40, 41, 43, 44, 47, 50, 51, 52, 57, 60, 64
• Suggested : 5, 6, 7, 9-16**, 19-24*, 25-30*, 31-36*, 37-42*, 45*, 46-54*, 55-58*, 59-62*
• Notes:
• For 31-36, the first step to your task is to find an appropriate interval [a,b] so that the Intermediate Value Theorem will apply.
• For 43-45, see the Question after Example 2.68
• For 55-58, the piecewise functions break the domain into intervals. When finding zeros or discontinuities of each piece, make sure that the resulting values are in the appropriate interval of the domain.
• For 59-62, unless the problem states otherwise, do not assume that the function is strictly increasing or decreasing. For example, Phil may have filled the tank at some point in between. (But you should still assume the functions are continuous.)
  

• Section 2.6
• Due Monday, October 11
• Required: 0, 2, 5, 6, 8, 12, 13, 14, 16, 19, 21, 23, 24, 25, 26, 30, 35, 36, 39, 42, 45, 46, 47, 50, 51, 53, 54, 55, 60, 62, 65, 67, 70, 72
• Suggested: 4, 7, 9-11*, 12-18*, 20*, 22, 28*, 29*, 31*, 32*, 33, 37, 40*, 41*, 43-44*, 48-49*, 52*, 56*, 57-59*, 61, 64*, 66*, 68*, 69*, 71*, 73*, 74*
• Notes:
• Because of the greater-than-usual number of problems and their level of difficulty for Sections 2.6 and 2.7, I am giving an extended time-period to work on these assignments. (Ask for help early.)
• It is important to understand both the conceptual problems as well as the skills problems.
• Don't be scared by the proofs. To show a function is continuous, you simply need to show that the function satisfies Definition 2.14. Use appropriate limit rules and simple statements to show this is true. That is, show the limit exists and that it is the same as you get by plugging in the value. Be sure to reference what rules you use.
  
  
• Section 2.5
• Due Wednesday, October 6
• Required: 0, 3, 7, 9, 13, 16, 19, 20, 21, 25, 26, 28, 31, 35, 38, 39, 40, 42
• Suggested: 1, 2, 6, 8, 10, 12*, 14, 15*, 17*, 18, 22, 23*, 24*, 27, 29*, 30*, 32*, 33, 34, 36*, 37*, 41*, 43*, 44*, 46, 47*, 48*
• Notes:
• Starred suggested problems correspond to those that require more practice: zero in the denominator and piecewise functions.
• I suggest looking at problems 23 and 24 together.
• Officially, we don't have a license to do problem 45. Why not?
• The proofs in problems 47 and 48 only take about 1 or 2 lines. Can you do them? Assume the result of 47 to do 48. To get 47, you really only need to find the right rule and then show that the limit corresponds to f(c).
  
• Section 2.4
• Due Monday, October 4
• Required: 0, 2, 5, 10, 14, 21, 25, 39, 45, 47, 50, 54, 57
• Suggested: 1, 3-12*, 13*, 15, 16-17*, 18, 19-28*, 29, 30, 31
• Notes:
• Problem 1 asks for statements we can't calculate using the basic rules. This may not be obvious: Think about such things as piecewise functions, functions with fractional powers, compositions of functions, infinite limits, one-sided limits, and so on.
• Problems 13/29 and 14/30 are essentially doing the same things, but for specific values.
• Problem 18: Hint! The individual limits may not exist, but when they are added to each other they do.
• Problems 19-28 involve writing down the epsilon-delta definition of the proposed limit. The instructions suggest using L and M for limits. That is, when you are defining limits, assuming that other limits exist (such as lim f(x)), make the statement that lim f(x)=L, and use that symbol in the implication "if 0<|x-c|<ɬ, then |h(x)-???|<É√" (to make the ??? appropriate).
• Problems 38-40. Part (a) directions should be understandable. Part (b) refers to first simplifying the function so that it looks like a linear function, and then only using the linear function rule.
  
• Section 2.3
• Due Friday, October 1
• Required: 0, 1, 4, 9, 14, 16, 18, 27, 29, 33, 35
• Suggested: 2, 3-8*, 10*, 11-16*, 17-19*, 20-23*, 24-29*, 30-32, 34-36*, 37-39, 4-42, 43.
  
• Comments:
• Problems 3-8 involve using a derived function, delta depends on epsilon, to create a punctured interval, such as what we did for Section 2.2.
• Problems 11-16 and 24-29 appear as pairs. Same for 17-19 and 34-36. Every limit statement appears twice, once to find the formula for delta, and once to write the proof. You may do these together, so long as you do all the parts.
• Problems 17-19 and 34-36 require choosing a bound on delta (see the hint in the problem). But 17 and 34 actually needs a bound stronger than delta <= 1, such as delta <= 1/2. (Why? See if you see where the problem is.)
• Problems 20-23 and 37-42 involve one-sided limits or limits involving infinity. See the book examples to see how our outline is slightly modified to take these problems into account. Even though this homework isn't going to grade such problems, you DO need to understand this.
  
• Section 2.2
• Due Wednesday, September 29
• Required: 0, 1, 10, 12, 16, 21, 25, 29, 37, 48, 53, 56, 60
  
• Suggested: 2, 3-5, 6*, 9-14*, 15-19*, 22*, 23, 24-32*, 33-36*, 38-39, 40-45*, 46-51, 52-55*, 57-58*, 59, 61
• Comments:
• (1) The required problems are intentionally too few for you to master the concept. Spend time learning the concepts by using the suggested problems, and then be sure to write up very clear solutions to the required problems.
  
  Trying to get the homework done fast will prove a bad choice in the long run.
  
  
• (2) Starred problem clusters correspond to key skills that you need to master
• a) 24-32 involve taking a statement about a limit, and rephrasing it into a statement involving epsilon/delta (or possibly bounds M/N).
• b) 33-36 involve reading a statement involving epsilon/delta and determining what is the corresponding limit.
  
• Section 2.1
• Due Monday, September 27
• Required: 0, 1, 4, 6, 9, 11, 12, 13, 16, 21, 24, 29, 32, 36, 39
• Suggested: 5*, 7*, 14, 15, 17, 18*, 19, 20*, 22*, 23, 25-28, 30, 31, 33-35, 38
• I strongly encourage you to try as many examples as you can before we move on to the formal definition of limits. Get comfortable with the concepts. (And avoid the temptation just to plug in the value--we don't have rules to allow this when it is allowed, yet.)
• HW Turn-In Make-up #1
• Play the graph transformation game
• Record the result of several turns and turn that in.
• Submit before October 1
  
• Section 1.7
• Due Monday, September 20
  
• Required: 0, 1, 5, 10, 11, 15, 18, 21, 27, 28, 32, 38, 43, 46, 47
  
• Suggested: 6*, 7-9, 14, 15-20*, 23*, 24-26*, 28-33*, 34-36, 39*, 44, 45*, 48, 49
• Hint: To use a theorem (such as Thm 1.13), you must first show that you satisfy the hypothesis. (This requires a mini proof that the hypothesis is true.) For Thm 1.13, this would mean you would first need to prove a function is monotonic before you could claim it is one-to-one.
• Section 1.6
• Due Friday, September 17
• Required: 0, 3, 14, 15, 18, 20, 27, 31, 33, 36, 45, 50, 51, 56, 62, 63, 71, 73, 77, 84
• Suggested: 1-7, 8-13, 16-17*, 22*, 23*, 24-29, 31*, 34*, 36-50, 51-58, 59*, 60-65, 66-68, 70, 74*, 79*, 80-85
• Note on Suggested problems: I listed nearly every section. I simply mean that you should be sure to feel confident in each of these categories of problems. Starred problems (*) indicate a special emphasis or that the problem is slightly difficult.
  
• Section 1.5
• Due Wednesday, September 15
• Required: 0, 4, 7, 14, 16, 22, 27, 33, 35, 41, 42, 48, 50, 61, 67, 71, 78, 84, 90, 98, 101, 109, 112
• Suggested: 5, 6, 8*, 11, 12, 15, 18*, 20*, 24*, 26, 28, 32, 34, 36, 38*, 39*, 43, 44*, 49*, 54, 59, 64, 68*, 73, 74, 75, 77, 79, 81*, 82, 83, 86, 88, 89*, 92, 93*, 94*, 97*, 99, 100, 102*, 104, 106, 108, 110*, 113
  
• Note: Only required problems need be turned in. See the Blackboard announcement for details.
• Starred (*) problems indicate problems that really test your understanding. You are strongly encouraged to try these.
  
• Warning: Be sure to do all parts described in instructions.
  
• Section 1.4
• Due Monday, September 13
• Problems: 0, 2, 7, 8, 9, 10, 12, 14, 15, 17, 19, 20, 21, 22, 24, 25, 27, 31, 34, 37, 40, 43, 44, 46, 49
• Note 1: Work on the following pairs of problems together: (a) 31 and 37, (b) 34 and 40, (c) 27 and 46
• Note 2: The directions for 37 and 40 might be a little confusing. The function g(x) simply refers to the function inside of the absolute values.
  
• Section 1.3
• Due Friday, September 10
• Problems: 0, 2, 3, 5, 9, 11, 12, 13, 16, 18, 19, 21, 22, 25, 27, 29, 31, 33, 34, 35, 39, 40, 44, 46, 47
• Note: You should look at every problem in the chapter and make sure you could do it.
• Section 1.2
• Due Wednesday, September 8
• Problems: 0, 2, 3, 4, 6, 8, 11, 13, 15, 16, 17, 18, 19, 23, 26, 28, 31, 34, 35, 38, 39
• Hint: In the Applications problems, you might want to imagine specific times when various events occur. Then ask yourself where the function should be at that time. How is the graph connected between these various times.
  
• Section 1.1
• Due September 6
• Problems:  0, 3, 6, 8, 9, 11, 16, 19, 21, 22, 24, 28, 31, 32, 34, 37, 40, 41
• Hint: To find the domain of a function with more than one square root, you must consider the INTERSECTION of the sets where each square root is separately defined. Also, don't forget to exclude individual values where a denominator is zero (0).
• Section 0.4 and 0.5
  
• Due September 3
  
• Section 0.4: 10, 13, 41, 66, 76, 81
• Section 0.5: 0, 2, 3, 6, 9, 11, 13, 16, 19, 20, 21, 25, 28, 34, 35, 36, 39, 46, 48
• Section 0.4
• Due August 31
• Problems: 0, 2, 4, 8, 11, 17, 18, 21, 26, 27, 31, 35, 36, 46, 47, 52, 56, 62, 63, 86
• Note: we will have more problems from this section on the next homework.
  
• Section 0.3
• Due August 30
• Problems: 0, 1, 2, 4, 6, 9, 11, 14, 18, 21, 25, 27, 35, 39, 43, 46, 52, 59, 66, 67, 72, 78, 79, 89
  
• Section 0.2
• Due August 27
• Problems: 0, 11, 14, 15, 16, 17, 20, 25, 32, 37, 39, 45, 49, 51, 56, 62, 64, 71, 73, 79, 83, 86, 88
  
• Section 0.1
• Due August 25
• Problems: 0, 3, 5, 6, 11, 12, 19, 29, 30, 33, 37, 42, 49, 60, 65, 68, 75, 80, 81, 85, 88, 92