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On Calculator Errors and EstimatesAll decimal answers in this course are rounded off to threesignificant digits. To obtain...
On Calculator Errors and EstimatesCalculator Errors
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine w...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                 6
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                 6Using 5 ≈ √27, ...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                 6Using 5 ≈ √27, ...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                 6Using 5 ≈ √27, ...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer.                  6Using 5 ≈ √27,...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P i...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8                                    ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5.                      ...
Power Equations and Calculator InputsNo calculator for part A, B and C.Exercise A. Estimate the following expressions. Fin...
Power Equations and Calculator Inputs Exercise D. Use a calculator to find lower estimates by using a smaller positive bas...
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13 on calculator mistakes and estimates

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13 on calculator mistakes and estimates

  1. 1. On Calculator Errors and EstimatesAll decimal answers in this course are rounded off to threesignificant digits. To obtain this,start from the 1st nonzero digit,round off to the third digit.For example, 12.35 ≈12.40.001235 ≈ 0.00124.
  2. 2. On Calculator Errors and EstimatesCalculator Errors
  3. 3. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes.
  4. 4. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine.
  5. 5. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge.
  6. 6. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.
  7. 7. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.
  8. 8. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.
  9. 9. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.Here is a short list of square roots. A square–root list
  10. 10. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.Here is a short list of square roots.25 < 30 < 36, hence √25 < √30 <√36, A square–root list
  11. 11. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.Here is a short list of square roots.25 < 30 < 36, hence √25 < √30 <√36, so5 < √30 < 6. A square–root list
  12. 12. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.Here is a short list of square roots.25 < 30 < 36, hence √25 < √30 <√36, so5 < √30 < 6.Since 30 is about half way between 25 and 36,so we estimate that √30 ≈ 5.5, half way between 5and 6. A square–root list
  13. 13. On Calculator Errors and EstimatesCalculator ErrorsAs mentioned in the last section that when we calculate with amachine we can make syntactic and/or semantic mistakes. Thesyntactic mistakes will be pointed out by the machine. But themachine will execute semantic mistakes and return wronganswers often without our knowledge. When possible, take thefollowing steps to reduce semantic mistakes.* Have an estimate of the final outcome. If needed,use the calculator to come up with an estimate.Example A.a. Estimate √30, then find its calculator answer.Here is a short list of square roots.25 < 30 < 36, hence √25 < √30 <√36, so5 < √30 < 6.Since 30 is about half way between 25 and 36,so we estimate that √30 ≈ 5.5, half way between 5and 6. In fact √30 ≈ 5.47722… ≈ 5.48 A square–root list
  14. 14. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6
  15. 15. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9,
  16. 16. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6
  17. 17. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.
  18. 18. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate.
  19. 19. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).
  20. 20. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.
  21. 21. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.If a whole Crazy Chicken and a drink cost $11.08, then $11 isa lower or under estimate.
  22. 22. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.If a whole Crazy Chicken and a drink cost $11.08, then $11 isa lower or under estimate. The true cost of two chickens andtwo drinks must be more than the lower estimate of $22.
  23. 23. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.If a whole Crazy Chicken and a drink cost $11.08, then $11 isa lower or under estimate. The true cost of two chickens andtwo drinks must be more than the lower estimate of $22.The true answer must not be less than any lower estimate.
  24. 24. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.If a whole Crazy Chicken and a drink cost $11.08, then $11 isa lower or under estimate. The true cost of two chickens andtwo drinks must be more than the lower estimate of $22.The true answer must not be less than any lower estimate.A semantic error had occurred if the answer falls outside ofthe range between a lower estimate an upper estimate.
  25. 25. On Calculator Errors and Estimatesb. Estimate –4 – √27 , then find its calculator answer. 6Using 5 ≈ √27, –4 – √27 ≈ –9, so –4 – √27 ≈ 2 . –3 6The calculator answer is –4 – √27 ≈ –1.53 which is 6the approximate, or the numeric answer.If a whole Crazy Chicken cost $8.99 then it’s easier to use$ 9 as an over or upper estimate. The real cost of twochickens must be less than $18 (an upper estimate).The true answer must not be more than any upper estimate.If a whole Crazy Chicken and a drink cost $11.08, then $11 isa lower or under estimate. The true cost of two chickens andtwo drinks must be more than the lower estimate of $22.The true answer must not be less than any lower estimate.A semantic error had occurred if the answer falls outside ofthe range between a lower estimate an upper estimate.Here are some basic about estimating arithmetic calculation.
  26. 26. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.
  27. 27. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.
  28. 28. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate.
  29. 29. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3,
  30. 30. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimate
  31. 31. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.
  32. 32. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.
  33. 33. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.Next we note that is 1/3 is smaller than 1/2,
  34. 34. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.Next we note that is 1/3 is smaller than 1/2, i.e. when sharing apizza, more people–less pizza (per person), or thatless people–more pizza (per person).
  35. 35. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.Next we note that is 1/3 is smaller than 1/2, i.e. when sharing apizza, more people–less pizza (per person), or thatless people–more pizza (per person).ii. Given a fraction N of two positive numbers, D
  36. 36. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.Next we note that is 1/3 is smaller than 1/2, i.e. when sharing apizza, more people–less pizza (per person), or thatless people–more pizza (per person).ii. Given a fraction N of two positive numbers, D{ if the denominator D increases, the new fraction is less. if the denominator D decreases, the new fraction is more.
  37. 37. On Calculator Errors and EstimatesWe note first that the answer for –3+5 can’t be more than 5 norbe less than –3.i. If P is positive and N is negative, then P+N is no more than P,and no less than N.If there are more terms, the sum of all the positive terms is anupper estimate and the sum of all the negative terms is a lowerestimate. So for –23+2(2)2–3, +2(2)2 = 8 is an upper estimateand –23–3 = –11 is a lower estimate.Next we note that is 1/3 is smaller than 1/2, i.e. when sharing apizza, more people–less pizza (per person), or thatless people–more pizza (per person).ii. Given a fraction N of two positive numbers, D{ if the denominator D increases, the new fraction is less. if the denominator D decreases, the new fraction is more. if the numerator N increases, the new fraction is more.{ if the numerator N decreases, the new fraction is less.
  38. 38. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.
  39. 39. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1
  40. 40. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one 4.8 + √34.8 6 + √2.1
  41. 41. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one 4.8 + √34.8 5 6 + √2.1 <
  42. 42. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one 4.8 + √34.8 5+6 6 + √2.1 <
  43. 43. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one 4.8 + √34.8 5+6 6 + √2.1 < 6
  44. 44. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6
  45. 45. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6The correct answer must be smaller than this upper estimate.
  46. 46. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6The correct answer must be smaller than this upper estimate. If we obtained an answer of 2, we would know 2 is awrong answer because 2 is above the upper estimate of 11/6.
  47. 47. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6The correct answer must be smaller than this upper estimate. If we obtained an answer of 2, we would know 2 is awrong answer because 2 is above the upper estimate of 11/6.Next we observe 24 < 34 or that for the positive power 4,the larger base 3 produces larger outcome then with base 2.
  48. 48. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6The correct answer must be smaller than this upper estimate. If we obtained an answer of 2, we would know 2 is awrong answer because 2 is above the upper estimate of 11/6.Next we observe 24 < 34 or that for the positive power 4,the larger base 3 produces larger outcome then with base 2.iii. Given bp where p > 0 is a fixed power and b > 0,if b increases, the outcome increases (is larger),
  49. 49. On Calculator Errors and EstimatesExample D. a. Given an upper estimate of 4.8 + √34.8 6 + √2.1Justify it.Any fraction with a larger numerator, and/or a smallerdenominator when compared to 4.8 + √34.8 is more. 6 + √2.1Hence by replacing the numerator with a larger one and theone with a small one, we obtain an upper estimate of 11/6. 4.8 + √34.8 5+6 11 6 + √2.1 < 6 = 6The correct answer must be smaller than this upper estimate. If we obtained an answer of 2, we would know 2 is awrong answer because 2 is above the upper estimate of 11/6.Next we observe 24 < 34 or that for the positive power 4,the larger base 3 produces larger outcome then with base 2.iii. Given bp where p > 0 is a fixed positive power and b > 0,if b increases, the outcome increases (is larger),if b decreases, the outcome decreases (is smaller).
  50. 50. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate.
  51. 51. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7
  52. 52. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.
  53. 53. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.
  54. 54. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.
  55. 55. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.
  56. 56. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.
  57. 57. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.However, because negative exponents means reciprocate,
  58. 58. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.However, because negative exponents means reciprocate,we have 2–4 < 3–4 or that for the negative power –4, or thatthe larger base 3 produces smaller outcome then with base 2.
  59. 59. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.However, because negative exponents means reciprocate,we have 2–4 < 3–4 or that for the negative power –4, or thatthe larger base 3 produces smaller outcome then with base 2.iii. Given bp where p < 0 is a fixed negative power and b > 0,
  60. 60. On Calculator Errors and Estimatesb. Use a calculator to find a lower estimate of (1.87 + 0.08 ) 25. √7Justify the estimate. 0.08The expression is of the form b25 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.825 ≈ 2,408,865 is a useful lower estimate.Now when we execute the original problem and the calculatorreturns an answer that’s smaller than 2,408,865, then we knowone or more semantic mistakes had occurred.However, because negative exponents means reciprocate,we have 2–4 < 3–4 or that for the negative power –4, or thatthe larger base 3 produces smaller outcome then with base 2.iii. Given bp where p < 0 is a fixed negative power and b > 0,if b increases, the outcome decreases (is smaller),if b decreases, the outcome increases (is larger),
  61. 61. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate.
  62. 62. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7
  63. 63. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.
  64. 64. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.
  65. 65. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.8 –5 ≈ 0.0529 when is a useful lower estimate.
  66. 66. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.8 –5 ≈ 0.0529 when is a useful lower estimate.If a suspicious calculator outcome seems to be too large, tryfinding an upper estimates that’s below the outcome to justifyour suspicion that the calculator answer was too large.
  67. 67. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.8 –5 ≈ 0.0529 when is a useful lower estimate.If a suspicious calculator outcome seems to be too large, tryfinding an upper estimates that’s below the outcome to justifyour suspicion that the calculator answer was too large.Likewise if the outcome seems to be too small, try find a lowerestimate that’s more than the suspiciously small calculatoranswer.
  68. 68. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.8 –5 ≈ 0.0529 when is a useful lower estimate.If a suspicious calculator outcome seems to be too large, tryfinding an upper estimates that’s below the outcome to justifyour suspicion that the calculator answer was too large.Likewise if the outcome seems to be too small, try find a lowerestimate that’s more than the suspiciously small calculatoranswer. For complicated calculation, we may estimate parts ofthe expression to check and isolate input mistakes.
  69. 69. On Calculator Errors and Estimatesc. Use a calculator to find a upper estimate of (1.87 + 0.08 ) –5. √7Justify the estimate. 0.08The expression is of the form b–5 where b = (1.87 + ) √7Let’s replace b with a smaller number that is easier to execute.By dropping the “extra bits” and keeping just the 1.8 we wouldhave an smaller and easier base to execute.Hence 1.8 –5 ≈ 0.0529 when is a useful lower estimate.If a suspicious calculator outcome seems to be too large, tryfinding an upper estimates that’s below the outcome to justifyour suspicion that the calculator answer was too large.Likewise if the outcome seems to be too small, try find a lowerestimate that’s more than the suspiciously small calculatoranswer. For complicated calculation, we may estimate parts ofthe expression to check and isolate input mistakes. And finally,When in doubt, insert ( )’s.
  70. 70. Power Equations and Calculator InputsNo calculator for part A, B and C.Exercise A. Estimate the following expressions. Find an upperestimate and a lower estimate. Justify.1. √7 2. √10 3. √15 4. √29 5. √47 6. √737. √17 + √5 8. √37 +√7 9. √24.5 –√4.2 10. √84.3 –√65.8Exercise B. Find the positive–term total as an upper estimateand the negative–term total as a lower estimate. Justify.11. –2 + 6 – 3 – 11 + 14 12. –2 –(–7) – 2 + 15 + 313. 42 – 62 – 23 + 2 14. (–2)(–3) – 4 +10 – 3215. –22 + 26 – 32 16. –2*33 – 42 + 4*2317. –4(–2)2 + 6 –(–3)2 18. (–2)*(–3)3 – 42 – 4*(2)3Exercise C. Estimate the following expressions. Find an upperestimate and a lower estimate. Justify.19. 1 + √10 20. –2 – √17 21. –5 + √24 2 3 3 –5.3 – √17.122. –√14 + √67 6.8 + √24.8 23. 6 – √4.1 24. 1.1 + √4.1 5
  71. 71. Power Equations and Calculator Inputs Exercise D. Use a calculator to find lower estimates by using a smaller positive base. 25. (3.11 + √4.12)5 26. (14.2 – √8.77)3 27. (√50.2 + 2.07)4 8.13 6 29. (√102.8 – 0.3 )7 30. (–√3.9 – √8.9)5 28. (√82.8 + √4.2 ) √9.04 Exercise F. Use a calculator to find upper estimates by using a larger positive base. 31. (2.91 + √8.92)5 32. (9.89 – √4.11)3 33. (√47.8 + 1.97)6 7.93 7 8.9 8.9 34. (√80.8 + √17 ) 35. (√24.8 + √9.1 )4 36. (√24.8 + √9.1 )4Exercise G. Use a calculator to find lower estimates by usinglarger positive base.37. (2.91 + √8.92)–5 38. (9.89 – √4.11)–3 39. (√47.8 + 1.97)–6Exercise H. Use a calculator to find upper estimates by using asmaller positive base.40. (3.11 + √4.12)–5 41. (14.2 – √8.77)–3 42. (√50.2 + 2.07)–4

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