MATH 2A Lecture Notes - Lecture 3: Logarithm, Inverse Function

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4 Oct 2018

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MATH 2A Lecture 1: Math 2A Sept28 Fri Lec1

MATH 2A Lecture 2: Math 2A (44095) - Lecture 2 - Functions(cont.)

MATH 2A Lecture Notes - Lecture 3: Logarithm, Inverse Function

Document Summary

Week 1 lecture 3 functions (cont. ) and compound interest. Imagine this scenario: 100% annual interest rate, what the bank pays you annually, there is an initial balance of . 2 (1+(100%/2))^2 (1+(100%/n): here, the first payment is initial 1, then balance at t = 0 is the initial, the balance at t = 1 is after the payment, for the n payments, per year. Inverse functions: definition, 1-1, also known as one-to-one, (cid:1858)(cid:4666)(cid:1876)(cid:4667) is 1-1 (cid:1858) does not take the same value (output) twice, on its domain for (cid:1876)(cid:2869) ,(cid:1876)(cid:2870) , think about it means if (cid:1876)(cid:2869) (cid:1876)(cid:2870, this means that (cid:1858)(cid:4666)(cid:1876)(cid:2869)(cid:4667) (cid:1858)(cid:4666)(cid:1876)(cid:2870)(cid:4667) In which a horizontal line can be created by picking any point on the y-axis: and is one-to-one. If you have (cid:1858) and (cid:1858) (cid:2869) acting on the same input, you will get back to: they are not the same.

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Problem 3

Math 151 - Sections 34, 35, 36 - Workshop 11 This workshop deals with an addition formula in section 5.1. This formula is related to an ld story about the great mathematician Gauss. Here is one version of the story Wh en Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math lecture period. To keep the students busy while he graded, he gave them the following task at the start of the period: Use your slate (the small personal blackboard that every student had in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate and yelled "Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the period Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer Then Gauss told everybody that the sum 1+2+3+ + 100 is ( 1 + 100) + (2+99)+ (2+98)+--+ (50+51) = 101 + 101 + 101 +. . . +10 101. 50-5050 Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prov the formula 1 +2+3+.+N) be an even number Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a+1.b) + (a+2 b) + (a+3.b)+. + (a + N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+(a+2.b)+ (a +3.b)+(a+ N b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this ve clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3++N, he could simply add N+1 to that sum, and that would give him the sum 1+2+3 Now assume that, by trial and error, Gauss discovered that the formula 1+2+3+ +N an even numbe +) Include the reason why it is important that +Â·Â·Â· + (N +1) very quickly is true for several values of N. How could he prove that the formula is really true for ALL N? The insight described in the previous paragraph might have led Gauss to think the following: All he had to do was prove N(N+1) Â± (N + 1) other words, he only needed to show: Formula for 1 + 2+3+.+ N plus N +1 equals formula for 1+2+3+(N+1). He would have discovered the principle of mathematical induction Problem 5. Prove of the formula for 1+2 +3+... + N (N +1)(+1X++1) Show how this leads to a proof N(N+1 N+)

Complete problem 3 using the information above.

Math 151 Sections 34, 35, 36 Workshop 11 ted to arn This workshop deals with an addition formula in section 5.1. This formula is rel oft told story about the great mathematician Gauss. Here is one version of the story When Gauss was ten years old, he attended a certain math class with other kids. One fine day, the math teacher discovered that he was running out of time to grade the latest batch of math homeworks. This is why he decided to finish the grading during his math period. To keep the students busy while he graded, he gave them the following task at th of the period: Use your slate (the small personal blackboard that every student had start in class) to add up the numbers 1,2,3,... , 100 Seconds after the teacher gave this in-class assignment, little Gauss pointed to his slate Here is the answer." Then Gauss did nothing for an hour, while the other students labored with the additions and the teacher graded. At the end of the perioc Gauss showed the correct answer 5050 on his slate. Only Gauss got the correct answer. Then Gauss told everybody that the sum 12 +3.+ 100 is (1+100)+(2+99) + (3498)+ Problem 1. Explain carefully what Gauss had done. Include the reason why it is impor tant that 100 happens to be an even number Problem 2. Let N be an even positive integer. Use the method of Problem 1 to prove the formula 1 +2 +3+.. . +N(N. Include the reason why it is important that N +(50+51) = 101 + 101 +101 + +101-101.50-5050 an even number. Problem 3. The formula is still valid when N is an odd integer. Prove this, using a variation on the argument given by Gauss The story continues: Gauss then gave the class a lecture on how his method could be used to sum an arithmeric series (a +1.b) (a +2.b) (a +3.b)+. (a +N b), where a and b are constants Problem 4. Find a formula for the sum (a+1.b)+ (a+2.b) + (a +3.b)+ +(a+ N.b) Prove this formula the way Gauss would have done it. You have to do the case when N is even and the case when N is odd Now let us consider a fictional alternate history where Gauss had not thought of this very clever trick. We can imagine the ten year old Gauss going home and doing the sequential computations 1+2 1+2+3 1+2+3+4 1+2+3+4+5 and so forth, just to see if he could find a pattern in the sequence of answers. He would have discovered somethin like this: once he had the sum 1+2+3+...+ N, he could simply add N +1 to that sum and that would give him the sum 1+2+3+ + (N +1) very quickly Now ass N(N+1 ume that, by trial and error, Gauss discovered that the formula 1+2+3+ is true for several values of N. How could he prove that the formula is r for ALL N? The insight described in the previous paragraph might have led Gauss to other words, he only needed to show: Formula for 1+2+3+ + N plus N +1 equals formula for 1+2+3++N+1). He would have discovered the principle of mathema -.. + N eally true 1)+1) think the following: All he had to do was prove mm) + (N + 1) NN+1 (N+1)(( In ered the principle of mathematical induction Problem 5. Prove N(N+1) (N + 1 of the formula for 1+2+3+.+N. (Nt1MNHHI) Show how this leads to a uCtion a proof

Problem 1

Part A: (Simple Interest) Rami has $1,500 in his bank account. He earns simple interest of 8% per year.

a- Deduce the value of the account (ðð¡ =?) after 3 years.

b- Suppose Rami wishes to buy a car for $4,500. Will he be able to buy it after 3 years? (No calculation needed)

Part B: (Interest compounded semi-annually) Calculate the present value (ð0 =? ) of $20,000 to be received in 5 yearsâ time when the discount rate is 5% per year, compounded semi-annually.

Problem 2 Part A: (Simple interest)

A savings account of $8,000 earns simple interest of 6% per annum.

a- Calculate the amount (I =?) received after 5 years. b- Deduce the value of the account (ðð¡ =?) after 5 years.

Part B: (Interest compounded semi-annually) Mohammad is saving $5,000 in a bank X for 3 years at 8% per annum compounded semiannually. Calculate the total value of the amount

Problem 3 (Interest compounded quarterly) As a graduation gift, your parents opened a bank account with $5,000 under your name. This account pays 5% interest rate per annum; compounded quarterly. If you decided to save the money over two years, by how much will your balance increase after two years?

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MATH 2A Lecture Notes - Lecture 3: Logarithm, Inverse Function

MATH 2A Lecture 1: Math 2A Sept28 Fri Lec1

MATH 2A Lecture 2: Math 2A (44095) - Lecture 2 - Functions(cont.)

MATH 2A Lecture Notes - Lecture 3: Logarithm, Inverse Function

Document Summary

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