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## Pre-Calc Lesson 5-4

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**Pre-Calc**Lesson 5-4 The number ‘e’ and the function ex ‘e’ is defined by this expression: limit (1 + 1/n)n n ф Complete the table : n (1 + 1/n)n 10 ? 100 ? 1,000 ? 10,000 ? 100,000 ? As n gets larger and larger, (1 + 1/n)n gets closer and closer to ??? 2.718**The Swiss mathematician Leonard Euler proved this**concept which is why this ‘limit’ is called ‘e’ in his honor. The function ex is called the natural exponential function: Its graph looks like such: Compound Interest and the number ‘e’. ‘e’ is a very important number when computing interest. When computing interest on money at various Intervals yearly, semi-annually, quarterly, monthly, even daily, we can use our basic formula for exponential**Growth---A(t) = A0(1 + r)t**Now a few adjustments need to be made -- in this formula, ‘r’ stands for a ‘yearly interest rate. So if we want to compute interest on money at say 12% per year compounded semiannually—which means Two times a year—then we take ‘r’ 12% divide by two To get 6%, but then we take that to the 2nd power to get 1.062 = 1.1236 Confused??**Interest period % growth growth factor**Amount each period during period . Annually 12% 1 + 0.12 1.121 = 1.12 1 . Semiannually 6% 1 + 0.12 1.062 = 1.1236 2 . Quarterly 3% 1 + 0.12 1.034 = 1.1255 4 . Monthly 1% 1 + 0.12 1.0112 = 1.1268 12 . Daily 12 % 1 + 0.12 (1 + 0.12)360 = 1.1275 (360 days) 360 360 360 . ‘k’ times 12 % 1 + 0.12 (1 + 0.12)k per year k k k . Through a series of mathematical manipulations, the expression (1 + 0.12)k can be written as [(1 + 1)n]0.12 k n Now we earlier stated that ‘e’ = (1 + 1)n n**So when we go to computing interest more often than**• ‘daily’ referred to as ‘continuously’ we just use • A(t) = A0ert. Now since we usually are calculating interest • On money which is called ‘principle’ we use the ‘pert’ • Formula: A(t) = Pert. • Example: Which plan yields the most interest? • Plan A: A 7.5% annual rate compounded monthly • for 4 years • Plan B: A 7.2 % annual rate compounded daily for • 4 years • Plan C: A 7% annual rate compounded continuously • for 4 years**Plan A: (1 + 0.075)12(4) = (1.00625)48 = 1.3486**• 12 • Plan B: (1 + 0.072)360(4) = (1.0002)1440 = 1.3337 • 360 • Plan C: e0.07(4) = (2.7183)0.28 = 1.3231 • Plan A yields the most! • Use your calculator to evaluate the following: • e2 2. e3.2 3. e- 4 4. e 5. e1 • 6. Which is larger e or e**If money is invested at 8% compounded semi-**• annually, then each year the investment is multiplied • by 1.042. What is the investment multiplied by if • the interest is compounded: • a) quarterly b) 12 times a year c) continuously • (1.02)4 • (1.0067)12 • e0.08 • A bank advertises that its 5% annual interest rate • compounded daily is equivalent to a 5.13 % effective • annual yield. What does this mean? • (1 + 0.05)360 = (1.000138889)360 = 1.051267… = 1.0513 • 360 • So 5 % compounded daily equates to approximately 5.13% • annually