Presentation on theme: "§11.1 The Constant e and Continuous Compound Interest."— Presentation transcript:
1 §11.1 The Constant e and Continuous Compound Interest. The student will be able to work with problems involving the irrational number eThe student will be able to solve problems involving continuous compound interest.
2 Do you remember how to find this on a calculator? The Constant e.ReminderDefinition: e = …Do you remember how to find this on a calculator?e is also defined as one of the following limits:
3 Compound InterestLet P = principal, r = annual interest rate, t = time in years, n = number of compoundings per year, and A = amount realized at the end of the time period.Simple InterestA = P ( 1 + r )tCompound interestContinuous compoundingA = P e rt.
4 Example: Generous Grandma Your Grandma puts $1,000 in a bank at 5% for you. Calculate the amount after 5 and 20 years.Simple interestA = 1000 ( ) 5 =$1,276.28A = 1000 ( ) 20 =$2,653.30Compound interest (daily)$1,284.00$2,718.09Continuous compoundingA = 1000 e (.05)(5) =$1,284.03A = 1000 e (.05)(20) =$
5 Show how to become a millionaire!! Example IRAAfter graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year. He plans to retire in 35 years.a. What will be its value at the end of the time period?A = P e rt =3000 e (.12)(35) =$200,058.99b. The second year he repeats the purchase of a Roth IRA. What will be its value in 34 years?$177,436.41Show how to become a millionaire!!
6 Example - Doubling Your Money After graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year.How long will it take for that investment to double?A = P e rt OR = 3000 e 0.12t AND solve for t.6000/3000 = e 0.12t or 2 = e 0.12tTake the ln of both sides yielding -But ln 2 = 0.12 t sot = ln 2/ .12 =yearsRemind the students of the “Rule of 72”.
7 Summary. The constant e occurs in natural situations. There are three different interest formulas.These applications can be of interest.
8 §11.2 Exponential Functions and Their Derivatives The student will learn about:the composite functions,the derivative of the exponential function, graphing strategies for these functions,and applications.
9 The Derivative of ex Step 2: Find f (x+h) – f (x+h) We will use (without proof) the fact thatWe now apply the four-step process from a previous section to the exponential function.Step 1: Find f (x+h)Step 2: Find f (x+h) – f (x+h)
10 The Derivative of ex (continued) Step 3: FindStep 4: Find
11 The Derivative of ex (continued) Result: The derivative of f (x) = ex is f ’(x) = ex.This result can be combined with the power rule, product rule, quotient rule, and chain rule to find more complicated derivatives.Caution: The derivative of ex is not x ex-1The power rule cannot be used to differentiate the exponential function. The power rule applies to exponential forms xn, where the exponent is a constant and the base is a variable. In the exponential form ex, the base is a constant and the exponent is a variable.
12 Examples Find derivatives for f (x) = ex/2 f (x) = 2ex +x2 f (x) = -7xe – 2ex + e2
13 f (x) = ex/2 f ’(x) = (1/2) ex/2 f (x) = 2ex +x2 f ’(x) = 2ex + 2x Examples (continued)Find derivatives forf (x) = ex/ f ’(x) = ex/2f (x) = ex/2 f ’(x) = (1/2) ex/2f (x) = 2ex +x2 f ’(x) = 2ex + 2xf (x) = -7xe – 2ex + e2 f ’(x) = -7exe-1 – 2exRemember that e is a real number, so the power rule is used to find the derivative of xe. The derivative of the exponential function ex, on the other hand, is ex. Note also that e2 is a constant, so its derivative is 0.
14 The Natural Logarithm Function ln x We summarize important facts about logarithmic functions from a previous section:Recall that the inverse of an exponential function is called a logarithmic function. For b > 0 and b 1Logarithmic form is equivalent to Exponential formy = logb x x = byDomain (0, ) Domain (- , )Range (- , ) Range (0, )The base we will be using is e. ln x = loge x
15 The Derivative of ln xWe are now ready to use the definition of derivative and the four step process to find a formula for the derivative of ln x. Later we will extend this formula to include logb x for any base b. Let f (x) = ln x, x > 0.Step 1: Find f (x+h)Step 2: Find f (x+h) – f (x)
16 The Derivative of ln x (continued) Step 3: FindStep 4: Find Let s = x/h.
17 Examples Find derivatives for f (x) = 5 ln x f (x) = x2 + 3 ln x
18 f (x) = x2 + 3 ln x f ’(x) = 2x + 3/x f (x) = 10 – ln x f ’(x) = – 1/x Examples (continued)Find derivatives forf (x) = 5 ln x f ’(x) = 5/xf (x) = x2 + 3 ln x f ’(x) = 2x + 3/xf (x) = 10 – ln x f ’(x) = – 1/xf (x) = x4 – ln x4 f ’(x) = 4 x3 – 4/xBefore taking the last derivative, we rewrite f (x) using a property of logarithms:ln x4 = 4 ln x
19 Other Logarithmic and Exponential Functions Logarithmic and exponential functions with bases other than e may also be differentiated.