Section 11.3: The Natural Base e
May 01, 2015
11.3 The Natural Base e
• e is an irrational number (much like π)
or
• crazy math notation aside, e ≈ 2.718 and it
used in a lot of real world and calculus based
calculations
• It uses Leonhard Euler's (1707-1783)
notation of e
• The function y = ex is one of the most
important exponential functions
Graph y = ex
Section 11.3: The Natural Base e
May 01, 2015
Practice Using e
Find f(x) for x = 0, 2, 10
f(x) = 2000e0.05x
2. Without further study, as time passes you forget things you
have learned. The Ebbinghaus model of human memory gives the
percent p of acquired knowledge that a person retains after t
weeks. The formula p = (100-a)e-bt + a, where a and b vary from one
person to another. If a = 18 and b = 0.6 for a certain student, how
much information will the student retain two weeks after learning a
new topic?
Section 11.3: The Natural Base e
May 01, 2015
Another look at the compound interest formula:
Ignoring the principal, the interest rate, and the number of
years by setting all these variables equal to " 1", and looking
only at the influence of the number of compoundings, we get:
Continuously Compounding Interest
A = Pert
A = final amount
P = initial amount
r = annual interest rate
t = time in years
3. Compare the balance after 25 year of a $10,000 investment
earning 6.75% interest compounded continuously to the same
investment compounded semiannually.
Section 11.3: The Natural Base e
4. Compare the balance after 30 years of a $15,000 investment
earning 12% interest compounded continuously to the same
investment compounded quarterly.
5. Sally wants to have $5,000 in two years so she can take a trip to
Paris. She finds an account that pay 5% interest compounded
continuously. How much should she invest now?
6. How much money should I save in an account paying 5% i nterest
compounded monthly if I want to have $6000 in 6 months?
7. How much money should you invest if you want to have $10,000
after 4 years if the bank's APR is 3.5% and is compounded
continuously?
May 01, 2015