Section 11.3
April 25, 2016
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
Pre­Calculus
Section 11.3
April 25, 2016
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?
Pre­Calculus
Section 11.3
April 25, 2016
Suppose you invest $100 with an annual
interest rate of 100%.
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:
Pre­Calculus
Section 11.3
April 25, 2016
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 years of a $10,000 investment
earning 6.75% interest compounded continuously to the same
investment compounded semiannually.
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?
Pre­Calculus
Section 11.3
April 25, 2016
6. How much money should I save in an account paying 5% interest
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?
Pre­Calculus