Lesson 8.2 - Part 2 Natural Exponential Function

Natural Exponential Function

• Any positive number can be used as the base for an exponential function.

• However, some are used more frequently than others.

– The bases 2 and 10 are convenient for certain applications.

– However, the most important is the number denoted by the letter e.

• The number e is defined as the value that (1 + 1/n)n approaches as n becomes large.

– In calculus, this idea is made more precise through the concept of a limit.

Natural Exponential Function

• The table shows the values of the expression (1 + 1/n)n for increasingly large values of n.

– It appears that, correct to five decimal places,

e ≈ 2.71828

Natural Exponential Function

• The approximate value to 20 decimal places is:

e ≈ 2.71828182845904523536

– It can be shown that e is an irrational number.– So, we cannot write its exact value in decimal

form.

Natural Exponential Function

The natural exponential function is

f(x) = ex

with base e.

It is often referred to as the exponential function.

Natural Exponential Function—Definition:

Since 2 < e < 3, the graph of the natural exponential function lies between the graphs of y = 2x and y = 3x.

Scientific calculators have a special key for the function f(x) = ex.

We use this key in the next example.

The graph of is upward-sloping,and increases faster as x increases. The x-axis is a horizontal asymptote. The inverse function is the natural logarithm ln(x);

54.5982 0.0498 1.6487

xy e

What are the 7 “Wonders” of the World?

1. Great Pyramid of Giza.2. Hanging Gardens of Babylon3. Temple of Artemis at Ephesus4. Statue of Zeus at Olympia5. Mausoleum of Halicarnassus6. Colossus of Rhodes7. Lighthouse of Alexandria8. Compound Interest

Continuously compounded interest is calculated by

A(t) = Pert

where:A(t) = amount after t yearsP = principalr = interest rate per yeart = number of years

Find the amount after 3 years if $1000 is invested at an interest rate of 12% per year, compounded continuously.

We use the formula for continuously compounded interest with:

P = $1000, r = 0.12, t = 3

Thus, A(3) = 1000e(0.12)3 = 1000e0.36

A = $1433.33