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Section 5.1 Introduction to Exponentials (EL1)

Subsection 5.1.1 Activities

Remark 5.1.1.

Linear functions have a constant rate of change - that is a constant change in output for every change in input. Let’s consider functions which do not fit this model - those which grow more rapidly and change by a varying amount for every change in input.

Activity 5.1.2.

You have two job offers on the horizon. One has offered to pay you $10,000 per month while the other is offering $0.01 the first month, $0.02 the second month, $0.04 the third month and doubles every month. Which job would you rather take?
(a)
Make a table representing how much money you will be paid each month for the first two years from the first job - paying $10,000 per month.
(b)
Make a table representing how much money you will be paid each month for the first two years from the second job - paying $0.01 the first month and doubling every month after.
(e)
According to your tables, does the second job ever earn more money per month than the first job?

Remark 5.1.3.

This idea of a function that grows very rapidly by a factor, ratio, or percent each time, like the second job in Activity 5.1.2, is considered exponential growth.

Definition 5.1.4.

Let a be a non-zero real number and b≠1 a positive real number. An exponential function takes the form
f(x)=abx
a is the initial value and b is the base.

Remark 5.1.6.

Notice that in Activity 5.1.5 part (a) the ouput value is larger than the base, while in part (b) the output value is smaller than the base. This is similar to the difference between a positive and negative slope for linear functions.

Remark 5.1.9.

An exponential function of the form f(x)=abx will grow (or increase) if b>1 and decay (or decrease) if 0<b<1.

Activity 5.1.10.

For each year t, the population of a certain type of tree in a forest is represented by the function F(t)=856(0.93)t.
(a)
How many of that certain type of tree are in the forest initially?

Activity 5.1.11.

To begin creating equations for exponential functions using a and b, let’s compare a linear function and an exponential function. The tables show outputs for two different functions r and s that correspond to equally spaced input.
x r(x)
0 12
3 10
6 8
9 6
x s(x)
0 12
3 9
6 6.75
9 5.0625
(e)
What is the ratio of consecutive outputs in the exponential function?
  1. 43
  2. 34
  3. βˆ’43
  4. βˆ’34

Remark 5.1.12.

In a linear function the differences are constant, while in an exponential function the ratios are constant.

Remark 5.1.14.

Recall the negative rule of exponents which states that for any nonzero real number a and natural number n
aβˆ’n=1an

Remark 5.1.16.

Similar to how Ο€ arises naturally in geometry, there is an irrational number called e that arises naturally when working with exponentials. We usually use the approximation eβ‰ˆ2.718282. e is also found on most scientific and graphing calculators.

Activity 5.1.17.

Use a calculator to evaluate the following exponentials involving the base e.
(a)
f(x)=βˆ’2exβˆ’2 for f(βˆ’2)
  1. βˆ’0.0366
  2. βˆ’2.2707
  3. βˆ’1.7293
  4. βˆ’16.778

Exercises 5.1.2 Exercises