Exponential Functions

Definition:

An exponential function is a mathematical function in which the variable appears in the exponent:

\[f(x) = a \cdot b^x\] where:

a is a non-zero constant (initial value, or coefficient),
b is the base of the exponential function (growth/decay factor), a positive real number not equal to 1
x is the exponent (time or count) and the variable.

if: b > 1 the function represents exponential growth;
if : 0 < b < 1 it represents exponential decay.

Explanation:
Exponential functions grow or decay at a rate proportional to their current value. That means the larger the value of 𝑥, the faster the function grows (or decays). This is different from linear functions, which grow at a constant rate.

Standard form:

$$ f(x) = a b^x $$

Exponential Growth ( 𝑏 > 1 ) : \[f(x)=a⋅b^x\] Example:
\[f(x)=2\cdot⋅3^x\]

a = 2, b = 3
This function triples for every unit increase in $x$.

$x$	$f(x)=a⋅b^x$
-1	$2⋅3^{−1}=0.666$
0	$2.1=2$
1	$2.3 =6$
2	$2.9=18$

This is exponential growth because the output increases rapidly.

Exponential Decay (0 < b < 1): \[f(x)=a⋅b^x\] Example:
$$ f(x) = 100 \cdot (0.5)^x $$

a = 100, b = 0.5
The function halves for each unit increase in 𝑥.

$x$	$f(x)$
0	100
1	50
2	25
3	12.5

This is exponential decay.

Examples in Real Life

1. Population Growth

If a population grows at a constant percentage rate, it follows an exponential function. $$ P(t) = P_0 \cdot (1 + r)^t $$ where, $P_0$ is initial population, 𝑟 is growth rate.

Example:
A town has 10,000 people in 2020. It grows at 3% per year. What will the population be in 2025?
Solution:
Formula: $$ P(t) = P_0 \cdot (1 + r)^t $$ where,

$P_0$ = 10,000 (initial population)
$r$ = 0.03 (growth rate = 3%)
$t$ = 5 years
$P(t)$ is the population after $t$ years.

Calculation

P(5) = 10,000 × (1 + 0.03)⁵ = 11,592.74

Answer: 11,593 people

2. Radioactive Decay

$$ N(t) = N_0 \cdot e^{-kt} $$ where $N_0$ is the initial amount, 𝑘 is a constant.

Example:
100g of a substance decays at 12% per year. How much remains after 3 years?
Solution:
Formula: $$ N(t) = N_0 \cdot e^{-kt} $$ $P(t)$ is the population after $t$ years. where

$N_0$= 100 grams,
$r$ = 0.12
$t$= 3,
$N(t)$ is the remaining substance.

Calculations:

N(3) = 100 × (0.88)³ = 68.15

Answer: 68.15 grams

3. Compound Interest

$5,000 at 5% annual interest, compounded quarterly, over 6 years: formula: $$ A = P(1 + \frac{r}{n})^{nt} $$ where,

P = 500 (initial investment)
r = 0.05 (annual rate)
n= 4 (compoundded quarterly)
A is the amount after t years

Calculation:

A = 5000 × (1 + 0.05/4)²⁴ = 6,744.25

Answer: $6,744.25

4. Bacterial Growth

Starts with 200 bacteria, doubles every 4 hours. How many after 12 hours?
Formula (doubling time)
$$ N(t) = N_0 \cdot 2^{\frac{t}{T}} $$ Where,

$N_0$ = 200
T = 4 hours (doubling time)
t = 1 hours

Calculation:

N(12) = 200 × 2³ = 1,600

Answer: 1,600 bacteria

5. Car Depreciation

A car worth $25,000 depreciates at 15% per year. What is the value after 4 years?
Formula:
$$ V(t) =V_0 \cdot (1 - r)^t $$ Where,

$V_0$= 25,000
r = 0.15
t = 4

Calculation:

V(t) = 25,000 × (0.85)⁴ = 13,050.16

Answer: $13,050.16

Graph Behavior

Exponential Growth $(b > 1)$: The graph rises steeply, showing rapid increase.
Exponential Decay $(0< b< 1)$: The graph falls steeply, approaching zero but never quite reaching it.

Natural Exponential Function

The natural exponential function is a special exponential function where the base is the mathematical constant (e), approximately equal to (2.718). Its general form is:

$f(x) = e^x$

Here, $x$ is the exponent, and $e$ is the base. The natural exponential function is particularly important in mathematics because it arises naturally in many contexts involving growth, decay, or rates of change.

Key properties include:

Its derivative and integral are equal to the function itself.
Rapid growth for $x > 0$, and approaches zero for $x< 0$.
Value at $x=0$: At $x = 0$, $f(0) = e^x$ passes through the point $(0, 1)$.

Applications include compound growth, radioactive decay, and phenomena in physics and engineering.

Compound Growth (Finance) Imagine you deposit $1,000 into a savings account that offers continuous compound interest at an annual rate of 5% ((r = 0.05)). The value of your investment (A(t)) after (t) years is given by: $$A(t) = 1000 \cdot e^{0.05t}$$ So, your $1,000 grows to $1,284.03 in 5 years due to continuous compounding.
Radioactive Decay (Physics) A radioactive substance has a half-life of 10 hours. If the initial amount is 100 grams, the remaining amount (M(t)) after (t) hours can be modeled using: $$M(t) = 100 \cdot e^{-0.0693t}$$ $-0.0693$ corresponds to the decay constant for a half-life of 10 hours. So, 50 grams of the substance remain after 10 hours.
Physics and Engineering (Cooling Process) Newton’s Law of Cooling states that the temperature $T(t)$ of an object changes exponentially over time. Suppose a cup of coffee starts at 80°C and is placed in a room at 20°C. The temperature (T(t)) after (t) minutes is: $$T(t) = 20 + (80 - 20) \cdot e^{-kt}$$ Here, $k$ is a cooling constant specific to the environment. The coffee cools to 38.6°C in 30 minutes.
Biology (Population Growth) In biology, $e^x$ can model bacterial growth under ideal conditions. If a bacterial colony doubles every hour and starts with 100 bacteria, the population $P(t)$ after $t$ hours is: $$P(t) = 100 \cdot e^{0.693t}$$ $0.693$ corresponds to the growth constant for doubling. So, the population grows to 800 bacteria in 3 hours.

Note: The natural exponential function beautifully describes processes involving continuous change, whether growth or decay.

\(x\)	\(f(x)=a⋅b^x\)
-1	\(2⋅3^{−1}=0.666\)
0	\(2.1=2\)
1	\(2.3 =6\)
2	\(2.9=18\)

Exponential Functions

Definition:

Standard form:

Examples in Real Life

1. Population Growth

2. Radioactive Decay

3. Compound Interest

4. Bacterial Growth

5. Car Depreciation

Graph Behavior

Natural Exponential Function

Interactive Real-Life Problems