Exponential Function

Exponential functions are mathematical functions in which the variable appears in the exponent.

• Describe quantities that increase or decrease rapidly.
• Their rate of growth or decay depends on their current value.
• Used to model population growth, compound interest, and radioactive decay.

It is generally written as:

f(x) = a^x

• a is a positive base, where a ≠ 1.
• x is the variable.

Key features:

• If a > 1, the function increases.
• If 0 < a < 1, the function decreases.
• The graph of an exponential function approaches but never touches the x-axis.

Exponential functions are widely used to model growth and decay, calculate investments, and study population changes.

Exponential Growth

In Exponential Growth, a quantity progresses rapidly. An exponentially growing function has an increasing graph. It can be used to illustrate economic growth, population expansion, compound interest, growth of bacteria in a culture, population increases, etc.

The formula for exponential growth is:

y = a(1 + r)^x

• r is the growth rate.

Exponential Decay

In Exponential Decay, a quantity decreases very rapidly at first and then fades gradually. An exponentially decaying function has a decreasing graph. The concept of exponential decay can be applied to determine half-life, mean lifetime, population decay, radioactive decay, etc.

The formula for exponential decay is:

y = a(1 - r)^x

Where r is the decay rate

Exponential Function Derivative

• For f(x) = eˣ, its derivative is, d/dx (eˣ) = eˣ.
• For f(x) = ax, its derivative is, d/dx (a^x) = a^x · ln a

Exponential Function Integration

• For f(x) = eˣ, its integration is ∫eˣ dx = eˣ + C
• For f(x) = ax, its integration is, ∫a^x dx = a^x / (ln a) + C

• Practice Questions on Exponential Functions
• Exponential Graphs