FUNCTIONS 08
Exponential Functions
An exponential function changes by a constant factor over equal changes in the input.
The main idea
A basic exponential function has the form
\[\boxed{f(x)=Ab^x,\qquad A\ne0,\quad b>0,\quad b\ne1}.\]The variable appears in the exponent. This is what makes exponential behaviour fundamentally different from polynomial behaviour.
1. Constant factor
For \(f(x)=2^x\), increasing \(x\) by one multiplies the output by \(2\):
\[1,\ 2,\ 4,\ 8,\ 16,\ldots\]So
\[\frac{f(x+1)}{f(x)}=2.\]\(x=1\to2\)
\(x=2\to4\)
\(x=3\to8\)
each step multiplies by \(2\)
2. Exponential growth
If \(b>1\), then \(b^x\) increases as \(x\) increases.
For example, \(2^x\) grows slowly at first and then increasingly rapidly.
↓
constant multiplicative growth
↓
increasingly steep curve
3. Exponential decay
If \(0 The graph compares \(2^x\) with \((1/2)^x\). Both pass exactly through \((0,1)\).
4. The role of \(A\)
Since
\[f(0)=Ab^0=A,\]the constant \(A\) gives the \(y\)-intercept.
For \(f(x)=5(2^x)\), the graph passes through \((0,5)\).
set \(x=0\)
\(f(0)=A\)
↓
vertical intercept \((0,A)\)
5. Percentage growth and decay
If a quantity grows by proportion \(r\) each period,
\[\boxed{P_n=P_0(1+r)^n}.\]For a \(5\%\) increase, the multiplier is \(1.05\). For a \(5\%\) decrease, it is \(0.95\).
decrease by \(5\%\): multiply by \(0.95\)
6. The number \(e\)
The constant
\[e\approx2.71828\]arises naturally when change occurs continuously rather than in separate steps.
The function \(e^x\) is the most important exponential function in calculus.
↓ increasingly frequent changes
continuous compounding
↓
\(e\)
7. Continuous exponential change
A common continuous model is
\[\boxed{P(t)=P_0e^{kt}}.\]If \(k>0\), the quantity grows. If \(k<0\), it decays.
\(k=0\) → constant
\(k<0\) → decay
8. Why \(e^x\) is special
The derivative of \(e^x\) is itself:
\[\boxed{\frac{d}{dx}e^x=e^x}.\]So the instantaneous rate of change is equal to the current value.
rate of change: \(e^x\)
same function
9. Exponential models and differential equations
If a quantity changes at a rate proportional to its current amount,
\[\frac{dP}{dt}=kP,\]then its solution is
\[P(t)=P_0e^{kt}.\]This structure appears in population growth, radioactive decay, finance and many biological models.
↓ determines
rate \(kP\)
↓ leads to
\(P_0e^{kt}\)
10. Linear versus exponential
Linear change adds a fixed amount, while exponential change multiplies by a fixed factor.
The graph compares \(4x+1\) and \(2^x\). The exponential function eventually grows much faster.
Exponential functions are characterised by constant multiplicative change. The base \(b\) controls growth or decay, \(A\) gives the value at zero, and \(e^x\) is central to calculus because its derivative is itself.
Why this matters for calculus
Exponential functions provide one of the clearest links between a function and its rate of change. They are therefore fundamental in differential equations and models of continuous growth and decay.