# Exponential Functions

##### 379 words — categories: calculus

An **exponential function** is given by

\[f(x) = a^x\]

where \(x\) is any real number, \(a > 0\), and \(a \neq 1\). The number \(a\) is called the **base**. Unlike power functions, which have the variable in the base, exponential functions have the variable in the exponent.

```
import numpy as np
import matplotlib.pyplot as plt
def f(x):
return 2**x
def g(x):
return (1/2)**x
def h(x):
return 0.4**x
x = np.linspace(0, 3)
plt.plot(x, f(x), label="$f(x) = 2^x$")
plt.plot(x, g(x), label="$g(x) = 1/2^x$")
plt.plot(x, h(x), label="$h(x) = 0.4^x$")
plt.legend(frameon=False)
```

\[e = \lim_{h \to 0}(1 + h)^{1/h} \approx 2.71\]

and we call \(e\) the *natural base*. The reason why it’s important in mathematics is that it represents the base rate of growth shared by all continually growing processes.

The derivative of the function \(f(x) = e^x\) is itself

\[f^\prime(x) = f(x)\]

or

\[\frac{d}{dx}e^x = e^x\]

The derivative of \(e\) to some power is the product of \(e\) to that power and the derivative of the power.

\[\frac{d}{dx}e^{f(x)} = e^{f(x)} \cdot f^\prime(x)\]

## Examples

Let’s find \(dy/dx\) for (a) \(y = 3e^x\) (b) \(y = e^{8x}\).

```
from sympy import *
from math import e
def f(x):
return 3 * e**x
def g(x):
return e**(8*x)
x = symbols("x")
diff(f(x), x)
```

`## 3.0*2.71828182845905**x`

`diff(g(x), x)`

`## 8.0*2.71828182845905**(8*x)`

## Worker Efficiency (Example)

It is reasonable for a manufacturer to expect the daily output of a new worker to start out slow and continue to increase over time, but then tend to level off, never exceeding a certain amount. A firm manufactures phones and determines that after working \(t\) days, the efficiency, in number of phones produced per day of most workers can be modeled by the function

\[N(t) = 80 - 70e^{-0.13t}\]

Find (a) \(N(0)\), \(N(1)\), \(N(5)\), \(N(10)\), \(N(20)\), \(N(30)\) (b) graph \(N(t)\) (c) find \(N^\prime(t)\) and interpret the derivative in terms of rate of change (d) what number of phones seems to determine where worker efficiency levels off?

```
def N(t):
return 80 - 70*e**(-0.13*t)
times = [round(N(t), 1) for t in [0, 1, 5, 10, 20, 30]]
times
```

`## [10.0, 18.5, 43.5, 60.9, 74.8, 78.6]`

```
t = np.linspace(0, 50)
plt.plot(t, N(t))
# level at which efficiency levels off
plt.axhline(80, linestyle="--", color="red")
```

```
t = symbols("t")
diff(N(t), t) # rate of change of phones after t days
```

`## 9.1*2.71828182845905**(-0.13*t)`